Quantum Communications Systems

Abstract

In this chapter the quantum decision theory, developed in the previous chapters, is applied to quantum communications systems by specifying the physical nature of the quantum states that carry the information. In practice, the states of a quantum communications system refer always to optical frequencies because at radio frequencies quantum phenomena are not appreciable. Specifically, quantum states represent coherent monochromatic radiations emitted by a laser and are called coherent states. Therefore the theory of coherent states, which is presented according to the formulation developed by Glauber in the 1980s of the last century, will play a fundamental role. Also squeezed states which represent an efficient form of optical radiation are considered. In the second part of the chapter, the most popular quantum communication systems are developed, starting from the simplest ones: The binary systems whose format may be the OOK (on–off keying) modulation and the 2-PSK (phase-shift keying) modulation. Then multilevel systems are developed with the formats: QAM (quadrature amplitude modulation), multilevel PSK modulation, and PPM (pulse position modulation). All these quantum systems will be compared with the classical counterparts, and the superiority in performance of quantum systems with respect to classical systems will be clearly stated.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

The quantum decision theory, developed in the previous two chapters, is now applied to quantum communications systems where the nature of the states that carry the information is specified. A constellation of $K$ quantum states, to which to commit a symbol belonging to a $K$-ary alphabet, corresponds, in the classical version, to a $K$-ary modulation format. We still consider states that operate at optical frequencies (optical quantum systems), because at radio frequencies quantum phenomena are not appreciable . In practice, the quantum states are usually treated as coherent states of a coherent monochromatic radiation emitted by a laser. For these states there exists a universal model, proposed by Glauber, that will be introduced in the next section .

Also squeezed states as a candidate carrier for quantum communications are considered. Squeezed light is an efficient form of optical radiation, which is obtained from a laser radiation in several ways, mainly based on parametric amplifiers.

In this chapter, we shall first examine binary systems, presenting the quantum versions of the OOK (on–off keying) and 2PSK (phase-shift keying) modulations . Then we shall move to multilevel systems, and examine the quantum versions of the QAM (quadrature amplitude), PSK, and PPM (pulse position) modulations . All the above-mentioned systems will be examined in the absence of thermal noise, which, instead, will be considered in the next chapter. Thus, in this chapter, the scheme of Fig. 7.1 will be followed, in which the channel is ideal and the received state is directly given by the transmitted state. As already observed, neglecting thermal noise does not mean that the analysis will be done in the absence of noise; because we shall take into account the fact that quantum measurements are affected by an intrinsic randomness, corresponding, in the classical model, to shot noise .

Fig. 7.1

Quantum communications system for digital transmission. $\{A_n\}$ is a sequence of classical symbols of information that Alice conveys into a sequence of quantum states $\{|\gamma _{A_n}\rangle \}$. Bob, in each symbol period, performs a quantum measurement to argue, from the result $m$ of the measurement, which symbol was transmitted

Organization of the Chapter

The next two sections deal with the definition and properties of coherent states and how to provide a constellation of coherent states. Section 7.5 develops the theory of classical optical systems where the decision is based on photon counting. The subsequent sections, from Sects. 7.9 to 7.13, develop the specific quantum communications system with the modulation format listed above.

In the two final sections, we will develop quantum communications with squeezed states with a comparison of the performance with that obtained with coherent states.

As explained in Chap. 4, only digital systems will be considered. For binary systems, we shall use the general theory of binary optimization, essentially Helstrom’s theory, developed in Sect. 5.4. For multilevel systems, for which an explicit optimization theory is not available, we shall use the square root measurements (SRM) decision developed in Chap. 6 and, when convenient, we compare SRM results with the ones obtained with convex semidefinite programming (CSP).

2 Overview of Coherent States

A general model of the quantum state created by an electromagnetic field at a certain (optical) frequency is given by a coherent quantum state according to Glauber’s theory. This model is now formulated in detail in a form suitable to deal with quantum communications systems, but without entering in theoretical considerations. In Chap. 11 coherent states will be fully developed in the framework of quantum information as continuous quantum states and also as Gaussian quantum states.

2.1 Glauber’s Representation

The coherent radiation emitted by a laser is modeled as a coherent state. It has been demonstrated [1–3] that the coherent states of a single mode can be represented in a Hilbert space of infinite dimensions, through an orthonormal basis $\{ |n\rangle , n = 0,1,2,\ldots \}$, where the states are called number states, because $|n\rangle$ contains exactly n photons. They are also called number eigenstates and Fock states.

To this basis, the number operator is associated, which is defined by

$$\begin{aligned} N=\sum _{n=0}^{\infty } n|n\rangle \langle n. \end{aligned}$$

(7.1)

Then $N$ has eigenvectors $|n\rangle$ with eigenvalues $n$ and the spectrum of $N$ is given by the set of naturals, $\sigma (N) = \{0,1,2,\ldots \}$.

In this mathematical context, a generic coherent state (or Glauber state) is expressed as follows:

(7.2)

where $\alpha$ is a complex amplitude whose meaning is

$$\begin{aligned} |\alpha |^2= \text{ average } \text{ number } \text{ of } \text{ photons } \text{ in } \text{ the } \text{ state }\; |\alpha \rangle . \end{aligned}$$

(7.3)

Therefore, according to (7.2), to each point $\alpha$ of the complex plane ${\mathbb C}$, a coherent state is associated whose physical meaning is given by (7.3). Thus, the more $\alpha$ moves away from the origin of ${\mathbb C}$, the higher becomes the photonic intensity associated to the state $|\alpha \rangle$.

The set of coherent states will be indicated by

$$\begin{aligned} \mathcal{G} = \{|\alpha \rangle , \alpha \in {\mathbb C}\} \quad :\quad \text {coherent states} \end{aligned}$$

(7.4)

and then the notation $|\alpha \rangle \in \mathcal{G}$ will be used to distinguish one of these specific kets from the other numerous kets that we will meet. It is interesting to observe that letting $\alpha =0$ in (7.2) we obtain

$$\begin{aligned} |\alpha \rangle _{\alpha =0} = |n\rangle _{n=0} \end{aligned}$$

(7.5)

that is, with $\alpha =0$ we obtain the state $|0\rangle$ of the Fock basis, called ground state .

Remark

The notations of Quantum Mechanics are powerful, but sometimes subtle. In this context, it is important to distinguish the complex number $\alpha \in {\mathbb C}$ from the coherent state $|\alpha \rangle \in \mathcal{G}$, which is a ket of the infinite dimensional Hilbert space $\mathcal{H}$, generated by the basis { $|n \rangle |n=0,1,2,\ldots \;$}. The fundamental relation (7.2) is a mapping ${\mathbb C}\rightarrow \mathcal{G}$, where $\mathcal{G}\subset \mathcal{H}$. For instance, $\alpha =3-\mathrm{i}4 \in {\mathbb C}$ is mapped onto the coherent state $| 3-\mathrm{i}4 \rangle \in \mathcal{G}$, whose full expression is

$$ | 3-\mathrm{i}4 \rangle =\mathrm{exp}\left[ -\frac{1}{2}| 3-\mathrm{i}4|^2\right] \sum _{n=0}^\infty \frac{(3-\mathrm{i}4)^n}{\sqrt{n}}|n\rangle . $$

2.2 Link with Poisson’s Regime

To find the relationship between the representation of a coherent state $|\alpha \rangle \in \mathcal{G}$ and Poisson’s regime, we set up a quantum measurement (Fig. 7.2) with the number operator $N$, interpreted as an observable (see Sect. 3.6). The outcome $m$ of the measurement gives the number of photons of the quantum system in the state $|\alpha \rangle$. Then the probability that the measurement gives the outcome $m=i$ turns out to be

$$\begin{aligned} \begin{aligned} \mathrm{P}[m=i|\alpha ]=&\, |\langle i|\alpha \rangle |^2 =\left| \sum _{n=0}^{\infty }{\mathrm{e}^{-\frac{1}{2}|\alpha |^2}} \frac{\alpha ^n}{\sqrt{n!}} \langle i | n \rangle \right| ^2\\ =&\left| \mathrm{e}^{- \frac{1}{2}|\alpha |^2} \frac{\alpha ^{i}}{\sqrt{i!}} \right| ^2 = \mathrm{e}^{-|\alpha |^2} \frac{|\alpha |^{2i}}{i !}. \end{aligned} \end{aligned}$$

(7.6)

Fig. 7.2

Quantum measurement in a Hilbert space of the coherent state $|\alpha \rangle$ with an observable given by the number operator $N$. The elementary projectors $|n\rangle \langle n|$ are formed by the number states $|n\rangle$

Therefore,

$$\begin{aligned} \mathrm{P}[m=i|\alpha ]=\mathrm{e}^{-N_\alpha }\;\frac{(N_\alpha )^i}{i!} \quad \text {with} \quad N_\alpha = |\alpha |^2. \end{aligned}$$

(7.7)

It can also be verified that the average of $m$ is

$$\begin{aligned} \mathrm{E}[m| \alpha ]= \langle \alpha |N| \alpha \rangle = |\alpha |^2=N_\alpha . \end{aligned}$$

(7.8)

In conclusion, the outcome of the measurement $m$ is a Poisson random variable with average $N_\alpha =|\alpha |^2$ .

2.3 Degree of Superposition of Coherent States

It is important to evaluate the degree of superposition of two distinct coherent states $|\alpha \rangle$ and $|\beta \rangle$, within the geometry given by the inner product . We have

Proposition 7.1

The inner product of two coherent states is given by

(7.9)

Hence two distinct coherent states are never orthogonal (Fig. 7.3).

In fact, from (7.2) we have

$$\begin{aligned} \langle \alpha | \beta \rangle =&\mathrm{e}^{-\frac{1}{2}(|\alpha |^2 + |\beta |^2)} \sum _{m=0}^{\infty }{\sum _{n=0}^{\infty }{ \frac{(\alpha ^*)^m \beta ^n}{\sqrt{m! n!}}\langle m | n \rangle }} \\ =&\mathrm{e}^{-\frac{1}{2}(|\alpha |^2 + |\beta |^2)} \sum _{m=0}^{\infty }{\frac{(\alpha ^* \beta )^m}{m!}}= \mathrm{e}^{-\frac{1}{2}(|\alpha |^2 + |\beta |^2)} \mathrm{e}^{\alpha ^* \beta }. \end{aligned}$$

and (7.9) follows    $\square$

Fig. 7.3

Two distinct coherent states are never orthogonal: $\langle \alpha |\beta \rangle \ne 0$

The (quadratic) degree of superposition of two states is expressed by

(7.10)

where $X=\langle {\alpha }|{\beta }\rangle$.

2.4 Tensor Product of Coherent States    $\Downarrow$

The tensor product of two or more coherent states will be particularly relevant to PPM modulation and in general for vector modulations .

Let $|\alpha \rangle$ be the tensor product of two coherent states

$$\begin{aligned} |\alpha \rangle =|\alpha _1\rangle \otimes |\alpha _2\rangle \,,\qquad |\alpha _1\rangle ,|\alpha _2\rangle \in \mathcal{G}. \end{aligned}$$

Then, for each of the two factors, the previous result holds: To the state $|\alpha _i\rangle$ a Poisson variable $m_i$ can be associated, with average $\mathrm{E}[m_i|\,\alpha _i]=|\alpha _i|^2$. The global number of photons $m$ associated to the composite state $|\alpha \rangle$ is given by the sum of the two random variables $m=m_1+m_2$, where $m_1$ and $m_2$ are statistically independent. Therefore, $m$ is again a Poisson variable with average $\mathrm{E}[m|\,\alpha ]=\mathrm{E}[m_1|\,\alpha _1]+\mathrm{E}[m_2|\,\alpha _2]= |\alpha _1|^2+|\alpha _2|^2$. This result can be easily generalized to the tensor product of $N$ Glauber states

$$\begin{aligned} |\alpha \rangle =|\alpha _1\rangle \otimes |\alpha _2\rangle \otimes \cdots \otimes |\alpha _N\rangle \end{aligned}$$

and we find, in particular, that the total number of photons $m=m_1+m_2+\cdots m_N$ associated to the composite state $|\alpha \rangle$ is still a Poisson variable with average given by

$$\begin{aligned} \mathrm{E}[m|\,\alpha ]=|\alpha _1|^2+|\alpha _2|^2+\cdots +|\alpha _N|^2. \end{aligned}$$

(7.11)

2.5 Coherent States as Gaussian States    $\nabla$

In Chap. 11, in the framework of continuous quantum variables, coherent states will be defined as eigenkets of the annihilator operator, acting in an infinite dimensional bosonic Hilbert space $\mathcal{H}$. Then, from this abstract definition, the infinite dimensional representation (7.2) is obtained. An alternative representation is considered in the so-called phase space, where a quantum state, pure or mixed, is represented by its Wigner function $W(x,y)$, a real function of two real variables, having properties similar to the joint probability density of two continuous random variables. Thus we pass from an infinite dimensional Hilbert space $\mathcal{H}$ to the two-dimensional real space ${\mathbb R}^2$, with notable advantages .

The Wigner function $W(x,y)$ allows us to define Gaussian quantum states, as the quantum states having as Wigner function the Gaussian bivariate form

$$\begin{aligned} W(x,y)=\frac{1}{2\pi \sqrt{\det V}} \mathrm{exp}\left[ -{1 \over 2}\frac{V_{22} (x-\overline{q})^2+ V_{11}(y-\overline{p})^2- 2V_{12}(x-\overline{q})( y-\overline{p})}{\det V}\right] \nonumber \\ \end{aligned}$$

(7.12)

where $V_{ij}$ are the covariances and $\overline{q}\;,\;\overline{p}$ are the mean values ($\det V=V_{11} V_{22}-V_{12}^2$). Hence a Gaussian state is completely specified by the mean vector and by the covariance matrix

$$\begin{aligned} \bar{X}=\left[ \begin{array}{l}\overline{q}\\ \overline{p}\end{array}\right] \,,\qquad V=\left[ \begin{array}{ll}V_{11}&{}V_{12}\\ V_{12}&{}V_{22} \end{array}\right] . \end{aligned}$$

(7.13)

To emphasize this property, a Gaussian state in general represented by a density operator is symbolized as $\rho (\bar{X},V)$.

We shall see that a coherent state $|\alpha \rangle$ is a special case of Gaussian states with the simple specification

$$\begin{aligned} \bar{X}=\left[ \begin{array}{l}\overline{q}\\ \overline{p}\end{array}\right] = \left[ \begin{array}{l}\mathfrak {R}\alpha \\ \mathfrak {I}\alpha \end{array}\right] \,,\qquad V= \left[ \begin{array}{ll}1&{}0\\ 0&{}1\end{array}\right] =I_2. \end{aligned}$$

(7.14)

Then the Wigner function of a coherent state results in

$$\begin{aligned} W(x,y)=\frac{1}{2\pi } \mathrm{exp}\left[ -{1 \over 2}\left( (x-\overline{q})^2+ (y-\overline{p})^2\right) \right] \end{aligned}$$

(7.15)

and in the $x,y$ plane it is often represented by a contour level obtained by the equation $W(x,y)=L$, where $L>0$ is a reference level. For a coherent state, this contour is a circle centered at $(\overline{q},\overline{p})$, as shown in Fig. 7.4.

Problem 7.1

$\star$    Prove that the inner product $X=\langle {\alpha }|{\beta }\rangle$ of two coherent states is real if and only if $\arg \alpha -\arg \beta =0$ or $\arg \alpha -\arg \beta =\pm \pi$.

Problem 7.2

$\star \star$    The map (7.2) gives for any $\alpha \in {\mathbb C}$ a coherent state $|\alpha \rangle$. Given $|\alpha \rangle$ is it possible to find the complex number $\alpha$?

Problem 7.3

$\star \star$    Examine the effect of the introduction of a phasor $z=\mathrm{e}^{i\varphi }$ into the complex parameter $\alpha$ that identifies the state $|\alpha \rangle$, that is, evaluate $|\mathrm{e}^{i\varphi }\alpha \rangle$.

Problem 7.4

$\star \star \star$    Let $|\alpha \rangle =|\alpha _1\rangle \otimes |\alpha _2\rangle$ be a two-mode coherent states. The number of photons $m_i$ associated to each component state is a Poisson variable with mean $\varLambda _i=|\alpha _i|^2$. Considering that $m_1$ and $m_2$ are statistically independent (see Sect. 3.10), prove that the total number of photons $m=m_1+m_2$ is a Poisson variable.

Hint: use the characteristic function given by (4.23).

Fig. 7.4

Contour level of the Wigner function $W(x,y)$ of a pure coherent state $|\alpha \rangle$ (in red). The mean vector $(\bar{q},\bar{p})=(\mathfrak {R}\alpha ,\mathfrak {I}\alpha )$ gives the center of the contour

3 Constellations of Coherent States

We recall that the target of a quantum communications system is the transmission of a sequence of classical symbols $\{A_n\}$ through a sequence of quantum states $\{|\gamma _{A_n}\rangle \}$, which in practice are often coherent states. Thus, in general, with a $K$-ary alphabet $\mathcal{A}=\{0,1,\ldots ,K-1\}$, Alice must be able to prepare a constellation of $K$ coherent states

$$\begin{aligned} \mathcal{S}=\{|\gamma _0\rangle ,|\gamma _1\rangle ,\ldots ,|\gamma _{K-1}\rangle \} \end{aligned}$$

(7.16)

to realize the c$\,\rightarrow \,$q mapping

$$ A_n\in \mathcal{A}\quad \rightarrow \quad |\gamma _{A_n}\rangle \in \mathcal{S}, $$

which must be bijective. This operation may be called quantum encoding.

Now a problem to investigate is the choice of the constellation, of course, with the purpose of realizing a high-performance quantum transmission system. One way to decide about the choice, as we shall see in this section, is to get the “inspiration” from the optical transmission systems that we shall briefly call classical systems. This approach has also the advantage of allowing us a comparison between the performances of two kinds of systems, classical and quantum.

In Sect. 4.4, we have seen that optical communications use two kinds of modulations, incoherent and coherent; but in the present context, the right comparison is with classical coherent modulations which make use just of a coherent radiation emitted by a laser, as done in quantum communications. A classical $K$-ary coherent modulation, in general nonlinear, is specified by $K$ complex waveforms

$$\begin{aligned} \gamma _{0}(t),\gamma _{1}(t),\ldots ,\gamma _{K-1}(t) \end{aligned}$$

(7.17)

of duration limited¹ to the signaling interval $[0,T]$, with the rule that if $A_n\in \mathcal{A}$ is the $n$th source symbol, the modulator forms a signal with complex envelope ² [4]

$$ c(t)=\gamma _{A_n}(t)\qquad 0\le t<T. $$

With a sequence of symbols $\{A_n\}$, the complete expression of the complex envelope becomes

$$\begin{aligned} c(t)=\sum _{n=-\infty }^{+\infty }\gamma _{A_n}(t-nT)\,, \end{aligned}$$

(7.18)

from which a real modulated signal is obtained as

$$\begin{aligned} v(t)=\mathfrak {R}\,c(t)\,\mathrm{e}^{\mathrm{i}2\pi \nu t} \end{aligned}$$

(7.19)

where $\nu$ is the carrier optical frequency. The comparison between a classical modulator and a quantum encoder is depicted in Fig. 7.5.

Fig. 7.5

Comparison of a classical modulator (left) with a quantum encoder

To proceed from the classical system, characterized by the waveforms $\gamma _{i}(t),\,i\in \mathcal{A}$, to the quantum system with coherent state constellation $|\gamma _{i}\rangle ,\,i\in \mathcal{A}$, we must “remove” in some way the time dependence, which is not present in coherent states. For some kinds of modulations the solution is straightforward; for others, it is less obvious .

3.1 State Constellations from Scalar Modulations

In some modulations, like PSK and QAM, the waveforms (7.17) are of the form

$$\begin{aligned} \gamma _{i}(t)=\gamma _{i}\,h(t)\,,\qquad i\in \mathcal{A}=\{0,1,\ldots ,K-1\}, \end{aligned}$$

(7.20)

where $h(t)$ is a real pulse; for example, rectangular between 0 and $T$, and $\gamma _i$ are complex numbers. The complex envelope $c(t)$ of the modulated signal is then produced by an encoder, mapping the symbols $i\in \mathcal{A}$ into the complex symbols $\gamma _i$, and by an interpolator with impulse response $h(t)$. The resulting complex envelope becomes

$$\begin{aligned} c(t)=\sum _{n=-\infty }^{+\infty }C_n\,h(t-nT)\,, \end{aligned}$$

(7.21)

where $\{C_n\}$ is the sequence of complex symbols obtained by the mapping $A_n=i\;\rightarrow \;C_n=\gamma _i$ (Fig. 7.6).

Fig. 7.6

Scheme of a classical scalar modulator. The encoder maps the source symbols $A_n\in \mathcal{A}$ into the complex symbols $C_n\in \mathcal{C}$. The interpolator maps the complex symbols into the complex envelope $c(t)$

In this way, a constellation of complex symbols is identified

$$\begin{aligned} \mathcal{C}=\{\gamma _0,\gamma _1,\ldots ,\gamma _{K-1}\}\,,\qquad \gamma _i\in {\mathbb C} \end{aligned}$$

(7.22)

from which one can form the constellation of coherent states

$$\begin{aligned} \mathcal{S} = \{|\gamma _0\rangle , |\gamma _1\rangle , \ldots , |\gamma _{K-1}\rangle \} \,,\qquad |\gamma _i\rangle \in \mathcal{G} \end{aligned}$$

(7.23)

that are in a one-to-one correspondence with constellation of complex symbols $\mathcal{C}$ (Fig. 7.7).

Fig. 7.7

Constellation of complex symbols $\mathcal{C}$ and corresponding constellation of coherent states $\mathcal{S}$: each complex symbol $\gamma \in \mathcal{C}$ is mapped into a coherent state $|\gamma \rangle \in \mathcal{G}$

3.2 State Constellations from Vector Modulations    $\Downarrow$

The previous procedure, consisting in directly creating the constellation of coherent states from the constellation of symbols, is not always possible, because in general the $K$ waveforms (7.17) cannot be expressed in the form (7.20). To remove the time dependence, we can proceed in the following way [4]. We take a basis of functions, $h_1(t),\ldots ,h_N(t)$, orthonormal in the interval $[0,T)$, where, in general, $N\le K$, and we expand the waveforms (7.17) on this basis, namely

$$\begin{aligned} {\gamma _i}(t) = \sum _{j=1}^{N}{\gamma _{ij}}\;h_j(t)\,,\qquad i=0,1,\ldots ,K-1 \end{aligned}$$

(7.24a)

where the coefficients are given by

$$\begin{aligned} {\gamma _{ij}} = \int _{0}^{T}{\gamma _i}(t)\; h_j^*(t)\;\mathrm{d}\,t\,,\qquad j=1,\ldots ,N. \end{aligned}$$

(7.24b)

The vectors of the complex coefficients

$$\begin{aligned} {\gamma _i} = ({\gamma _{i1}},\ldots ,{\gamma _{iN}})\,,\qquad i=0,1,\ldots ,K-1 \end{aligned}$$

(7.25)

uniquely identify the waveform ${\gamma _i}(t)$.

The classical modulator can be implemented as in Fig. 7.8, where the encoder makes the map

$$ A_n=i\in \mathcal{A}\quad \rightarrow \quad C_n=\gamma _i\in {\mathbb C}^N $$

with

$$ C_n=[C_{n1},\ldots ,C_{nN}]\,,\qquad \gamma _i=[\gamma _{i1},\ldots ,\gamma _{iN}]. $$

Then from the vector $C_n$, a bank of interpolators forms the complex envelope $c(t)$ of the modulated signal, as

$$\begin{aligned} c(t)=\sum _{n=-\infty }^{\infty } \sum _{i=1}^NC_{ni}\,h_i(t-nT). \end{aligned}$$

(7.26)

Fig. 7.8

Scheme of a classical vector modulator. The encoder maps the source symbols $A_n\in \mathcal{A}$ into a vector of complex symbols $C_n=[C_{n1},\ldots ,C_{ni},\ldots ,C_{nN}]^{{\text {T}}}$. The bank of interpolators maps the vectors $C_n$ into the complex envelope $c(t)$

This generalizes the scalar modulation, which is obtained with $N=1$.

The general procedure just described allows us to identify a constellation of complex vectors $\{\gamma _i, i=0,\ldots ,K-1\}$ with $\gamma _i\in {\mathbb C}^N$. Now, to introduce the coherent states, we must consider a composite Hilbert space, given by the tensor product $\mathcal{H}=\mathcal{H}_0\otimes \mathcal{H}_0\otimes \cdots \otimes \mathcal{H}_0$ of $N$ equal Hilbert spaces $\mathcal{H}_0$. In this composite space, the states become the tensor product of coherent states and, through (7.25), to each symbol $i\,\in \,\mathcal{A}$ the tensor product of coherent states is associated

$$\begin{aligned} |\gamma _{i}\rangle =|\gamma _{i1}\rangle \otimes |\gamma _{i2}\rangle \otimes \cdots \otimes |\gamma _{iN}\rangle \end{aligned}$$

(7.27)

that, with $i$ varying in $\mathcal{A}$, forms the desired constellation of coherent states. In the context of continuous variables of Chap. 11, the tensor product of $N$ coherent states (7.27) is called $N$-mode coherent state.

An example in which we use this method of forming a composite constellation of coherent states will be seen in PPM modulation developed in Sect. 7.13 (see also Problem 7.5).

3.3 Construction of a Symmetric Constellation

An innovative way to obtain a state constellation is based on the geometrically uniform symmetry (GUS), introduced in Sect. 5.13. To this end, it is sufficient finding a unitary operator $S$ having the property of a symmetry operator $S$

$$\begin{aligned} S^K= I_{\mathcal{H}} \end{aligned}$$

(7.28)

that is, $S$ must be a $K$th root of the identity operator. Then, fixing an arbitrary state $|\gamma _0\rangle \in \mathcal{H}$, one gets a $K$-ary constellation of states as

$$\begin{aligned} |\gamma _i\rangle =S^i\,|\gamma _0\rangle \,,\qquad i=0,1,\ldots ,K-1. \end{aligned}$$

(7.29)

More generally, one can fix an arbitrary density operator $\rho _0$ acting on the Hilbert space $\mathcal{H}$ to get a constellation of density operators as (see (5.123))

$$\begin{aligned} \rho _i=S^i\,\rho _0\,(S^i)^*\,,\qquad i=0,1,\ldots ,K-1. \end{aligned}$$

(7.30)

In this way, we can generate infinitely many constellations having the very useful property represented by the GUS. After the choice of $S$ and of the reference state $|\gamma _0\rangle$ or $\rho _0$, one achieves “interesting practical properties” for the quantum communications system based on the corresponding constellation. A nontrivial problem is finding a unitary operator with the property (7.28), especially in the case of infinite dimensions, as is for coherent states.

Problem 7.5

$\star \star$    Show that the PPM must be considered a vector modulation. Find explicitly the waveform $\gamma _i(t)$ and the vector $\gamma _i$ of the coefficients.

Problem 7.6

$\star \star$    The $n$-DFT matrix $W_{[n]}$ is unitary and has the property $W_{[n]}^n=I_n$. Then it allows for the construction of $n$-ary constellations in $\mathcal{H}={\mathbb C}^n$. Find a quaternary constellation using $S=W_{[4]}$ and reference state $|\gamma _0\rangle =[1,1,0,0]^{\text {T}}$. Also prove that the four states are linearly independent.

4 Parameters in a Constellation of Coherent States

In the previous section, we have investigated how to form interesting constellations of coherent states for quantum communications systems. In this section, we want to clarify how a given constellation format can be parametrized to modify the photonic flux therein, expressed, e.g., in terms of the number of signal photons per symbol. In fact, we are interested in the evaluation of the system performance in a given range of this parameter.

Note that the constellation of coherent states $\mathcal{S}$ given by (7.23) can be structured in matrix form as

$$\begin{aligned} \varGamma =[|\gamma _0\rangle ,|\gamma _1\rangle ,\ldots ,|\gamma _{K-1}\rangle ] \end{aligned}$$

(7.31)

and becomes the state matrix. In practical modulation formats (that will be considered further on), the states of $\mathcal{S}$ are always independent (in the sense of vector spaces), and then the state matrix has always full rank, i.e., $\text {rank}(\varGamma )=K$. From the state matrix, we obtain the Gram’s matrix, a $K\times K$ matrix formed by the inner products between the couples of states

$$\begin{aligned} G = \varGamma ^*\varGamma =\left[ \langle \gamma _i|\gamma _j\rangle \right] \,,\qquad \qquad |\gamma _i\rangle ,|\gamma _j\rangle \in \mathcal{G} \end{aligned}$$

that can be calculated using (7.9). Also $G$ has always full rank and, because the states are not orthogonal, all the entries of $G$ are different from zero.

Even in the $N$-dimensional case, when the states are given by the tensor product of $N$ component states (see (7.27)), to calculate the state superposition, we evaluate the inner products

$$\begin{aligned} \langle \gamma _i|\gamma _j\rangle =\langle \gamma _{i1}|\gamma _{j1}\rangle \;\langle \gamma _{i2}|\gamma _{j2}\rangle \;\cdots \; \langle \gamma _{iN}|\gamma _{jN}\rangle . \end{aligned}$$

(7.32)

In this relation we have borne in mind that the inner product of states, given by a tensor product, is obtained as a product of the inner products of the component states (see relation (2.100)). Each of the inner products of the component states is evaluated from (7.9).

4.1 Number of Signal Photons in a Constellation

From (7.8) we have that the average number of photons associated to the coherent state $|\gamma \rangle \in \mathcal{G}$ is given by the squared norm of the complex amplitude $\gamma$

$$ N_\gamma = |\gamma |^2. $$

In a constellation of coherent states, we introduce the signal photons per symbol. To this end, we observe that the generic symbol of the constellation, $C\in \mathcal{C}$, must be considered as a random variable with probability $\mathrm{P}[C=\gamma ],\gamma \in \mathcal{C}$, and also the average number of photons $N_C$ associated to $C$ becomes a random variable; the statistical average of $N_C$,

$$\begin{aligned} N_s = \mathrm{E}[N_C]=\sum _{\gamma \in \mathcal{C}}\mathrm{P}[C=\gamma ]N_\gamma =\sum _{\gamma \in \mathcal{C}} \mathrm{P}[C=\gamma ]|\gamma |^2, \end{aligned}$$

(7.33)

defines the average number of photons per symbol, briefly number of signal photons per symbol. Now, given the one-to-one correspondence $A=i\Leftrightarrow C=\gamma _i$, the probability of these two events turns out to be equal to the prior probability $q_i$. Therefore, we have

$$ N_s=\sum _{i\in \mathcal{A}}q_i|\gamma _i|^2\quad \text {(photons/symbol)}. $$

In particular, with equally likely symbols, the number of signal photons per symbol becomes

(7.34)

Finally, remembering that, with equiprobable symbols, there are $\log _2 K$ bit/symbol, we find that the number of signal photons per bit is given by

$$\begin{aligned} N_R = \frac{N_s}{\log _2 K}\quad \text {(photons/bit)}. \end{aligned}$$

(7.35)

$\Downarrow$    We have seen above that in an $N$-dimensional constellation $\mathcal{C}$, whose states are $N$-mode coherent states, $|\gamma \rangle =|\gamma _{1}\rangle \otimes |\gamma _{2}\rangle \otimes \ldots \otimes |\gamma _{N}\rangle$, the average number of photons associated to the composite state $|\gamma \rangle$ results in (see (7.11))

$$\begin{aligned} N_{\gamma }=|\gamma _1|^2+|\gamma _2|^2+\cdots +|\gamma _N|^2 \end{aligned}$$

(7.36)

where $\gamma =[\gamma _1,\gamma _2,\ldots ,\gamma _N]$. Consequently, the number of signal photons per symbol must be evaluated according to

$$\begin{aligned} N_s=\sum _{\gamma \in \mathcal{C}}\mathrm{P}[C=\gamma ]\,N_{\gamma } \end{aligned}$$

(7.37)

with $N_{\gamma }$ given by (7.36), and the sum is extended to the $N$-dimensional constellation. Of course, with equiprobable symbols we have $\mathrm{P}[C=\gamma ]=1/K$ and the number of signal photons per bit is still given by (7.35).

Sensitivity of a receiver. In telecommunications an important parameter is the sensitivity, which is defined as the minimum value of a parameter of the receiver that guarantees a given value of the error probability $P_e$, typically $P_e=10^{-9}$. In optical communications (classical or quantum), the parameter is often given by the number of photons per bit $N_R$. Thus we say, e.g., that a quantum receiver has the sensitivity of $N_R= 11.5$ photons/bit.

4.2 Scale Factor and Shape Factor of a Constellation

The constellation (7.22) of complex symbols $\mathcal{C}$, from which we can directly obtain the constellation (7.23) of coherent states $\mathcal{S}$, contains a scale factor linked to the photonic intensity, but modulation formats are usually specified in a normalized form. In the evaluation of a system’s performance, it is worthwhile to underline this aspect by expressing the symbols $\gamma _i$ in the form $\bar{\gamma }_i\varDelta$, where $\bar{\gamma }_i$ are normalized symbols and $\varDelta$ is the scale factor. Then it is convenient to introduce a normalized constellation

$$\begin{aligned} \mathcal{C}_0=\{\bar{\gamma }_0,\bar{\gamma }_1,\ldots ,\bar{\gamma }_{K-1}\} \end{aligned}$$

from which one obtains the scaled constellation as $\mathcal{C}=\{\bar{\gamma }_0\varDelta ,\bar{\gamma }_1\varDelta ,\ldots , \bar{\gamma }_{K-1}\varDelta \}$ and hence the constellation of coherent states as

$$ \mathcal{S}=\{|\bar{\gamma }_0\varDelta \rangle ,|\bar{\gamma }_1\varDelta \rangle ,\ldots , |\bar{\gamma }_{K-1}\varDelta \rangle \}. $$

The scale factor appears in the number of signal photons per symbol, given by (7.34), which can be written in the form

$$\begin{aligned} N_s=\frac{1}{K}\sum _{\gamma \in \mathcal{C}}|\gamma |^2=\varDelta ^2\frac{1}{K}\sum _{\bar{\gamma } \in \mathcal{C}_0}|\bar{\gamma }|^2=\mu _K\varDelta ^2 \end{aligned}$$

(7.38)

where

$$\begin{aligned} \mu _K:=\frac{1}{K}\sum _{\bar{\gamma }\in \mathcal{C}_0}|\bar{\gamma }|^2 \end{aligned}$$

(7.39)

is a characteristic parameter of the constellation, which we call shape factor. For example, in the PSK modulation, the normalized constellation consists of $K$ points on the unit circle

$$ \mathcal{C}_0=\{\mathrm{e}^{\mathrm{i}2\pi m/K}|\,m=0,1,\ldots ,K-1\} $$

whereas the scaled constellation is given by $K$ points on the circle of radius $\varDelta$

$$ \mathcal{C}=\{\varDelta \,\mathrm{e}^{\mathrm{i}2\pi m/K}|\,m=0,1,\ldots ,K-1\}. $$

In this case, the shape factor is $\mu _K=1$. An example where $\mu _K\ne 1$ is given by the QAM modulation.

4.3 Summary of Constellation Formats

To conclude these two sections on constellations of coherent states, it is convenient to recall that the target of a quantum communications system is the transmission of a classical information, encoded in a classical symbol sequence $\{A_n\}$, through a sequence of quantum states $\{|\gamma _{A_n}\rangle \}$, as illustrated in Fig. 7.1. Thus, a key operation is the c$\;\rightarrow \;$q mapping $A_n\;\rightarrow \;|\gamma _{A_n}\rangle$. This finally explains why we have constellations both classical and quantum domain.

Here, we wish to summarize the constellations introduced above which are all useful to proceed on. Starting from a symbol alphabet, which was indicated in the form $\mathcal{A}=\{0,1,\ldots ,K-1\}$, we have introduced:

• a constellation of normalized complex symbols $\mathcal{C}_0=\{\bar{\gamma }_0,\bar{\gamma }_1,\ldots , \bar{\gamma }_{K-1}\}$,

• a constellation of scaled complex symbols $\mathcal{C}=\{\gamma _0,\gamma _1,\ldots ,\gamma _{K-1}\}$, with $\gamma _i =\bar{\gamma }_0 \varDelta$, where $\varDelta$ is a scale factor,

• a constellation of coherent states $\mathcal{S}=\{|\gamma _0\rangle ,|\gamma _1\rangle ,\ldots ,|\gamma _{K-1}\rangle \}$, where $|\gamma _i\rangle$ is the coherent state uniquely determined by the scaled complex symbol $\gamma _i$, according to relation (7.2).

Note that $\mathcal{C}_0$ and $\mathcal{C}$ live in the field of complex numbers ${\mathbb C}$, while $\mathcal{S}$ lives in the infinite dimensional Hilbert space $\mathcal{H}$.

Problem 7.7

$\star \!\star \nabla$    Find the shape factor $\mu _k$ of the 16-QAM constellation (see Fig. 7.28).

5 Theory of Classical Optical Systems

We want to compare the performance of a quantum communications system with that of the corresponding classical communications system, i.e., not based on quantum measurements, but on an optical detection (see semiclassical detection in Chap. 4).

In the formulation of the transmitter and the receiver, it is convenient to introduce two distinct schemes: One working at the level of instantaneous optical power and the other one working on the complex envelope. In fact in the semiclassical theory of an optical system, both the optical power and the complex envelope must be jointly considered, as remarked in Sect. 4.7.

5.1 Scheme for Instantaneous Optical Powers

We recall that a monochromatic radiation at the optical frequency $\nu$ can be modeled as an instantaneous optical power, which is formed by the energy quanta of size $h\nu$ and has the impulsive expression

$$\begin{aligned} p(t)= \sum _k(h\nu )\,\delta (t-t_k) \end{aligned}$$

(7.40)

where the arrival instants $\{t_k\}$ are represented by a doubly stochastic Poisson process, specified by its random intensity $\lambda (t)$.

Referring to digital systems, the information to be transmitted is first conveyed in a sequence of symbols $\{A_n\}\;,\;A_n\in \{0,1,\ldots ,K-1\}$ and then, for convenience, in a sequence of complex symbols $\{C_n\}$ belonging to a given (normalized) constellation $\mathcal{C}_0$. Then the first part of the transmitter is an encoder, which provides the map $A_n\;\rightarrow \;C_n$. The task of a digital modulator is to modify the laser beam in each symbol period $(nT,nT+T)$ in dependence of the symbol $C_n$ falling in this period.³ If there are no further processing, as we suppose, the output of the modulator gives the instantaneous transmitted power $p_T(t)$, as shown in Fig. 7.9.

Fig. 7.9

Scheme of a classical modulator and demodulator for the instantaneous optical power. In the figure, the initial encoder $A_n\;\rightarrow \; C_n$ and the final decoder $\widehat{C}_n\;\rightarrow \;\widehat{A}_n$ are omitted

In the receiver, the incoming instantaneous power $p_R(t)$, an attenuated version of $p_T(t)$, is combined with the instantaneous power $p_L(t)$ of a local laser tuned at the same frequency $\nu$ as the laser in the transmitter (homodyne detection) or at a different frequency (etherodyne detection). The task of the demodulator is the production of two distinct instantaneous powers $p_a(t)$ and $p_b(t)$ to feed two photon counters, which count the photon numbers in each symbol period as

$$ n_a=\frac{1}{h\nu }\int _{nT}^{nT+T} {p_a}(t)\;\mathrm{d}\,t\,,\qquad n_b=\frac{1}{h\nu }\int _{nT}^{nT+T} {p_b}(t)\;\mathrm{d}\,t. $$

The reason of this double path is due to the fact that $n_a$ and $n_b$ are real (integer) and the receiver has to give an estimated version $\{\widehat{C}_n\}$ of the complex sequence $\{C_n\}$.

In the case of binary systems, where the symbols $C_n$ are real, the double path is not necessary and the detection is based only on a single photon counting. In the following, for brevity, we will consider only homodyne detection.

5.2 Scheme for Complex Envelopes

In an optical system, the complex envelope $V(t)$ (denoted by $c_v(t)$ in Sect. 4.7) contains all the information useful both for the signal analysis and the statistical analysis. In fact from $V(t)$, we can obtain the signal $v(t)$, present in the form of electric field, as

$$\begin{aligned} v(t)=\mathfrak {R}V(t)\,\mathrm{e}^{\mathrm{i}\,2\pi \,\nu \,t}\,. \end{aligned}$$

(7.41)

Also, the average power $P(t)$ is proportional to $|V(t)|^2$ and, by appropriate normalization of the electric field, it can be directly written as

$$\begin{aligned} P(t)=|V(t)|^2\,. \end{aligned}$$

(7.42)

On the other hand, the average power is connected to the instantaneous power, modeled as a doubly stochastic filtered Poisson process, through Campbell’s theorem according to

$$\begin{aligned} P(t)=\mathrm{E}[p(t)|\lambda ] =(h\nu )\,\lambda (t)\,, \end{aligned}$$

(7.43)

where $\mathrm{E}[\cdot |\lambda ]$ denotes the conditional expectation “with a given $\lambda (t)$.” This holds for the powers $p_T(t)$, $p_R(t)$, $p_a(t)$, and $p_b(t)$ in the scheme of Fig. 7.9. The fundamental remark is that from the complex envelope $V(t)$, we can obtain the intensity $\lambda (t)$ which gives the full statistical description of the doubly stochastic Poisson processes involved.

At the level of complex envelope, the modulator scheme is essentially the one anticipated in Fig. 7.6, where, starting from the complex sequence $\{C_n\}$, the complex envelope of the modulated signal is obtained with an interpolator according to (7.21), that is,

$$\begin{aligned} V_T(t)=\sum _{n=-\infty }^{+\infty }\;C_n V_0\;h(t-nT) \end{aligned}$$

(7.44)

where $V_0$ is the amplitude of the carrier produced by the laser. This corresponds to the transmitter instantaneous power $p_T(t)$.

The scheme of the demodulator is extremely simple. To the incoming complex envelope $V_R(t)$, corresponding to the received instantaneous power $p_R(t)$, the amplitude $V_L$ is added for the upper path and the amplitude $\mathrm{i}\,V_L$ to the lower path to get

$$\begin{aligned} V_a(t)=V_R(t)+V_L, \quad V_b(t)=V_R(t)+\mathrm{i}\,V_L \end{aligned}$$

(7.45)

as shown in Fig. 7.10.

Fig. 7.10

Scheme of a classical modulator and demodulator for the complex envelopes

In the following, we make the assumption that the interpolator impulse response $h(t)$ is unitary in $(0,T)$, so that $V_T(t)$ is simplified as

$$\begin{aligned} V_T(t)=C_0\,V_0,\quad 0<t<T. \end{aligned}$$

(7.46)

Correspondingly (7.45) become

$$\begin{aligned} V_a(t)=C_0\,V_R+V_L,\quad V_b(t)=C_0\,V_R+\mathrm{i}V_L \end{aligned}$$

(7.47)

where $V_R$ is the amplitude of the received carrier.

5.3 Scheme for Signals. Quadrature Modulator

The scheme for the complex envelope is sufficient for the analysis of an optical system. Now we consider the scheme for the signal, which is more detailed and may have interest for the implementation of the system.

Signals are obtained from complex envelopes according to relation (7.41). Letting $C_n=A_n+\mathrm{i}B_n$, from (7.44); we find that the modulated signal results in

$$\begin{aligned} v_T(t)=\,&\mathfrak {R}\sum _{n=-\infty }^{+\infty }(A_n+\mathrm{i}B_n)\,h(t-nT)\;V_0 \,\mathrm{e}^{\,\mathrm{i}\,2\pi \nu t} \nonumber \\ =&\sum _{n=-\infty }^{+\infty } \left[ A_nV_0\,h(t-nT)\,\cos 2\pi \nu \,t- B_nV_0\,h(t-nT)\,\sin 2\pi \nu \,t\right] . \end{aligned}$$

(7.48)

The interpretation of these relations leads to the scheme of Fig. 7.11, called quadrature modulator. The carrier $V_T\cos 2\pi \nu t$ is produced by a laser tuned at the frequency $\nu$ and the quadrature carrier $-V_T\sin 2\pi \nu t$ is obtained by shifting the carrier $V_T\cos 2\pi \nu t$.

With the simplification of (7.46), (7.48) gives

$$\begin{aligned} v_T(t)=A_0V_0\cos 2\pi \nu t -B_0V_0\sin 2\pi \nu t\,,\qquad 0<t<T. \end{aligned}$$

(7.49)

Fig. 7.11

Implementation of a coherent optical system based on a quadrature modulator. On the left the transmitter and on the right the homodyne receiver

At reception, the modulated signal becomes

$$\begin{aligned} v_R(t)=A_0V_R\cos 2\pi \nu t-B_0V_R\sin 2\pi \nu t\,,\qquad 0<t<T, \end{aligned}$$

(7.50)

and the constant complex envelopes $V_L$ and $\mathrm{i}\,V_L$ give

$$ \mathfrak {R}\;V_L\mathrm{e}^{\mathrm{i}\,2\pi \,\nu \,t} =V_L\cos 2\pi \nu t\,,\qquad \mathfrak {R}\;\mathrm{i}\,V_L\mathrm{e}^{\mathrm{i}\,2\pi \,\nu \,t} =-V_L\sin 2\pi \nu t. $$

These carriers are provided by a local laser, tuned with the transmission laser (homodyne reception). Finally, (7.47) gives for $\;0<t<T$

$$\begin{aligned} v_a(t)&=\mathfrak {R}\;\left[ {C_0}{V_R}+V_L \right] \mathrm{e}^{i2\pi \nu t} =(A_0V_R+V_L)\cos 2\pi \nu t-B_0V_R\sin 2\pi \nu t\nonumber \\ v_b(t)&=\mathfrak {R}\;\left[ {C_0}{V_R}+i\,V_L)\right] \mathrm{e}^{i2\pi \nu t} =A_0V_R\cos 2\pi \nu t-(B_0V_R+V_L)\sin 2\pi \nu t. \end{aligned}$$

(7.51)

These signals feed the photon counters.

5.4 Photon Counting and Detection

The count in the interval $(0,T]$ yields two values, $n_a$ and $n_b$, from which a decision must be taken on the transmitted symbol $C_0$; $n_a$ and $n_b$ are conditioned Poisson variables and therefore characterized by their averages $\bar{n}_a(C_0):=\mathrm{E}[n_a|C_0]$ and $\bar{n}_b(C_0):=\mathrm{E}[n_b|C_0]$, the condition being “given a transmitted symbol $C_0$.” These averages are obtained dividing the corresponding energies in a symbol period $T$ by the quantum $h\nu$. Considering that the complex envelopes are constant in $(0,T)$, we have $E_a=P_a T=|V_a |^2T$ and $E_b=P_b T=|V_b |^2T$, and then

$$\begin{aligned} \bar{n}_a(C_0)&=H|C_0\, V_R +V_L|^2=H\left[ (A_0V_R+V_L)^2+(B_0V_R)^2\right] \nonumber \\ \bar{n}_b(C_0)&=H|C_0\, V_R +\mathrm{i}\,V_L|^2=H\left[ (A_0V_R)^2+(B_0V_R+V_L)^2\right] \end{aligned}$$

(7.52)

where $H=T/(h\nu )$.

At this point, we assume that the local carrier has an amplitude $V_L$ much greater than $V_R$, which allows us to get the following approximations

$$\begin{aligned} \bar{n}(A_0)=H(2A_0V_RV_L+V_L^2)\,,\qquad \bar{n}(B_0)=H(2B_0V_RV_L+V_L^2), \end{aligned}$$

(7.53a)

where now the upper counting depends only on $A_0$ and the lower counting only on $B_0$. The averages can be expressed in “numbers” by letting $N_L=HV_L^2$ and $N_R=HV_R^2$ to get

$$\begin{aligned} \bar{n}(A_0)=2\sqrt{N_L\,N_R}\,A_0+N_L\,,\qquad \bar{n}(B_0)=2\sqrt{N_L\,N_R} \,B_0+N_L. \end{aligned}$$

(7.53b)

The numbers photons $n_a$ and $n_b$ can be decomposed as

$$\begin{aligned} \left. \begin{array}{l}n_a=\bar{n}(A_0)+u_a=A_0\,U_0+N_L+u_a\\ n_b=\bar{n}(B_0)+u_b=B_0\,U_0+N_L+u_b\end{array}\right. \,,\qquad \quad U_0:=2\,\sqrt{N_L\,N_R} \end{aligned}$$

(7.54)

where

• $U_0\,A_0$ and $U_0\,B_0$ are the useful signals,

• $N_L$ is a bias,

• $u_a$ and $u_b$ are the shot noises.

We compose for convenience the two countings into a complex one to get

$$\begin{aligned} z_0=n_a+\mathrm{i}n_b=C_0\,U_0+N_L+\mathrm{i}\,N_L+u_a+\mathrm{i}u_b \end{aligned}$$

(7.55)

which is the standard form of the “signal at the decision point” in a quadrature modulator. Note that

$$ \bar{n}(C_0):=\bar{n}(A_0)+\mathrm{i}\,\bar{n}(B_0)=C_0\,U_0+ N_L(1+\mathrm{i}) \,,\qquad C_0\in \mathcal{C}_0 $$

generates a constellation of “received values”, with center the point $N_L(1+\mathrm{i})$ of the complex plane ${\mathbb C}$. The constellation is illustrated in Fig. 7.12 in the case of 8-PSK.

Fig. 7.12

Constellation of “received values” in the complex plane of optical 8-PSK. The points of the constellations are given by $N_L(1+\mathrm{i})+U_0\,\mathrm{e}^{\mathrm{i}\,2\pi k/8}\;,\; k=0,1\ldots ,7$

5.5 Correct Decision Probability

In principle, it is possible to evaluate the correct decision probability $P_c {\,=\,} \mathrm{P}[\widehat{C}_0 = C_0]$ from the statistical description of the integer random variables $n_a$ and $n_b$. These variables can be considered statistically independent, and therefore described by two conditioned Poisson distributions $p_{n_a}(k|A_0)$ and $p_{n_b}(k|B_0)$, which in turn are specified by their averages $\bar{n}(A_0)$ and $\bar{n}(B_0)$ given by (7.53). The preliminary step is the choice of the decision regions $\{\mathcal{R}(\gamma )|\gamma \in \mathcal{C}_0\}$, which has to form a partition of the set of integer pairs $\{(k_1,k_2)|k_1,k_2=0,1,2,\ldots \}$. Then we have the decision criterion

$$\begin{aligned} \widehat{C}_0= \gamma \quad \text {if}\quad (n_a,n_b)\in \mathcal{R}(\gamma ). \end{aligned}$$

(7.56)

Correspondingly, the transition probabilities are given by

$$\begin{aligned} p(\gamma '|\gamma ):=\mathrm{P}[\hat{C}_0=\gamma {\,'}|\;C_0= \gamma ]\sum _{(k_1,k_2)\in \mathcal{R}(\gamma ')} p_{n_a}(k_1|\mathfrak {R}\gamma )\;p_{n_b}(k_2|\mathfrak {I}\gamma ) \end{aligned}$$

(7.57)

and the correct decision probability, with equally likely symbols, by

$$\begin{aligned} P_c=\frac{1}{K}\sum _{\gamma \in \mathcal{C}}\,p(\gamma |\gamma ). \end{aligned}$$

(7.58)

The decision regions should be optimized to maximize $P_c$.

This procedure will be applied in the next chapter (Sect. 8.6) to a specific case (a BPSK system). In general, it is cumbersome and does not give readable results because only numerical evaluations are possible. The alternative is the Gaussian approximation, where it is assumed that the photon numbers $n_a$ and $n_b$ are independent Gaussian random variables. This allows us to simplify the analysis and to arrive at very simple results.

Note that $n_a$ and $n_b$ are Poisson random variables and it may appear to be strange that discrete random variables, described by (mass) probability distributions, are approximated by continuous random variables, described by probability densities. The approximation does not work in counting, but in the evaluation of the transition probabilities and of the error probability. In Sect. 8.6, we will compare the exact evaluation of probabilities, obtained with the Poisson statistics, and the approximate evaluation, obtained with the Gaussian assumption. We will see that the Gaussian approximation gives a very accurate evaluation of the exact probabilities. This conclusion holds in general in the presence of a strong photonic intensity [5] (here ensured by the assumption $V_L\gg V_0$).

With the Gaussian approximation, $n_a$ and $n_b$ become specified by their conditional means $\overline{n}(A_0):=\mathrm{E}[n|A_0]$ and $\overline{n}(B_0):=\mathrm{E}[n_b|B_0]$ and by their variances $\sigma ^2(A_0)=\overline{n}(A_0)$ and $\sigma ^2(B_0)=\overline{n}(B_0)$. For the latter, a further simplification⁴ can be introduced by neglecting in (7.53) $\;2A_0\sqrt{\,N_R\,N_L}$ and $2B_0\sqrt{\,N_R\,N_L}$ with respect to $N_L$, so that they become equal, $\sigma ^2_n:=\sigma ^2(B_0)=\sigma ^2(B_0)=N_L$, and independent of the symbols. Then their joint probability density results in

$$\begin{aligned} f_{n_a}(a|A_0)\,f_{n_b}(a|B_0)&= \frac{1}{2\pi \sigma _n^2}\mathrm{exp}\left[ -\frac{(a-\overline{n}(A_0))^2+(b-\overline{n}(B_0))^2}{2 \sigma _n^2}\right] \nonumber \\&=\frac{1}{\sigma _n}\phi \left( \frac{a-\overline{n}(A_0)}{\sigma _n}\right) \frac{1}{\sigma _n}\phi \left( \frac{b-\overline{n}(B_0)}{\sigma _n}\right) . \end{aligned}$$

(7.59)

The decision regions $\{R(\gamma )\,|\;\gamma \in \mathcal{C}_0\}$ become a partition of the complex plane. Then the transition probabilities are given by

$$\begin{aligned} {p_c}(\gamma {\,'}|\gamma ):=\mathrm{P}[\hat{C}_0=\gamma {\,'}|\;C_0=\gamma ]= \int _{R(\gamma ^{\,'})}{f_{n_a}}(a|\mathfrak {R}\gamma )\,{f_{n_b}}(a|\mathfrak {I}\gamma )\,\mathrm{d}a\,\mathrm{d}b. \end{aligned}$$

(7.60)

The correct decision , with equally likely symbols, is still given by (7.58).

The above probabilities depend only on the SNR, which results in

$$\begin{aligned} \varLambda =\frac{U_0^2}{\sigma ^2_n}=4\,N_R \end{aligned}$$

(7.61)

and is related to the number of signal photons contained in the received power $P_R=V_R^2$. In fact, considering that $V_R(t)=C_0 \,V_R$, the received power is given by $P_R=|C_0|^2V_R^2$, and therefore the number of signal photons associated to the symbol $C_0$ is $|C_0|^2HV_R^2=|C_0|^2 \,N_R$. This can be related to the number of signal photons per symbol $N_s$ as (see (7.34))

$$\begin{aligned} N_s=\frac{1}{K}\sum _{\gamma \in \mathcal{C}_0}|\gamma |^2\,N_R=\mu _K\,N_R \end{aligned}$$

(7.62)

where $\mu _K$ is the shape factor of the constellation (see (7.39)). Then $\varLambda =4\,N_s/\mu _K$. As we will see, in the cases of interest, with an optimized choice of the decision region, the error probability is a function of the SNR $\varLambda$ expressed by function $Q(x)$ (see Problem 7.8).

The above theory on classical optical systems is quite long and contains a lot of relations, but the net result for the evaluation of the performance is extremely simple.

Proposition 7.2

In a classical optical system, where the local carrier has an amplitude $V_L$ much greater than the received carrier amplitude $V_R$, the shot noise may be considered Gaussian. With equally likely symbols and optimized decision regions $\{R(\gamma ),\gamma \in \mathcal{C}_0\}$, the minimum error probability turns out to be a simple function of the SNR

(7.63)

expressed through the complementary normalized Gaussian distribution $Q(x)$. In (7.63) $N_s$ is the number of signal photons per symbol and $\mu _K$ is shape factor of the constellation.

Problem 7.8

$\star$    Consider the 4-QAM (which is equivalent to 4-PSK) where the normalized constellation is $\mathcal{C}_0=\{\gamma =\pm 1+\pm \mathrm{i}\}$ and the constellation of received values is given by

$$\begin{aligned} \{(\pm 1+\pm \mathrm{i})U_0+(1+\mathrm{i})N_L\}. \end{aligned}$$

Find the optimal decision regions and prove that the minimum error probability $P_c$ is given by $P_c=1-\left( 1-Q(\sqrt{\varLambda })^2\right)$ with $\varLambda =4N_R$.

6 Analysis of Classical Optical Binary Systems

In classical optical systems, the transmitted symbol $C_0$ has in general a complex format; but in the binary case, without restrictions, we can assume a real format. Then the general schemes of the previous section (Figs. 7.9, 7.10, and 7.11) are simplified because the double path is reduced to a single path.

For the sake of comparison with other schemes, it is convenient to express the system performance (error probability) in terms of the average number of photons per bit $N_R$, which is given in general by

$$ N_R=q_0\,N_R(0)+ q_1\,N_R(1)\,,\qquad \quad q_0+q_1=1 $$

where $q_i=\mathrm{P}[A_0=i]$ are the a priori probabilities and $N_R(i)=\mathrm{E}[n|A_0=i]$ the average number of photons associated to the symbol $A_0=i$. Usually we will consider equiprobable symbols, so that the average number of photon per bit becomes

$$\begin{aligned} N_R=\frac{1}{2}\,N_R(0)+ \frac{1}{2}\,N_R(1). \end{aligned}$$

(7.64)

In the following analysis, we will use the notations for signals and complex envelopes:

• $v_0(t)$   $V_0\quad$   optical carrier at the transmitter,

• $v_T(t)$   $V_T(t)$   transmitted optical signal,

• $v_R(t)$   $V_R(t)$   received optical signal,

• $v_L(t)$   $V_L\quad$   local optical carrier at the reception.

We assume that the channel is ideal, so that

$$ v_R(t)=v_T(t). $$

6.1 Binary System with Incoherent Detection (OOK Modulation)

In the classical formulation, a monochromatic wave at frequency $\nu$ emitted by a laser can be represented by a sinusoidal signal

$$\begin{aligned} v_0(t)=V_0 \cos 2 \pi \nu t \end{aligned}$$

(7.65)

where the amplitude $V_0$ gives the optical power as (see Sect. 7.5)

$$\begin{aligned} P = V_0^2. \end{aligned}$$

(7.66)

The simplest optical communications system uses amplitude modulation (OOK) and incoherent detection, as shown in Fig. 7.13. The OOK modulator is a special case of the general modulator of Fig. 7.10 with the encoding mapping the identity, $A_0\;\;\rightarrow C_0=A_0$, which gives the modulated signal

$$ v_T(t)=\mathfrak {R}C_0\,V_0\,\mathrm{e}^{\mathrm{i}\, 2\pi \nu \,t}=A_0\,V_0\cos 2\pi \nu t\,,\qquad 0<t<T. $$

In practice, in the symbol period $(0,T)$ the transmitter associates a zero field to the symbol $A_0=0$ and the field $V_0\cos 2\pi \nu t$ to the symbol $A_0=1$.

Figure 7.14 shows a sequence of binary symbols and the corresponding modulated signal. This is obtained by amplitude modulating the laser beam of frequency $\nu$ or, more simply, by switching on and off the laser itself according to the source symbol to be transmitted. At the receiver, a photodetector transforms the incident field into an electrical current from which a photon counting can be obtained, as seen in Sect. 4.8.

Considering that with the transmission of the symbol $A_0=0$, the number of photons is null, $n=0$, in (7.64) we have $N_R(0)=0$, and therefore

$$ N_R= \frac{1}{2}N_R(1). $$

At reception the photon count receiver uses the decision criterion

Fig. 7.13

A realization of a binary sequence and corresponding OOK signal

Fig. 7.14

Binary optical system with amplitude on–off modulation and uncoherent detection

$$\begin{aligned} \widehat{A}_0=\left\{ \begin{array}{cc} \;0 &{} \quad \text {if}\,\; n=0\\ \;1 &{} \quad \;\text {if}\,\; n\ge 1,\\ \end{array} \right. \end{aligned}$$

(7.67)

where $n$ is the number of photons counted in a symbol period. Then, with the transmission of the symbol $A_0=0$, we always have a correct decision

$$\begin{aligned} P_e(0)=0. \end{aligned}$$

(7.68a)

When $A_0=1$ the number of arrivals $n$ is a Poisson variable with average $N_R(1)$, and therefore with (conditioned) distribution

$$ p_n(k|1)=\mathrm{e}^{-N_R(1)}\frac{N_R(1)^k}{k!}\,,\qquad k=0,1,\ldots $$

and we have an error when $n=0$, which occurs with probability

$$\begin{aligned} P_e(1)=p_n(0|1)=\mathrm{e}^{-N_R(1)}=\mathrm{e}^{-2N_R}. \end{aligned}$$

(7.68b)

The average error probability in the classical system is therefore

(7.69)

where equally likely symbols are assumed.

In optical communications this probability is called the quantum limit [6] or shot noise limit, and it is the optimum for any detection that does not exploit the coherence property of the optical beam. Notice, in fact, that in this classical context the decision criterion (7.67) is optimal (see Problem 5.4). The receiver scheme is called direct detection of the incident light pulses. The main advantage of this approach is its simplicity. In particular, phase and frequency instabilities of the laser source are well tolerated. Moreover, at the receiver detection is used and phase sensitive devices are avoided.

6.2 Quantum Interpretation of Photon Counting in OOK

The above scheme, known as on–off keying (OOK) modulation, has a simple quantum equivalent, employing the coherent states $|0\rangle$ and $|\alpha \rangle$, with $\alpha >0$, where the photon counting can be treated as a quantum measurement with not optimal measurement operators .

The quantum measurement realized by the photon counter is obtained with the elementary projectors $|n\rangle \langle n|$, where $|n\rangle$ is the number state (see Fig. 7.2), and the outcome of the measurement is given by the number of photons $n$. The transition probabilities in the measurement are

$$ p(i|\alpha )=\mathrm{P}[n=i|\alpha ]=\mathrm{e}^{-|\alpha |^2} \frac{|\alpha |^{2i}}{i!} \,,\qquad p(i|0)=\mathrm{P}[n=i|0]=\delta _{i0}. $$

The alphabet of the measurement is then $\mathcal{M}=\{0,1,2,\ldots \}$, and it is different from the alphabet $\mathcal{A}=\{0,1\}$ of the source (see Sect. 5.2). To find the global performance, we must introduce a decision criterion consisting in the partitioning of $\mathcal{M}$ into two decision regions $\mathcal{M}_0$ and $\mathcal{M}_1$ to obtain two global measurement operators (see Sect. 5.2.3). The optimal partition is $\mathcal{M}_0=\{0\}$ and $\mathcal{M}_1=\{1,2,\ldots \}$ (Fig. 7.15) and so we have the global projectors

$$\begin{aligned} Q_0=|0\rangle \langle 0|\qquad Q_1=\sum _{n=1}^\infty |n\rangle \langle n|. \end{aligned}$$

(7.70)

The global transition probabilities, from (5.15), are $p_c(0|0)= \text {Tr}[\rho _0\,Q_0]$ and $p_c(0|1)=\text {Tr}[\rho _1\,Q_0]$, where $\rho _1=|\alpha \rangle \langle \alpha |$ and $\rho _0=|0\rangle \langle 0|$. Then

$$\begin{aligned} p_c(0|0)&=\langle 0|0\rangle \langle 0|0\rangle =1\nonumber \\ p_c(0|1)&=\langle \alpha |Q_0|\alpha \rangle =|\langle \alpha |0\rangle |^2= \mathrm{e}^{-|\alpha |^2}=\mathrm{e}^{-2N_R}. \end{aligned}$$

(7.71)

Fig. 7.15

Quantum interpretation of the decision made via a photon counter in an OOK system. The outcome of quantum measurement is given by the number of photons $n$ present in the state $|\alpha \rangle$ or $|0\rangle$. The decision converts the measurement alphabet $\mathcal{M}=\{0,1,2,\ldots \}$ into the binary alphabet $\mathcal{A}=\{0,1\}$, thus realizing a binary channel

Fig. 7.16

Decision with a photon counter (left) and optimal decision

The performance is lower than that of the quantum version of the OOK, which will be seen in Sect. 7.9, because the projectors (7.70) are suboptimal. We recall, in fact, that with pure states, the optimal measurement operators must be elementary with measurement vectors arranged symmetrically with respect to the coherent states (Fig. 7.16); whereas (7.70) $Q_1$ has infinite rank and $Q_0$ is elementary with measurement vector $|\mu _0\rangle$ coinciding with the state $|0\rangle$.

6.3 Binary System with Coherent Detection (BPSK Modulation)

A more sophisticated scheme of classical optical communications uses binary phase-shift keying (BPSK) modulation (Fig. 7.17). The BPSK modulator is a special case of the general modulator of Fig. 7.10 with the encoding mapping

$$ A_0\;\rightarrow \;C_0=\mathrm{e}^{\mathrm{i}A_0\,\pi \,}={\left\{ \begin{array}{ll}\;+1&{}\quad A_0=0\\ \;-1&{}\quad A_0=1\end{array}\right. } $$

Fig. 7.17

Scheme of a binary coherent optical system with BPSK modulation. The receiver is called homodyne because the frequency of the local laser is the same as the frequency of modulation carrier

Fig. 7.18

A realization of a binary sequence and corresponding BPSK signal

which gives the modulated signal

$$\begin{aligned} v_T(t)=\mathfrak {R}\;C_0\,V_0\;\mathrm{e}^{\,\mathrm{i}\, 2\pi \,\nu \, t}=V_0\cos (2\pi \nu t+A_0 \pi )\,,\qquad 0<t<T \end{aligned}$$

(7.72)

where $V_0$ is the amplitude of the carrier $v_0(t)=V_0\,\cos 2\pi \nu t$. Figure 7.18 shows a sequence of binary symbols and the corresponding BPSK signal, which in the interval $(nT,nT+T)$ is given by $V_0\cos (2\pi \nu t+A_n\,\pi )$.

BPSK with Homodyne Detection

Since the modulated signals $v_R(t)=v_T(t)$ for different symbols have the same optical energy, and hence the same photon counting, direct detection cannot discriminate between them. Then the receiver adds to the incoming field $v_R(t)=V_R\cos (2\pi \nu t+A_n\,\pi )$ the field $V_L\cos 2\pi \nu t$, generated by a “local” laser tuned at the same frequency as $v_0(t)$, to get the signal

$$\begin{aligned} v(t) = V_R \cos \left( 2 \pi \nu t + A_0 \pi \right) + V_L \cos \left( 2 \pi \nu t \right) . \end{aligned}$$

(7.73)

As in Sect. 7.5.4, we assume that the local carrier has an amplitude $V_L$ much greater than that of the received signal, $V_L\gg V_0$. Since $\cos \left( 2 \pi \nu t + A_0 \pi \right) = \cos A_0\pi \; \cos 2 \pi \nu t$, the power becomes

Fig. 7.19

Example of the optical power $P_R(t)$ after the introduction of the local carrier in a homodyne receiver

$$\begin{aligned} P_v (t) = \left( V_R \cos \pi A_0 + V_L \right) ^2 = V_R^2 + V_L^2 + 2 V_R V_L \cos A_0\pi \end{aligned}$$

(7.74)

which is illustrated in Fig. 7.19 for a sequence of source symbols. Applying this power to a photon counter, we obtain a number of arrivals $n$ in a symbol period, which can be decomposed in the form

$$ n=\bar{n}(A_0)+u $$

where $\bar{n}(A_0)=\mathrm{E}[n|A_0]$ is the useful signal and the fluctuation $u$ is the shot noise. Now, from the theory of semiclassical detection developed in the previous section, the number of signal photons is given by the photonic intensity $P_v (t)/ h \nu$ integrated over $(0,T)$, and therefore it results in

$$\begin{aligned} \bar{n}(A_0)= H\left( V_L^2 + V_R^2 + 2V_R V_L \cos \pi A_0 \right) =N_L+N_R + U_0\,\cos \pi A_0 \end{aligned}$$

(7.75)

where $N_L+N_R = H(V_L^2 + V_R^2)$ is a bias term, $U_0=2\sqrt{N_LN_R}$, and $U_0\;\cos \pi A_0$ is the symbol-dependent part. The variance, coinciding with the average, is

$$\begin{aligned} \sigma _n^2(A_0) = N_L+N_R + U_0\,\cos \pi A_0 \cong N_L, \end{aligned}$$

(7.76)

where the approximation follows from the hypothesis $V_L \gg V_0$. In conclusion, the decision on the transmitted symbol $A_0$ is made on the value

(7.77)

At this point we introduce the Gaussian approximation, where it is assumed that the photon number $n$ is a Gaussian random variable and hence specified by the mean $\bar{n}(A_0)$ and by the variance $\sigma ^2(A_0)= N_L$, which is independent of the symbol $A_0$. As seen in the previous section, the Gaussian assumption allows us to simplify the analysis and to arrive at a very simple result.

By choosing the decision rule as

$$\begin{aligned} \widehat{A}_0={\left\{ \begin{array}{ll}\;1&{}\quad n\le N_L+N_R\\ \;0&{}\quad n>N_L+N_R \end{array}\right. } \end{aligned}$$

(7.78)

we obtain the error probability

$$\begin{aligned} P_e=Q\left( \frac{U_0}{\sigma _n} \right) =Q \left( \sqrt{\varLambda } \right) , \end{aligned}$$

(7.79)

where the SNR is given by $\varLambda =U_0^2/\sigma ^2_n =4\,N_R$. Note that $N_R=N_R(0)=N_R(1)$ gives the number of signal photons per bit. In conclusion, the error probability in the classical BPSK with homodyne receiver is given by

(7.80)

This error probability is known as the standard quantum limit. The result is in agreement with Proposition 7.2.

Comparison with incoherent detection (OOK) shows that the performances of the homodyne detection are better, as illustrated in Fig. 7.20, where the error probability $P_e$ is plotted versus the average number of signal photons per bit $N_R$. On the other hand, the implementation of an efficient homodyne scheme implies some complications, in that it requires the presence of a local laser that must be accurately tuned in frequency and phase with the source laser.

Fig. 7.20

Comparison of error probability $P_e$ versus average number of signal photons per bit $N_R$ in classical binary optical systems

BPSK with Superhomodyne Reception

We suppose that at the reception we have available a laser (local oscillator) producing a radiation $v_L (t)$ with the same amplitude, frequency and phase of the carrier at the transmitter, that is,

$$\begin{aligned} v_L (t) = V_L \cos \left( 2 \pi \nu t \right) \quad \text {with}\quad V_L=V_0. \end{aligned}$$

(7.81)

This local carrier is added to the received modulated signal, yielding (Fig. 7.21)

$$\begin{aligned} v(t)&=V_0 \cos \left( 2 \pi \nu t + A_0 \pi \right) + V_0 \cos \left( 2 \pi \nu t \right) \\&={\left\{ \begin{array}{ll} 2 V_0 \cos \left( 2 \pi \nu t \right) &{}\;\;\; \qquad A_0=0 \\ \qquad \qquad \; 0 &{} \;\;\; \qquad A_0=1 . \end{array}\right. } \end{aligned}$$

(7.82)

Then the number of signal photons in a symbol period becomes

$$\begin{aligned} N_v(0) =\frac{4 P_R T}{h \nu } \,,\qquad N_v(1)=0. \end{aligned}$$

(7.83)

Using a photon counter, the decision is based on the number of arrivals $n$ by the rule

$$\begin{aligned} \widehat{A}_0={\left\{ \begin{array}{ll}\;1&{}\quad \mathrm{if} \; n=0\\ \;0&{}\quad \mathrm{if} \; n>0.\end{array}\right. } \end{aligned}$$

(7.84)

Then, similarly to relation (7.69), we get the error probability

$$\begin{aligned} P_{e,\text {classical}} =\frac{1}{2} P_e (0) =\frac{1}{2} \mathrm{e}^{- N_v (0)} \end{aligned}$$

(7.85)

The interesting thing is that the number of signal photons per bit at the reception, i.e., before adding the carrier, is $N_R = P_R\,T/(h \nu )$ and it is equal to one fourth of $N_v(0)$, so that the relation (7.85) becomes

Fig. 7.21

Signals in classical BPSK superhomodyne reception

(7.86)

which represents the super quantum limit [6]. So we have a great improvement over the homodyne detection, as shown in Fig. 7.20; because the power introduced by the local oscillator creates a more favorable situation for a correct decision. But the implementation of superhomodyne is very difficult in that it requires the presence of a local laser that must be accurately tuned with the source laser, not only in frequency and phase but also in amplitude .

Problem 7.9

$\star \!\star \!\star$    The error probability in classical homodyne BPSK has been evaluated assuming equiprobable symbols. When the symbols are not equiprobable the number of signal photons per bit $N_R$ is still independent of the symbols and gives the SNR as $\varLambda =4N_R$. The only change is in the evaluation is the decision element, given for equiprobable symbol by (7.78), as

$$ \widehat{A}_0={\left\{ \begin{array}{ll}\;1&{}\quad n\le S\\ \;0&{}\quad n>S \end{array}\right. } $$

where $S$ is the threshold to be optimized.

Find the optimal decision threshold and prove that the minimum error probability is given by

$$\begin{aligned} P_e=q_1 \;Q\left( \sqrt{\varLambda }+\frac{1}{2\sqrt{\varLambda }}\log \frac{q_1}{q_0}\right) + q_0\;Q\left( \sqrt{\varLambda }-\frac{1}{2\sqrt{\varLambda }}\log \frac{q_1}{q_0}\right) . \end{aligned}$$

(7.87)

7 Quantum Decision with Pure States

In a $K$-ary quantum communications system, in the absence of thermal noise, we use the quantum decision theory developed in the previous two chapters, limited to pure states. We recall the main ideas and the available methods.

The source (Alice) is characterized by a constellation of coherent states $\mathcal{S}=\{|\gamma _0\rangle , |\gamma _1\rangle ,...,|\gamma _{K-1}\rangle \}$, which can be collected in the state matrix

(7.88)

where $n$ is the dimension of the underlying Hilbert space ($n$ may be infinite, and it really is infinite in Glauber’s representation). The specification of the source requires also the definition of the prior probabilities $q_i=\mathrm{P}[A_0=i]=\mathrm{P}[C_0=\gamma _i]$, but usually throughout the chapter we shall assume equiprobable symbols $q_i=1/K$.

The goal is to find an optimal system of measurement operators $Q_i,i\in \mathcal{A}$ , that is, minimizing error probability. Kennedy’s theorem (Theorem 5.3) states that, with pure states, the optimal measurement operators must be elementary, i.e., in the form $Q_i=|\mu _i\rangle \langle \mu _i|$. We can then limit our search to the measurement vectors, specified by the measurement matrix

(7.89)

These vectors are given as a linear combination of the states, as established by the relation

(7.90)

where $A$ is a $K\times K$ complex matrix.

Kennedy’s theorem states also that the optimal measurement vectors $|\mu _i\rangle$ must be orthogonal, and therefore the corresponding measurement operators $Q_i=|\mu _i\rangle \langle \mu _i|,i\in \mathcal{A}$ must form a system of projectors. Unfortunately, Kennedy’s theorem, as well as Holevo’s theorem, do not provide explicit solutions. To compute optimal solutions, we could resort to the numeric programming methods outlined in Sect. 5.8, but, luckily enough, we can get help from the geometrically uniform symmetry (GUS), which is verified in the majority of quantum communications systems. In fact, square root measurement (SRM) decision, which is, in general, suboptimal, with pure states and in the presence of GUS becomes optimal (see Sect. 6.5.4). It is then appropriate to explicitly recall the SRM methodology, that gives good results even in the absence of GUS.

From the measurement vectors, we calculate the transition probabilities $p(j|i)=|\langle \mu _j|\gamma _i\rangle |^2$ and then the probability of correct decision.

7.1 Recall of SRM Approach

We summarize the main steps of the SRM theory developed in Sect. 6.3.

Starting from the constellation of $K$ coherent states $\mathcal{C}=\{|\gamma _0\rangle ,\ldots ,|\gamma _{K-1}\rangle \}$, we evaluate in sequence

• (1) Gram’s matrix of the inner products $G=[\langle \gamma _i|\gamma _j\rangle |]$, $i,j=0,1, \ldots ,K-1$, calculated according to (7.9). In the cases of interest, the matrix $G$, which is $K\times K$, has rank $K$.

• (2) The spectral decomposition (EID) of $G$

  $$\begin{aligned} G=V\varLambda _GV^*=\sum _{i=0}^{K-1}\sigma _i^2\, |v_i\rangle \langle v_i|\,. \end{aligned}$$

  (7.91)

  From this EID we find the eigenvalues $\sigma ^2_i$ and the orthonormal basis $\{|v_i\rangle \}$.

• (3) The square roots of $G$

  $$\begin{aligned} G^{\pm \frac{1}{2}}=V\varLambda _G^{\pm \frac{1}{2}}V^*. \end{aligned}$$

  (7.92)

• (4) The transition probabilities according to (see (6.29))

  $$\begin{aligned} p_c(i|j)=\left| \bigl (G^{\frac{1}{2}}\bigr )_{ij}\right| ^2 \end{aligned}$$

  (7.93)

  and the error probabilities (with equiprobable symbols)

  $$\begin{aligned} P_e=1-\frac{1}{K}\sum _{i=0}^{K-1} \left| \bigl (G^{\frac{1}{2}}\bigr )_{ii}\right| ^2. \end{aligned}$$

  (7.94)

• (5) The measurement vectors as linear combination of the states according to

  $$\begin{aligned} M=\varGamma \,G^{\frac{1}{2}}\quad \rightarrow \quad |\mu _i\rangle =\sum _{j=0}^{K-1} \bigl (G^{-\frac{1}{2}}\bigr )_{ij}|\gamma _j\rangle . \end{aligned}$$

  (7.95)

With geometrically uniform symmetry (GUS). If the states $|\gamma _i\rangle$ have the GUS, Gram’s matrix becomes circulant and its EID is given by

$$\begin{aligned} G=W_{[K]}\,\varLambda _G\,W_{[K]}^*=\sum _{i=0}^{K-1}\sigma _i^2|w_i\rangle \langle w_i|, \end{aligned}$$

(7.96)

where the vectors $|w_i\rangle$ are the columns of the DFT matrix $W_{[K]}$

$$\begin{aligned} |w_i\rangle =\frac{1}{\sqrt{K}} \left[ W_K^{-i},W_K^{-2i},\ldots ,W_K^{-i(K-1)}\right] ^{\text {T}} \,,\qquad i=0,1,\ldots ,K-1 \end{aligned}$$

(7.97)

and the eigenvalues are given by the DFT of the first row $[r_0,r_1,\ldots ,r_{K-1}]$ of the matrix $G$

$$\begin{aligned} \lambda _i=\sigma ^2_i=\sum _{k=0}^{K-1}r_kW_K^{-ki}\,,\qquad r_k=\langle \gamma _0|\gamma _k\rangle . \end{aligned}$$

(7.98)

The square roots of $G$ have elements

$$\begin{aligned} \bigl (G^{\pm \frac{1}{2}}\bigr )_{ij}=\frac{1}{K}\sum _{p=0}^{K-1}\lambda _p^{\pm \frac{1}{2}}\,W_K^{-p(i-j)} \end{aligned}$$

(7.99)

and in particular the diagonal elements are all equal. Therefore, the error probability is simply given by

$$\begin{aligned} P_e=1-\left| \bigl (G^{\frac{1}{2}}\bigr )_{00}\right| ^2\,. \end{aligned}$$

(7.100)

8 Quantum Binary Communications Systems

We develop the analysis of a quantum binary system, in which the information is carried by two coherent states. In this section, we assume that the constellation of the two states be generic; whereas, in the subsequent sections, two specific modulation formats will be developed (OOK and BPSK) .

In the binary case, the optimal decision can be obtained in explicit form by Helstrom’s theory and also by the geometric approach, seen in Sect. 5.4. With equiprobable symbols, we can also use the SRM method, which provides an optimal decision (see Sect. 6.5).

8.1 Binary Systems with Coherent States

To implement a Quantum Communications binary system, the transmitter (laser) is placed in one of two distinct coherent states $|\gamma _0\rangle \;,\;|\gamma _1\rangle \;\in \mathcal{G}$, which can be collected in the state matrix $\varGamma =[|\gamma _0\rangle ,|\gamma _1\rangle ]$ . The geometry is completely specified by the inner product $X:=\langle \gamma _0|\gamma _1\rangle$, which can be calculated explicitly from (7.9).

The optimal decision is based on two measurement operators $Q_0$ and $Q_1$, with $Q_0+Q_1=I$, which, by Kennedy’s theorem (see Sect. 5.11), must be in the form $Q_0=|\mu _0\rangle \langle \mu _0|$ e $Q_1=|\mu _1\rangle \langle \mu _1|$, and therefore are identified by two measurement vectors $|\mu _0\rangle$ and $|\mu _1\rangle$, which form the measurement matrix

$$ M_{\text { opt}}=[|\mu _0\rangle ,|\mu _1\rangle ]. $$

Still by Kennedy’s theorem, the two measurement vectors must be orthogonal, $\langle \mu _0|\mu _1\rangle =0$, so that the quantum measurement is always projective.

8.2 Recall of Helstrom’s Theory and of Geometric Approach

We briefly recall the theory of optimal binary decision developed in Sects. 5.3 and 5.4. The results of this theory are completely specified by the a priori probabilities $q_0$ and $q_1$ and by the (quadratic) superposition degree of the states $|X|^2=|\langle \gamma _0|\gamma _1 \rangle |^2$, which can be calculated in explicit form from (7.9), obtaining

$$\begin{aligned} |X|^2=\mathrm{e}^{-|\gamma _0-\gamma _1|^2} . \end{aligned}$$

(7.101)

The optimal measurement matrix is related to the state matrix as $M=\varGamma \,A$ where the matrix $A$ is given explicitly by (5.39).

The error probability, known as the Helstrom bound, is given by

(7.102)

Case of Equiprobable Symbols

When the symbols are equiprobable, which is the case of main interest, we have a few simplifications. The matrix $A$ becomes

$$\begin{aligned} A={1 \over 2}\left[ \begin{array}{llll}\frac{1}{\sqrt{1+|X|}}+\frac{1}{\sqrt{1-|X|}}&{} \frac{1}{\sqrt{1+|X|}}-\frac{1}{\sqrt{1-|X|}}\\ \frac{1}{\sqrt{1+|X|}}-\frac{1}{\sqrt{1-|X|}}&{} \frac{1}{\sqrt{1+|X|}}+\frac{1}{\sqrt{1-|X|}}\;\end{array}\right] . \end{aligned}$$

(7.103)

The error probability is simplified as

(7.104)

Problem 7.10

$\star \star$    Prove that with the optimization the a posteriori probabilities $q(i|i):=\mathrm{P}[A_0=i|\hat{A}_0=i]$ are equal and coincide with the correct decision probability $P_c$.

Problem 7.11

$\star$    Prove that in a binary system with equiprobable symbols, the error probability can be expressed as function of $N_R(0)$, $N_R(1)$, and of the relative phase of the complex parameters $\gamma _0$ and $\gamma _1$. that determine the coherent states.

Fig. 7.22

Constellation of symbols and states in OOK modulation

9 Quantum Systems with OOK Modulation

The constellation consists of the states (Fig. 7.22)

$$\begin{aligned} |\gamma _0\rangle =|0\rangle \;,\;|\gamma _1\rangle =|\varDelta \rangle \;\in \mathcal{G}\; \end{aligned}$$

(7.105)

where $|0\rangle$ is the ground state and the state $|\varDelta \rangle$ is determined by the number $\varDelta$ which is not restrictive to assume real and positive. The quadratic superposition of the two states is $|\langle 0|\varDelta \rangle |^2=\mathrm{e}^{-\varDelta ^2}$. The number of signal photons associated to the symbol $A_0=0$ is $N_R(0)=0$, while the one associated to the symbol $A_0=1$ is $N_R(1)=\varDelta ^2$. The number of signal photons per bit is then

$$ N_R={\textstyle \frac{1}{2}}\,N_R(0)+{\textstyle \frac{1}{2}}\,N_R(1)={\textstyle \frac{1}{2}}\,N_R(1) $$

and the quadratic superposition of the two states can be written in the meaningful form

$$ |X|^2=\mathrm{e}^{-2N_R}. $$

From (7.104), we obtain that the error probability of the OOK quantum system with equiprobable symbols becomes

(7.106)

The evaluation of the measurement vectors does not exhibit any specific simplification in (7.103), where now the inner product $X$ can be expressed in the form $X=\mathrm{e}^{-N_R}$.

9.1 Comparison with Classical OOK Optical Systems

The classical OOK system was developed in Sect. 7.6.1, where we found that the error probability, with equiprobable symbol, is given by

(7.107)

The comparison between the $P_{e,\text {classical}}$ of the classical receiver, given by (7.107), and the $P_e$ of the quantum receiver, given by (7.106), is shown in Fig. 7.23 as a function of the average number of photons per bit $N_R$. The asymptotic behavior of (7.106) becomes (by the approximation $1-\sqrt{1-x}\simeq \frac{1}{2}x$ for $x$ small)

$$ P_e={\textstyle \frac{1}{4}}\mathrm{e}^{-2N_R}\qquad N_R\gg 1 $$

Fig. 7.23

Comparison of quantum and classical OOK

that is, one half of the classical case. Thus we not have a great improvement in quantum OOK with respect to classical OOK (in error probability a relevant improvement is expressed in decades). The sensititvities in the two kinds of OOK are

$$ N_R=9.668 \text { photons/bit}\,,\qquad N_{R,\text {classical}}= 10.015 \text { photons/bit}. $$

Another comparison is between the channels realized by the two kinds of receiver: With the quantum receiver (optimized with equiprobable symbols) we obtain a symmetric channel, notwithstanding that the constellation is unbalanced, while, with the receiver based on photon count, the channel turns out to be very asymmetric (Fig. 7.24).

Fig. 7.24

Symmetric binary channel realized by the quantum optimal decision and asymmetric binary channel realized by photon count decision

Fig. 7.25

Constellation of symbols and states in 2-PSK modulation

10 Quantum Systems with BPSK Modulation

In the BPSK quantum system, the symbol $A_0=0$ (phase $\varphi =0$) is encoded into a coherent state $|\varDelta \rangle$ with a given amplitude $\varDelta$ and the symbol $A_0=1$ (phase $\varphi =\pi$) into the coherent state $|-\varDelta \rangle$ (Fig. 7.25)

$$\begin{aligned} |\gamma _0\rangle =|\varDelta \rangle \;,\;|\gamma _1\rangle =|-\varDelta \rangle \;\in \mathcal{G}. \end{aligned}$$

(7.108)

Obviously, the number of signal photons associated to the two states is equal

$$ N_R(0)=N_R(1)=|\varDelta |^2=N_R $$

and the (quadratic) superposition degree of the two states becomes

$$ |X|^2=\mathrm{e}^{-|\varDelta -(-\varDelta )|^2}=\mathrm{e}^{-4|\varDelta |^2}=\mathrm{e}^{-4N_R} $$

which yields the error probability

(7.109)

Compared to the quantum OOK modulation, we have an improvement, because the term at the exponent $4N_R$ in place of $2N_R$.

10.1 Comparison with Classical BPSK Optical System

The classical BPSK system was developed in Sect. 7.6.2, where we found that the error probability, with equiprobable symbols, is given by

(7.110)

where $Q(x)$ is the normalized complementary Gaussian distribution. The Fig. 7.26 shows the comparison between the $P_e$ of the classical homodyne receiver and the $P_e$ of the quantum receiver.

Fig. 7.26

Comparison between quantum and classical BPSK (with equal a priori probabilities)

In this case, the improvement is relevant. For instance for $N_R=5$ photons per bit, we have $P_e=0.515 \;10^{-9}$ and $P_{e,\text {classical}}=0.387\;10^{-5}$, and the improvement is of the order of four decades! The sensititvities in the two kinds of BPSK are

$$ N_R=4.837 \text { photons/bit}\,,\qquad N_{R,\text {classical}}= 8.913 \text { photons/bit}. $$

Generic a Priori Probabilities

Usually we consider equally probable symbols, but it may be interesting to see the comparison when the a priori probabilities $q_0$ and $q_1=1-q_0$ are different.

In both quantum and classical BPSK we have $N_R(0)=N_R(1)$ and then the number of signal photons per bit $N_R=q_0N_R(0)+q_1N_R(1)$ is independent of $q_0$. In the classical BPSK, the error probability is given by (7.87) of Problem 7.9, that is,

$$\begin{aligned} P_{e,\text {classical}}= q_1 \;Q\left( \sqrt{4N_R}+\frac{1}{2\sqrt{4N_R}}\log \frac{q_1}{q_0}\right) + q_0\;Q\left( \sqrt{4N_R}- \frac{1}{2\sqrt{4N_R}}\log \frac{q_1}{q_0}\right) . \end{aligned}$$

(7.111)

In the quantum BPSK (7.109) becomes

$$\begin{aligned} P_e=\frac{1}{2}\left[ 1-\sqrt{1-q_0q_1\mathrm{e}^{-4N_R}}\right] . \end{aligned}$$

(7.112)

Fig. 7.27

Comparison between quantum and classical BPSK with non equiprobable symbols. The error probability $P_e$ is reduced when $q_0$ becomes smaller

The comparison is shown in Fig. 7.27. Note in particular that for $N_R=0$ in both systems the error probability becomes $P_e=q_0$, so that it reduces with $q_0$ and also for $N_R>0$ it is reduced when $q_0$ becomes smaller. This may lead to think that the performance of a quantum BPSK improves when the a priori probabilities are unbalanced. This is not true because the performance of a system is given not only by the error probability, but also by the entropy and by the capacity (see Chap. 12). With $q_0={1 \over 2}$ the entropy $H$ of a binary source is $H=1$ bit per symbol, while with $q_0=0.01$ the entropy is reduced to $H=0.08$ bit per symbol.

11 Quantum Systems with QAM Modulation

Quadrature amplitude modulation (QAM) is one of the most interesting format in radio frequency (RF) transmission, and can also be proposed for coherent optical modulation (classical system) and for quantum modulation.

QAM is the first example of multilevel modulation we are considering, with typically $K=L^2$ levels, that is, $K=4$, $K=9$, $K=16$, and so on. For this format, the optimal quantum detection is not available, and suboptimal solutions must be adopted. We will apply the SRM technique which gives a good overestimate of the error probability [7], with a check by convex semidefinite programming (CSP).

11.1 Classical and Quantum QAM Formats

The alphabet of QAM modulation consists of a constellation of $K=L^2$ equally spaced points on a square grid of the complex plane, which can be defined starting from the $L$–ary balanced alphabet

$$ \mathcal{A}_L=\{-(L-1)+2(i-1)|\;i=1,2,\ldots ,L\}\qquad \text {with}\quad L=2,3,4,\ldots $$

In particular

$$\begin{aligned}&\mathcal{A}_3=\{-2,0,+2\}\\&\mathcal{A}_4=\{-3,-1,+1,+3\}\\&\mathcal{A}_5=\{-4,-2,0,+2,+4\}. \end{aligned}$$

The $K$-ary QAM constellation is then formed by the complex numbers

$$ \mathcal{C}=\{\varDelta (u+\mathrm{i}v)|\;u,v\in \mathcal{A}_L\} $$

where $\varDelta$ is the scale factor and $2\varDelta$ gives the spacing of symbols in the constellation, with $\varDelta$ real and positive. Figure 7.28 illustrates the constellation for $L=4$ (16-QAM). Notice that the 4-QAM, obtained with $L=2$, is equivalent to the 4-PSK, which will be developed in the next section.

To obtain the constellation of the coherent states in quantum QAM it suffices to assign to each symbol $\gamma$ of the constellation $\mathcal{C}$ the corresponding coherent state $|\gamma \rangle$. Then the constellation of the coherent states becomes

$$ \mathcal{S}=\left\{ |\gamma _{uv}\rangle = |\varDelta (u+\mathrm{i}v)\rangle \bigm |\; u,v\in \mathcal{A}_L\right\} . $$

Fig. 7.28

Constellation of 16-QAM with scale factor $\varDelta$

For example, for the 16-QAM, where $L=4$ and $\mathcal{A}_4=\{-3,-1,+1,+3\}$, we have the following coherent states listed in lexicographic order (see Sect. 2.13)

$$\begin{aligned} u&=-3\qquad v=-3\qquad |\gamma _0\rangle =|\gamma _{-3,-3}\rangle =|\varDelta (-3-3\mathrm{i})\rangle \\ u&=-3\qquad v=-1\qquad |\gamma _1\rangle =|\gamma _{-3,-1}\rangle =|\varDelta (-3-\mathrm{i})\rangle \\ u&=-3\qquad v=+1\qquad |\gamma _2\rangle =|\gamma _{-3,+1}\rangle =|\varDelta (-3+\mathrm{i})\rangle \\ u&=-3\qquad v=+3\qquad |\gamma _3\rangle =|\gamma _{-3,+3}\rangle =|\varDelta (-3+3\mathrm{i})\rangle \\ u&=-1\qquad v=-3\qquad |\gamma _4\rangle =|\gamma _{-1,-3}\rangle =|\varDelta (-1-3\mathrm{i})\rangle \\&\,\vdots \quad \qquad \quad \vdots \\u&=+3\qquad v=-3\qquad |\gamma _{12}\rangle =|\gamma _{+3,-3}\rangle =|\varDelta (3-3\mathrm{i})\rangle \\ u&=+3\qquad v=-1\qquad |\gamma _{13}\rangle =|\gamma _{+3,-1}\rangle =|\varDelta (3-\mathrm{i})\rangle \\ u&=+3\qquad v=+1\qquad |\gamma _{14}\rangle =|\gamma _{+3,+1}\rangle =|\varDelta (3+\mathrm{i})\rangle \\ u&=+3\qquad v=+3\qquad |\gamma _{15}\rangle =|\gamma _{+3,+3}\rangle =|\varDelta (3+3\mathrm{i})\rangle \\ \end{aligned}$$

11.2 Performance of Quantum QAM Systems

We consider the decision based on the SRM method, recalled in Sect. 7.7.1. We start from the construction of Gram’s matrix $G$, whose elements are the inner products

$$ \langle \gamma _{uv}|\gamma _{u'v'}\rangle = \langle \varDelta (u+\mathrm{i}v)|\varDelta (u'+\mathrm{i}v')\rangle . $$

Remembering (7.9), we get

$$\begin{aligned} \langle \gamma _{uv}|\gamma _{u'v'}\rangle =\mathrm{exp}\{-{\textstyle \frac{1}{2}}\varDelta ^2[(u'&-u)^2+ (v'-v)^2-2\mathrm{i}(u'v-v'u)]\}\nonumber \\&\;u,v,u',v'\in \mathcal{A}_L. \end{aligned}$$

(7.113)

The only problem in building the Gram matrix $G$ is the ordering of the four-index elements in a standard (bidimensional) matrix. To this end, we can use the lexicographic order indicated above.

The main point of the SRM technique is the spectral decomposition of $G$, according to (7.91), namely,

$$ G=V\varLambda _GV^*=\sum _{i=0}^{K-1}\sigma _i^2\,|v_i\rangle \langle v_i| $$

which identifies the eigenvalues $\sigma ^2_i$ and the orthonormal basis $|v_i\rangle$, $i=1,2,\ldots ,K$, and also the square roots $G^{\pm \frac{1}{2}}=V\varLambda _G^{\pm \frac{1}{2}}V^*$. We can then compute the transition probabilities from (7.93) and the error probability from (7.94), that is,

$$\begin{aligned} p(j|i)=|(G^{1 \over 2})_{ij}|^2\,,\qquad P_e=1-\frac{1}{K}\sum _{i=0}^{K-1}\left[ (G^{\frac{1}{2}})_{ii}\right] ^2. \end{aligned}$$

(7.114)

As usual, the performance is evaluated as a function of the number of signal photons per symbol $N_s$, given in general by (7.34). For the QAM we find

$$\begin{aligned} N_s&=\frac{1}{K}\sum _{i=0}^{K-1}|\gamma _i|^2=\frac{1}{K} \sum _{u\in \mathcal{A}_L}\sum _{v\in \mathcal{A}_L}|\gamma _{uv}|^2\\&=\frac{1}{K}\varDelta ^2\sum _{u\in \mathcal{A}_L}\sum _{v\in \mathcal{A}_L}(u^2+v^2)=\frac{2L}{K}\varDelta ^2\sum _{u\in \mathcal{A}_L}u^2\\&=\frac{2L}{K}\varDelta ^2\sum _{i=1}^L[-(L-1)+2(i-1)]^2. \end{aligned}$$

The result is⁵

$$\begin{aligned} N_s=\frac{2}{3}(L^2-1)\varDelta ^2=\frac{2}{3}(K-1)\varDelta ^2 \end{aligned}$$

(7.115)

so that the shape factor (7.39) of the QAM constellation is given by

$$\begin{aligned} \mu _K=\frac{2}{3}(K-1). \end{aligned}$$

(7.115a)

For example, for the 16-QAM we have $N_s=10\varDelta ^2$ and $\mu _K=10$.

Finally, from $N_s$, we get the number of signal photons per bit as

$$ N_R=N_s/\log _2K. $$

Remark

As noted above, the 4-QAM may be viewed as a 4-PSK, for which an exact (non-numerical) evaluation of the SRM is possible. This exact evaluation can be used as a test to check the numerical accuracy of higher order QAM.

11.3 An Alternative Evaluation Using the Generalized GUS

The QAM modulation has not the ordinary GUS, but it verifies the first form of generalized GUS introduced in Sect. 5.13, where there are $L$ reference states $|\gamma _0\rangle , \ldots ,|\gamma _L\rangle$, instead of a single state $|\gamma _{0}\rangle$, and the $K$-ary constellation is subdivided into $L$ subconstellations generated by a common symmetry operator $S$ in the form $|\gamma _{ik}\rangle = S^i|\gamma _{k}\rangle$, $k=1,\ldots ,L,\;\; i=0,1, \ldots ,K/L-1$.

Fig. 7.29

Constellation of 16-QAM and its decomposition into four 4-PSK constellations

In particular, the 16-QAM constellation can be decomposed into 4-PSK constellations, as shown in Fig. 7.29. The reference states are the states belonging to the first quadrant of the complex plane, namely

$$\begin{aligned} |\gamma _0\rangle =|\varDelta (1+\mathrm{i})\rangle \;,\;|\gamma _1\rangle =|\varDelta (3+3\mathrm{i})\rangle \;,\;|\gamma _2\rangle =|\varDelta (3+\mathrm{i})\rangle \;,\;|\gamma _3\rangle =|\varDelta (1+3\mathrm{i})\rangle . \end{aligned}$$

(7.116)

The symmetry operator is the rotation operator of 4-PSK, $S=R(\pi /2)$, which allows us to generate all the other states of the 16-QAM by rotating the reference states into the other three quadrants. In fact, if we consider the state vector of the reference states $\tilde{\gamma }_0=[|\gamma _0\rangle ,|\gamma _1\rangle , |\gamma _2\rangle ,|\gamma _3\rangle ]$ and apply the rotation operator in the form

$$ \tilde{\varGamma }=[\tilde{\gamma }_0, S\tilde{\gamma }_0, S^2\tilde{\gamma }_0, S^3\tilde{\gamma }_0] $$

we obtain a $16\times 16$ Gram matrix $\tilde{G}=\tilde{\varGamma }^*\tilde{\varGamma }$, which contains the same inner products of the Gram of the previous approach. But, for the different ordering, $\tilde{G}$ turns out to be block circulant.

At this point, although we are dealing with pure states, we can represent the 16 states through four density operators, $\rho _0$, $\rho _1$, $\rho _2$, and $\rho _3$, where $\rho _0$ collects the four reference states with a fictitious probability $1/4$, that is,

$$ \rho _0=\gamma _0\,\gamma _0^*={\textstyle \frac{1}{4}} (|\gamma _0\rangle \langle \gamma _0|+|\gamma _1\rangle \langle \gamma _1|+|\gamma _2\rangle \langle \gamma _2|+ |\gamma _3\rangle \langle \gamma _3|) $$

and the other density operators are obtained by the GUS relation

$$ \rho _i=S^i\,\rho _0\,S^{-i}\,,\qquad i-1,2,3. $$

Hence we can apply the theory of the SRM with GUS for mixed states (see Proposition 6.3), which requires the evaluation of the matrices (where now $K$ becomes $L$)

$$\begin{aligned} D_k= \sum _{i=0}^{L-1}\gamma _0^*\gamma _i W_L^{-ki} \end{aligned}$$

(7.117)

and of their square roots $D_k^{1/2}$, where in the present case $L=4$. Finally, one gets the the correct decision probability as

$$ P_c=\text {Tr}\,\left[ \frac{1}{L} \sum _{k=0}^{L-1}D_k^{1/2}\right] ^2\,. $$

This new approach gives exactly the same performance that we find with the previous SRM approach where the generalized GUS was not considered, with the advantage of a reduced computational complexity. In the specific case of 16-QAM, the evaluation is confined to the square roots of $4\times 4$ matrices, instead of the square root of a $16\times 16$ matrix.

The technique developed for the 16-QAM can be applied to constellations of any order. In particular, the constellation of 64-QAM can be decomposed into 16-PSK constellations, and the evaluation of the square roots is still confined to $4\times 4$ matrices , instead of the square root of a $64\times 64$ matrix.

11.4 Performance of the Classical Optical QAM System

The scheme of modulation and demodulation falls under the general classical scheme of Fig. 7.11. We have seen that the signal at the decision point is (see (7.55))

$$ z=C_0\,U_0+u_a+\mathrm{i}\,u_b $$

where $C_0=A_0+\mathrm{i}\,B_0$ is the transmitted symbol, $U_0$ is the amplitude, $u_a$ and $u_b$ are statistically independent Gaussian noises with null average and the same variance $\sigma _u^2$.

To calculate the error probability, we must choose the decision regions on the complex plane. With equiprobable symbols, the optimal decision regions are found in a straightforward way, as illustrated in Fig. 7.30 for the 16-QAM. In particular for the inner symbols of the constellation, the decision regions are squares with sides of length $2U_0$, centered in the corresponding symbols. Using the procedure outlined in Sect. 7.5, one obtains the following expression for the error probability [9]

$$\begin{aligned} P_{e,\text {classical}}=1-\left[ 1-2\left( 1-\frac{1}{L}\right) Q\left( \frac{U_0}{\sigma _u}\right) \right] ^2 \end{aligned}$$

(7.118)

where $Q(x)$ is the normalized complementary Gaussian distribution.

The result depends on the cardinality $K=L^2$ and on the SNR ratio $\varLambda =U_0^2/{\sigma _u^2}$, which can be expressed as a function of the average number of photons per symbol $N_s$ (see (7.63)), that is,

(7.119)

This result is in agreement with the conclusions of Proposition 7.2.

Fig. 7.30

Decision regions for the 16-QAM system constellation

11.5 Comparison of Quantum and Classical QAM Systems

We are now able to compare the two QAM systems: The classical optical version, in which the error probability is given by (7.118) and the quantum optical version, in which $P_e$ is evaluated numerically from (7.114) by the SRM procedure. In both cases, the parameters are the number of levels $K=L^2$ and the number of signal photons/symbol $N_s$.

The comparison, made in Fig. 7.31 for $K=16$ and $K=64$, shows the clear superiority of the quantum QAM system with respect to the classical one. For instance in 16-QAM with $N_s=50$ photons/symbols we find $P_{e,\text {classical}}=1.161\;10^{-5}$, while in the quantum system $P_e=1.546\;10^{-9}$; in 64-QAM with $N_s=200$ photons/symbol we find $P_{e,\text {classical}}=2.231\;10^{-5}$ and $P_e=4.674\;10^{-9}$. In both cases the improvement obtained with the quantum system is of about four decades.

In Fig. 7.32 the 16-QAM is compared to the 64-QAM as a function of the number of signal photons per bit $N_R$. The sensitivity at $P_e=10^{-9}$ is $N_R=12.783$ photons/bit in 16-QAM and $N_R=36.035$ photons/bit in 64-QAM.

Fig. 7.31

Comparison of quantum and classical 16–QAM and 64–QAM

Fig. 7.32

Error probability of quantum QAM versus the number of signal photons per bit $N_R$

11.6 Comparison of CSP and SRM Evaluation

The QAM format does not have the GUS, and therefore the SRM approach does not give the minimum error probability. For this reason, we have evaluated the minimum error probability also by convex semidefinite programming, implemented in MatLab by the CVX procedure, which gives (numerically) this minimum. The results of the two approaches are shown in the following table for the 16-QAM

$N_s$	$P_e$ with CSP	$P_e$ with SRM
0.1	0.86919	0.884949
0.5	0.764902	0.784679
1	0.675115	0.690197
1.5	0.597074	0.608958
2.5	0.461457	0.467108
4.5	0.239407	0.240096
6.5	0.0913599	0.0910951
9.5	0.0188974	0.0188975

The two evaluations are very close, especially for large values of $N_s$, and cannot be distinguished in a log plot (recall that in the evaluation of $P_e$, decades are relevant, not decimals),

The conclusion is that the SRM approach is recommended also for the QAM (for the other formats the SRM gives the minimum of $P_e$).

12 Quantum Systems with PSK Modulation

Also PSK (phase-shift keying) modulation is one of the best known and most often used formats at radio frequency and at optical frequencies. The BPSK  $=$  2-PSK format has been already seen in Sect. 7.10 as a special case of quantum binary systems.

Quantum $K$-ary PSK systems were analyzed by several authors and in particular by Kato et al. [7], using the SRM technique. In this case, the constellation of the states enjoys the geometrically uniform symmetry and then the SRM technique gives an optimal quantum receiver .

12.1 Classical and Quantum PSK Format

The constellation of the PSK modulation consists of $K$ points uniformly distributed along a circle of the complex plane

$$\begin{aligned} \mathcal{C}=\{\varDelta \, W_K^m\;|\;m=0,1,\ldots ,K-1\} \end{aligned}$$

(7.120)

where the scale factor $\varDelta$ is given by the radius of the circle and $W_K=\mathrm{e}^{\mathrm{i}2\pi /K}$. The constellation is illustrated in Fig. 7.33 for some values of $K$.

Fig. 7.33

Constellations of PSK modulation

In the quantum version, the states are obtained by simply associating to every complex symbol $\gamma$ of the constellation (7.120) the corresponding coherent state, which is given by

$$\begin{aligned} |\gamma _m\rangle =|\varDelta \;W_K^m\rangle =\mathrm{e}^{{-{\textstyle \frac{1}{2}} \varDelta ^2}}\sum _{n=0}^{\infty }\frac{(\varDelta \; W_K^m)^n}{\sqrt{n!}}|n\rangle \,,\qquad m=0,1,\ldots ,K-1 . \end{aligned}$$

(7.121)

In this constellation, all the coherent states have the same number of signal photons given by

$$\begin{aligned} N_s=\varDelta ^2. \end{aligned}$$

(7.122)

Constellation (7.121) enjoys the GUS, that is, it satisfies the conditions (5.120a) and (5.120b) for an appropriate unitary operator $S$. In this case $S$ is obtained from the rotation operator, which is defined as

$$ R(\phi ):=\mathrm{exp}(\mathrm{i}\phi N) $$

where $N$ is the number operator given by (7.1). Specifically we have

$$\begin{aligned} S=R\Bigl (\frac{2\pi }{K}\Bigr )= \mathrm{exp}\Bigl (\frac{\mathrm{i}2\pi }{K}N\Bigr )=W_K^{N}. \end{aligned}$$

(7.123)

The GUS property is verified for all constellations of Gaussian states generated by the rotation operator, as we will prove in Sect. 11.20. In Problem 7.9 we propose a specific proof for the PSK constellations where the key is that the operator $R(\phi )$ rotates a coherent state $|\alpha \rangle$ in the form (see 11.20)

$$\begin{aligned} R(\phi )\,|\alpha \rangle =|\mathrm{e}^{\mathrm{i}\phi }\alpha \rangle \,, \end{aligned}$$

(7.124)

and the result is again a coherent state. In other words, the class of coherent states is closed under rotations.

12.2 Performance of Quantum PSK Systems

For the decision we apply the SRM, which gives an optimal result. Then, for the performance evaluation, we follow the procedure described in Sect. 7.8.3, taking into account that the PSK constellation satisfies the GUS.

The generic element $p,q$ of Gram’s matrix $G = [G_{pq}]$ is the inner product $G_{pq} = \langle \gamma _p|\gamma _q\rangle$ obtained from (7.9) with $\alpha =\varDelta W_{K}^p$ and $\beta =\varDelta W_{K}^q$, namely,

$$\begin{aligned} G_{pq} =\mathrm{exp}[-\varDelta ^2(1-W_{K}^{q-p})] \,,\qquad p,q=0,1,\ldots ,K-1 . \end{aligned}$$

(7.125)

As predicted (by the GUS), the element $p,q$ depends only on the difference $q-p$; and therefore Gram’s matrix becomes circulant. The eigenvalues are obtained computing the DFT of the first row of Gram’s matrix, that is,

$$\begin{aligned} \lambda _i = \sum _{k=0}^{K-1}G_{0k}W_{K}^{-ki} \end{aligned}$$

(7.126)

and the corresponding eigenvectors are given by the columns of the DFT matrix

$$ |w_{i}\rangle = \frac{1}{\sqrt{K}} \left[ 1,W_{K}^{-i}, W_{K}^{-2i},\ldots ,W_{K}^{-(K-1)i}\right] ^{\text {T}} . $$

Thus the matrices $G^{\pm \frac{1}{2}}$ are obtained from (7.99), where the element $i,j$ is given by

$$ (G^{\pm \frac{1}{2}})_{ij} = \frac{1}{K} \sum _{p=0}^{K-1}\lambda _p^{\pm \frac{1}{2}}W_K^{(j-i)p}. $$

The measurement vectors are computed as linear combination of the states according to (7.95), i.e.,

$$ |\mu _i\rangle =\sum _{j=0}^{K-1}\bigl (G^{-\frac{1}{2}}\bigr )_{ij}|\gamma _j\rangle . $$

Finally, the error probability with equiprobable symbols is simply

$$\begin{aligned} P_e = 1-\frac{1}{K^2}\left( \sum _{i=0}^{K-1}\sqrt{\lambda _i}\right) ^2. \end{aligned}$$

(7.127)

Therefore, to calculate $P_e$ it suffices to evaluate the eigenvalues according to (7.126) and to apply (7.127). As usual, $P_e$ can be expressed as a function of the number of signal photons per symbol $N_s$, given by (7.122), and of the number of levels $K$. In fact, the Gram matrix $G$ depends only on $\varDelta ^2=N_s$ and $K$, and so is for the square root of $G$ and subsequent relations. This conclusion is in agreement with Proposition 7.2.

12.3 Performance of Classical PSK Systems and Comparison

The classical optical PSK system falls into the general model of quadrature modulation (with homodyne receiver) seen in Sect. 7.5. The signal at the decision point becomes

$$ z_0 =C_0\,U_0+u_a+\mathrm{i}u_b $$

where $C_0$ is the transmitted symbol, $C_0\in \mathcal{C}_0=\{W_K^i\;|\;i=1,\ldots ,K\}$, $u_a$ and $u_b$ are independent zero-mean Gaussian components with the same variance $\sigma _u^2$. At this point, we introduce the count parameters, recalling that

$$ U_0=(2V_RV_L)H\qquad \sigma _u^2=H\,V_L^2 $$

and that in this case the number of signal photons contained in the received power is

$$ N_R=N_s=H\,V_R^2. $$

So the SNR becomes

$$ \varLambda =\frac{U_0^2}{\sigma _u^2}=4N_s $$

the shape factor $\mu _K$ being unitary.

To find the error probability, we must partition the complex plane into $K$ decision regions. Even in this case, with equiprobable symbols, such regions are easily found, as illustrated in Fig. 7.34 for the 8-PSK, and then we apply the general relation (7.60). Given the nature of the constellation, it is convenient to do a coordinate change in the probability density, from Cartesian to polar coordinates. The exact computation is only known for $K=2$ and $K=4$ and it yields (see (7.80))

$$\begin{aligned}&K=2\qquad P_{e,\text {classical}}=Q(2\sqrt{N_s})\nonumber \\&K=4\qquad P_{e,\text {classical}}=1-\left[ 1-Q(\sqrt{2N_s})\right] ^2 \end{aligned}$$

(7.128)

Fig. 7.34

Decision regions in 8-PSK

where we recall that for $K=4$ the PSK coincides with the QAM. For $K>4$ the exact computation is not known, and we resort to the inequality [10, Chap.10]

$$\begin{aligned} P_{e,\text {classical}}<P_e'=\mathrm{exp}\left( -\frac{1}{2}\frac{U_0^2}{\sigma _u^2}\sin ^2\frac{\pi }{K}\right) =\mathrm{exp}\left( -2N_s\sin ^2\frac{\pi }{K}\right) . \end{aligned}$$

(7.129)

The comparison between the classical and the quantum system has been done in Fig. 7.26 for the 2-PSK. The comparison of 4-PSK and 8-PSK is done in Fig. 7.35, where the error probability is plotted as a function of the number of signal photons per symbol $N_s$. Even in this case we notice a striking superiority of the quantum system. For instance in 4-PSK with $N_s=10$ photon/symbol we find $P_e=1.030 \;10^{-9}$ and $P_{e,\text {classical}}=7.742\;10^{-6}$ ; in 8-PSK with $N_s=30$ photon/symbol we find $P_e=1.166 \;10^{-8}$ and $P_{e,\text {classical}}=7.742\;10^{-6}$. In both cases, the improvement of the quantum system is of several decades .

Fig. 7.35

Comparison of quantum and classical 4-PSK and 8-PSK

Fig. 7.36

Error probability of quantum PSK versus the number of signal photons per bit $N_R$

Figure 7.36 compares quantum 4-PSK and quantum 8-PSK as a function of the number of signal photons per bit $N_R$. The sensitivity at $P_e=10^{-9}$ is $N_R=5.001$ photons/bit in 4-PSK and $N_R=11.402$ photons/bit in 8-PSK.

Problem 7.12

$\star \star \star$    Prove that the operator $S$ defined by (7.123) is the symmetry operator of the $K$-PSK modulation.

Problem 7.13

$\star$    Find explicitly the formula for the error probability $P_e$ of quantum 4-PSK system, with the target to show that $P_e$ depends only on $\varDelta ^2=N_s$.

13 Quantum Systems with PPM Modulation

Pulse position modulation (PPM) is widely adopted in free space optical transmission, and is a candidate for deep-space transmission, also in quantum form [11, 12].

The analysis of a quantum PPM system has been done in a famous article [4] by Yuen, Kennedy and Lax, who found the optimal elementary projectors using an algebraic method developed “ad hoc” for this kind of modulation. Here we shall propose an original method based on the SRM and on the property of quantum PPM of verifying the GUS [13]. It seems odd that such property has not been remarked by other authors; because, on one hand, it is very intuitive, and, on the other hand, it makes it possible to directly achieve the same optimal result.

13.1 Classical PPM Format

In the classical version, the symbol period $T$ is subdivided into $K$ parts with spacing $T_0=T/K$, obtaining $K$ “positions.” Then, to the symbol $i\in \mathcal{A}=\{0,1,\ldots ,K-1\}$ the waveform is associated consisting of a rectangle in the $i$th position $iT_0$ of the symbol period

$$\begin{aligned} \gamma _i(t)=\left\{ \begin{array}{cc} \varDelta &{}\quad iT_0<t<(i+1)T_0 \\ 0 &{} \quad \text {elsewhere}\\ \end{array} \right. \qquad i=0,1,\ldots ,K-1 \end{aligned}$$

(7.130)

where $\varDelta >0$ is the scale factor (see Fig. 4.12). But, we will adopt the specular format

$$\begin{aligned} \gamma _i(t)=\left\{ \begin{array}{cc} \varDelta &{}\quad (K-1-i)T_0<t<(K-i)T_0 \\ 0 &{} \quad \text {elsewhere}\\ \end{array} \right. \qquad i=0,1,\ldots ,K-1 \end{aligned}$$

(7.131)

where the $i$th position becomes $(K-1-i)T_0$ instead of $iT_0$, as illustrated in Fig. 7.37 for $K=4$. The reason of this choice is due to the fact that it simplifies the formulation of the symmetry operator in the quantum version.

To waveforms (7.131), $K$ binary words can be associated of length $K$

$$ \gamma _i=[\gamma _{i,K-1},\ldots ,\gamma _{i,1},\gamma _{i,0}]\,,\qquad i=0,1,\ldots ,K-1 $$

where $\gamma _{ij}=\varDelta \,\delta _{ij}$. For example, for $K=4$ the words are

$$\begin{aligned}&\gamma _0=\left[ \begin{matrix} 0&\quad 0&\quad 0&\quad \varDelta \end{matrix}\right] \qquad \gamma _1=\left[ \begin{matrix} 0&\quad 0&\quad \varDelta&\quad 0\end{matrix}\right] \\&\gamma _2=\left[ \begin{matrix}0&\quad \varDelta&\quad 0&\quad 0\end{matrix}\right] \qquad \gamma _3= \left[ \begin{matrix}\varDelta&\quad 0&\quad 0&\quad 0\end{matrix}\right] . \end{aligned}$$

As outlined in Problem 7.5, it can be verified that PPM modulation is a special case of vector modulation seen in Sect. 7.3.2.

13.2 Quantum PPM Format

We have seen in Sect. 7.3.2 that the quantum formulation of a vector modulation must be done over a composite Hilbert space, given by the tensor product $\mathcal{H}=\mathcal{H}_0\otimes \mathcal{H}_0\otimes \cdots \otimes \mathcal{H}_0$ of $K$ equal Hilbert spaces $\mathcal{H}_0$, into each of which Glauber’s representation must have been introduced, and the states are given by the tensor product of $K$ coherent states and become $K$ –mode Gaussian states.

In the specific case of PPM, the states become

$$\begin{aligned} |\gamma _i\rangle =|\gamma _{i,K-1}\rangle \otimes \cdots \otimes |\gamma _{i,1} \rangle \otimes |\gamma _{i,0}\rangle \,,\qquad i=0,1,\ldots K-1 \end{aligned}$$

(7.132)

with

$$\begin{aligned} |\gamma _{ij}\rangle =\left\{ \begin{array}{cc} |\varDelta \rangle \quad &{} i=j\\ |0\rangle \quad &{} i\ne j\\ \end{array} \right. \end{aligned}$$

(7.132a)

where $|\varDelta \rangle$ is a coherent state with parameter $\varDelta$, and $|0\rangle$ is the “ground state”. For example, for $K=4$ we have the four states

$$\begin{aligned} |\gamma _0\rangle&=|0\rangle \otimes |0\rangle \otimes |0\rangle \otimes |\varDelta \rangle \qquad | \gamma _1\rangle =|0\rangle |\otimes 0\rangle \otimes |\varDelta \rangle \otimes |0\rangle \nonumber \\ |\gamma _2\rangle&=|0\rangle \otimes |\varDelta \rangle \otimes |0\rangle \otimes |0\rangle \qquad | \gamma _3\rangle =|\varDelta \rangle \otimes |0\rangle \otimes |0\rangle \otimes |0\rangle . \end{aligned}$$

(7.133)

Fig. 7.37

Realization of transmitted symbols and corresponding optical power in classic 4-PPM modulation

13.3 Geometrically Uniform Symmetry (GUS) of PPM

As we shall see in Sect. 11.20, the constellations of Gaussian states generated by the rotation operator $R(\phi )$ verify the GUS, also in the $K$–mode. This is the case of $K$-ary PPM. But the parameter $\phi$ becomes a $K\times K$ Hermitian matrix and the exponential defining $R(\phi )$ becomes difficult to handle. Here we prefer finding directly the symmetry operator $S$, and in Chap. 11 we will prove that $S$ can be expressed by the $K$–mode rotation operator.

The symmetry operator $S$ of the quantum PPM format (7.132) can be defined as follows: $S$ is an operator of the composite Hilbert space $\mathcal{H}$ that causes a shift to the left by one position (modulo $K)$ of the factors of the tensor product of the states, moving the first factor to second position, the second to third , and the $K$th factor to first. For example, for $K=4$, the action of $S$ is as follows

$$\begin{aligned} S|\gamma _{i3}\rangle \otimes |\gamma _{i2}\rangle \otimes |\gamma _{i1}\rangle \otimes |\gamma _{i0}\rangle = |\gamma _{i2}\rangle \otimes |\gamma _{i1}\rangle \otimes |\gamma _{i0}\rangle \otimes |\gamma _{i3}\rangle . \end{aligned}$$

(7.134)

Then, going on with $K=4$, with the states of (7.113) we can see that, starting from the state $|\gamma _0\rangle =|0\rangle \otimes |0\rangle \otimes |0\rangle \otimes |\varDelta \rangle$, the other states can be obtained in the following way:

$$\begin{aligned}&S\;|0\rangle \otimes |0\rangle \otimes |0\rangle \otimes |\varDelta \rangle =|0\rangle \otimes |0\rangle \otimes |\varDelta \rangle \otimes |0\rangle = |\gamma _1\rangle \\&S^2|0\rangle \otimes |0\rangle \otimes |0\rangle \otimes |\varDelta \rangle = |0\rangle \otimes |\varDelta \rangle \otimes |0\rangle \otimes |0\rangle =|\gamma _2\rangle \\&S^3|0\rangle \otimes |0\rangle \otimes |0\rangle \otimes |\varDelta \rangle = |\varDelta \rangle \otimes |0\rangle \otimes |0\rangle \otimes |0\rangle =|\gamma _3\rangle \end{aligned}$$

while the application of $S^4$ brings back to the initial state, and therefore $S^4=I_{\mathcal{H}}$.

The considerations we just made “in words” can be translated to “formulas,” but this is not so simple, because the symmetry operator $S$ is not separable, but it operates between the $K$ factors of the composite Hilbert space $\mathcal{H}=\mathcal{H}_0^{\otimes K}$. For the symmetry operator, the following result applies, recently demonstrated in [13].

Proposition 7.3

Let $n$ be the dimension of the component Hilbert spaces $\mathcal{H}_0$, and therefore $N=n^K$ is the dimension of $\mathcal{H}_0^{\otimes K}$. Then the symmetry operator of the $K$-ary PPM has the following expression

$$\begin{aligned} S=\sum _{k=0}^{n-1}w_n(k)\otimes I_L\otimes w^*_n(k) , \end{aligned}$$

(7.135)

where $\otimes$ is Kronecker’s product, $w_n(k)$ is a column vector of length $n$, with null elements except for one unitary element at position $k$, and $I_L$ is the identity matrix of order $L=n^{K-1}$.    $\square$

For example, for $n=2$ and $K=3$ (3-PPM) we have

$$\begin{aligned} w_2(0)=\left[ \begin{matrix}1\\ 0\end{matrix}\right] \,,\qquad w_2(1)=\left[ \begin{matrix} 0\\ 1\end{matrix}\right] \,,\qquad I_4=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}1\end{array}\right] \end{aligned}$$

and therefore from (7.135)

$$\begin{aligned} S=\,&w_2(0)\otimes I_4\otimes w^*_2(0)+w_2(1)\otimes I_4\otimes w^*_2(1)\\ =&\left[ \begin{matrix}1\\ 0\end{matrix}\right] \otimes \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}1\end{array}\right] \otimes \left[ \begin{matrix}1&0\end{matrix}\right] + \left[ \begin{matrix} 0\\ 1\end{matrix}\right] \otimes \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1&{}0&{}0&{}0\\ 0&{}1&{}0&{}0\\ 0&{}0&{}1&{}0\\ 0&{}0&{}0&{}1\end{array}\right] \otimes \left[ \begin{matrix}0&1\end{matrix}\right] \end{aligned}$$

and, developing the products, we obtain the $16\times 16$ matrix (remember that the tensor product for matrices becomes Kronecker’s product, see Sect. 2.13)

$$\begin{aligned} {S}=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} 1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0\\ 0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}0&{}1\end{array}\right] . \end{aligned}$$

(7.136)

We leave it to the reader to check that, using the properties of Kronecker’s product of Sect. 2.13, from (7.135) we obtain that the matrix $S$ has dimensions $n^K\times n^K$, it is unitary, and has the property $S^K=I_{n^K}$.

For later use it is important to evaluate explicitly the EID of the symmetry operator $S$ in the form (5.128)

$$\begin{aligned} S=\sum _{i=0}^{K-1}W_K^i\;Y_i\,Y_i^* \end{aligned}$$

(7.137)

where the columns of the matrices $Y_k$ are formed by the eigenvectors corresponding to the eigenvalues $\lambda _i=W_K^i$. The explicit evaluation of such eigenvectors is long and cumbersome, and is developed in [13], using the fact that for the PPM the operator $S$ is a permutation matrix. The important thing is that this evaluation can be done “analytically” for every $n$ and $K$, without resorting to numeric evaluation, which could be prohibitive for high values of $N=n^K$. For example, for $K=4$, $n=2$, the eigenvalues $1,W_4,W_4^2,W_4^3$ have multiplicities respectively 6, 3, 4, and 3, and the corresponding eigenvectors form the matrices

$$\begin{aligned} \left. \begin{array}{cccc} Y_{0}&{} Y_{1} &{}Y_{2} &{} Y_{3}\\ \left[ \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \end{array} \right] &{} \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ -\frac{\mathrm{i}}{2} &{} 0 &{} 0 \\ -\frac{1}{2} &{} 0 &{} 0 \\ 0 &{} -\frac{\mathrm{i}}{2} &{} 0 \\ \frac{\mathrm{i}}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} -\frac{1}{2} &{} 0 \\ 0 &{} 0 &{} -\frac{\mathrm{i}}{2} \\ \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} \frac{\mathrm{i}}{2} &{} 0 \\ 0 &{} 0 &{} \frac{\mathrm{i}}{2} \\ 0 &{} 0 &{} -\frac{1}{2} \\ 0 &{} 0 &{} 0 \end{array} \right] &{} \left[ \begin{array}{cccc} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -\frac{1}{2} &{} 0 &{} 0 \\ 0 &{} \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} -\frac{1}{2} &{} 0 \\ 0 &{} -\frac{1}{2} &{} 0 &{} 0 \\ -\frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} -\frac{1}{2} \\ 0 &{} \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{2} &{} 0 \\ \frac{1}{\sqrt{2}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 0 &{} -\frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 &{} -\frac{1}{2} \\ 0 &{} 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} 0 &{} 0 &{} 0 \end{array} \right] &{} \left[ \begin{array}{ccc} 0 &{} 0 &{} 0 \\ \frac{\mathrm{i}}{2} &{} 0 &{} 0 \\ -\frac{1}{2} &{} 0 &{} 0 \\ 0 &{} \frac{\mathrm{i}}{2} &{} 0 \\ -\frac{\mathrm{i}}{2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} -\frac{1}{2} &{} 0 \\ 0 &{} 0 &{} \frac{\mathrm{i}}{2} \\ \frac{1}{2} &{} 0 &{} 0 \\ 0 &{} \frac{1}{2} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \frac{1}{2} \\ 0 &{} -\frac{\mathrm{i}}{2} &{} 0 \\ 0 &{} 0 &{} -\frac{\mathrm{i}}{2} \\ 0 &{} 0 &{} -\frac{1}{2} \\ 0 &{} 0 &{} 0 \end{array} \right] \,. \end{array}\right. \end{aligned}$$

13.4 Performance of Quantum PPM Systems

Considering that the PPM states have the GUS, the SRM gives an optimal detection, then, in performance evaluation, the same results will have to be found as those of Yuen et al. [4] with a different methodology.

Applying the method summarized in Sect. 7.7.1, about the SRM detection in the presence of GUS, the performance evaluation is articulated as follows. Gram’s matrix $G$ has as element $i,j$ the inner product $G_{ij} = \langle \gamma _i|\gamma _j\rangle$, where now the states $|\gamma _i\rangle$ are composite. We recall that the inner product of two states, each generated by the tensor product of $K$ component states, is given by the product of the $K$ inner products of the component states, that is,

$$\begin{aligned} \langle \gamma _i|\gamma _j\rangle =\langle \gamma _{i0}|\gamma _{j0}\rangle \;\langle \gamma _{i1}|\gamma _{j1} \rangle \;\ldots \;\langle \gamma _{iK-1}|\gamma _{jK-1}\rangle . \end{aligned}$$

(7.138)

Then, from (7.9), we get

$$\begin{aligned} \langle \gamma _i|\gamma _j\rangle ={\left\{ \begin{array}{ll}1 &{} \quad i=j\\ \; \mathrm{e}^{-|\varDelta |^2}&{}\quad i\ne j.\end{array}\right. } \end{aligned}$$

(7.139)

For example, in the case $K=4$ the inner product $\langle \gamma _0|\gamma _2 \rangle$ results in

$$\begin{aligned} \langle \gamma _0|\gamma _2\rangle&=\langle \varDelta |0\rangle \;\langle 0| 0\rangle \;\langle 0|\varDelta \rangle \;\langle 0|0\rangle \\&=\mathrm{e}^{-\frac{1}{2}|\varDelta |^2}\;\;1\;\;\;\mathrm{e}^{-\frac{1}{2}|\varDelta |^2}\;\;1=\mathrm{e}^{-|\varDelta |^2}. \end{aligned}$$

We observe that the same energy $E$ is associated to all symbols, and, according to (7.132), to each composite state the same signal photons are associated, given by

$$\begin{aligned} N_s=|\varDelta |^2= \text {number of signal photons/symbol}. \end{aligned}$$

(7.140)

Therefore Gram’s matrix becomes

$$\begin{aligned} G=\left[ \begin{matrix}1 &{} |X|^2 &{} \ldots &{} |X|^2\\ |X|^2 &{} 1 &{} \ldots &{} |X|^2\\ \vdots &{} &{} \ddots &{} \\ |X|^2 &{} |X|^2 &{} \ldots &{} 1 \end{matrix}\right] \end{aligned}$$

where $|X|^2$ is the quadratic superposition degree of the component states ($X=\langle {\gamma _i}{\gamma _j}\rangle$, $i\ne j$), given by

$$ |X|^2=|\langle {\varDelta }{0}|^2=\mathrm{e}^{-|\varDelta |^2}=\mathrm{e}^{-N_s}\rangle . $$

Notice that $G$ is a circulant matrix, as a consequence of the GUS .

Considering the GUS, the eigenvalues of $G$ are given by the DFT of the first row $[1,|X|^2,\ldots ,|X|^2]$, and therefore

$$ \lambda _i=\sum _{k=0}^{K-1}G_{0k}W_K^{-ki} =1+|X|^2\sum _{k=1}^{K-1}W_K^{-ki}. $$

Recalling the orthogonality condition

$$\begin{aligned} \sum _{k=0}^{K-1}W_K^{-ki}={\left\{ \begin{array}{ll} K &{} \qquad i=0\\ 0 &{} \qquad i\ne 0\end{array}\right. } \end{aligned}$$

(7.141)

we have

$$\begin{aligned} \lambda _i={\left\{ \begin{array}{ll}1+(K-1)|X|^2 &{} \quad i=0\\ 1-|X|^2 &{}\quad i=1,\ldots ,K-1.\end{array}\right. } \end{aligned}$$

The square roots of $G$ become

$$ G^{\pm \frac{1}{2}}=\sum _{i=0}^{K-1}\lambda _i^{\pm \frac{1}{2}}|w_i\rangle \langle w_i|. $$

The transition probabilities are computed from (7.93) and result in

$$ p_c(i|j)=\frac{1}{K^2}\Bigl |\sum _{p=0}^{K-1}\lambda _p^{1/2}W_K^{-p(i-j)}\Bigr |^2\ =\frac{1}{K^2}\Bigl |\lambda _0^{1/2}+\lambda _1^{1/2}\sum _{p=1}^{K-1}W_K^{-p(i-j)}\Bigr |^2 $$

and, from (7.141),

$$\begin{aligned} p_c(j|i)={\left\{ \begin{array}{ll} K^{-2}\left( \lambda _0^{1/2}-\lambda _1^{1/2}\right) ^2 &{} i\ne j\\ K^{-2}\left( \lambda _0^{1/2}+(K-1)\lambda _1^{1/2}\right) ^2 &{} i=j\end{array}\right. } \end{aligned}$$

(7.142)

where

$$ \lambda _0=1+(K-1)|X|^2\,,\qquad \lambda _1=1-|X|^2\,,\qquad \text {with}\quad |X|^2=\mathrm{e}^{-N_s}. $$

The error probability is computed from (7.94) and becomes

(7.143)

in perfect agreement with the results of [4].

Finally, the measurement vectors are obtained from (7.95), namely,

$$ |\mu _i\rangle =\sum _{j=0}^{K-1}a_{ij}|\gamma _j\rangle \quad \,,\qquad a_{ij}={\left\{ \begin{array}{ll} K^{-2}\left( \lambda _0^{-1/2}-\lambda _1^{-1/2}\right) ^2 &{} i\ne j\\ K^{-2}\left( \lambda _0^{-1/2}+(K-1)\lambda _1^{-1/2}\right) ^2 &{} i=j.\end{array}\right. } $$

13.5 Performance of Classical PPM Systems

We use the notations

• $A_0\;$ transmitted word,

• $B_0\;$ received word,

• $\widehat{A}_0\;$ decided word.

Let us consider, for simplicity, the case $K=4$ with the standard (non specular) format and with a unitary scale factor ($\varDelta =1$), in which the possible transmitted words $A_0$ are

$$\begin{aligned} \gamma _0&=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l}1&0&0&0\end{array}\right] \,,\qquad \gamma _1=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&1&0&0\end{array}\right] \nonumber \\ \gamma _2&=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&0&1&0\end{array}\right] \,,\qquad \gamma _3=\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&0&0&1\end{array}\right] . \end{aligned}$$

(7.144)

With a photon counter, the symbol 0 is always received correctly, whereas the symbol 1 may be received as 0, with an error probability $\mathrm{e}^{-N_s}$. Then we have five possible received words: the four correct words (7.144) and the wrong word $\left[ \begin{matrix}0&0&0&0\end{matrix}\right]$, and we have to decide to which correct word the wrong word should be associated. The optimum criterion (with equiprobable symbols) is to associate the wrong word $\left[ \begin{matrix} 0&0&0&0\end{matrix}\right]$ to whatever correct word, for example to $\gamma _0$ (Fig. 7.38).

Fig. 7.38

Channel and decision criterion of a classical 4-PPM. $A_0$ is the transmitted word, $B_0$ the received word, and $\widehat{A}_0$ the decided word

Then the decision criterion becomes

$$\begin{aligned} \widehat{A}_0&=\gamma _0\qquad \text {if}\quad B_0 =\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1&0&0&0\end{array}\right] \qquad \text {or}\quad B_0 =\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&0&0&0\end{array}\right] \\ \widehat{A}_0&=\gamma _1\qquad \text {if}\quad B_0 =\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&1&0&0\end{array}\right] \\ \widehat{A}_0&=\gamma _2\qquad \text {if}\quad B_0 =\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&0&1&0\end{array}\right] \\ \widehat{A}_0&=\gamma _3\qquad \text {if}\quad B_0 =\left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&0&0&1\end{array}\right] \end{aligned}$$

and we can get an error only in the last three cases, each with probability $\mathrm{e}^{-N_s}$. Thus,

$$\begin{aligned} P_{e,\text {classical}}&=\frac{1}{4}[P_e(\gamma _1)+P_e(\gamma _2) +P_e(\gamma _3)+P_e(\gamma _4)]\\&=\frac{1}{4}[0+\mathrm{e}^{-N_s}+\mathrm{e}^{-N_s}+\mathrm{e}^{-N_s}]=\frac{3}{4}\mathrm{e}^{-N_s}. \end{aligned}$$

The general result is

(7.145)

13.6 Comparison in the Binary Case

For binary quantum PPM, from (7.143) we get

$$ P_c=\frac{1}{4} \left[ \sqrt{1-|X|^2} + \sqrt{1+|X|^2}\right] ^2= \frac{1}{2} \left[ 1+ \sqrt{1-|X|^4}\right] $$

where $|X|^4=\mathrm{e}^{-2N_s}=\mathrm{e}^{-2N_R}$, with $N_R=N_s$ the number of signal photons per bit. The error probability is therefore

$$\begin{aligned} P_e=\frac{1}{2} \left[ 1-\sqrt{1-\mathrm{e}^{-2N_R}}\right] , \end{aligned}$$

(7.146)

the same result found for the OOK format (see (7.106)).

Instead, in the classical case, from (7.145) we have $P_{e,\text {classical}}=\frac{1}{2} \mathrm{e}^{-N_R}$. The comparison of these results is shown in Fig. 7.39.

Fig. 7.39

Comparison of quantum and classical 2-PPM

Fig. 7.40

Comparison between quantum and classical 8-PPM and 64-PPM in terms of number of signal photons per symbol $N_s$

13.7 Comparison in the $K$-ary Case

In the quantum case, the error probability is given by (7.143), which can be rewritten in the form

$$ P_e=\frac{K-1}{K^2} \left[ K-(K-2)(1-|X|^2)+2\sqrt{(1-|X|^2)(1+(K-1)|X|^2)}\right] $$

where the superposition degree $|X|^2$ can be expressed as a function of the number of signal photons per symbol $N_s$, or of the number of signal photons per bit $N_R$

$$ |X|^2=\mathrm{e}^{-N_s}=\mathrm{e}^{-N_R\log _{2}K}. $$

In the classical case the error probability is given by (7.145).

The comparison between the two systems is illustrated for the 8-PPM and 64-PPM in Fig. 7.40 as a function of the number of signal photons per symbol $N_s$. Even in this case we notice a striking superiority of the quantum system. For instance in 8-PPM with $N_s=10$ photon/symbol we find $P_e=3.660 \;10^{-9}$ and $P_{e,\text {classical}}= 3.972\;10^{-5}$; in 64–PPM with $N_s=10$ photon/symbol we find $P_e=3.421\;10^{-9}$ and $P_{e,\text {classical}}=4.469\;10^{-5}$. In both cases the improvement of the quantum system is of several decades.

In Fig. 7.41 the error probability of the quantum PPM is plotted as a function of the number of signal photons per bit $N_R$ for four values of $K$. We realize that quantum PPM receivers have an extraordinary sensitivity, specifically

Fig. 7.41

Error probability of quantum PPM as a function of the number of signal photons per bit $N_R$

2-PPM	$\qquad N_R= 9.66849 \;\;$ photons/bit
4-PPM	$\qquad N_R= 5.10889 \;\;$ photons/bit
8-PPM	$\qquad N_R= 3.54713 \;\;$ photons/bit
16-PPM	$\qquad N_R= 2.75561 \;\;$ photons/bit
32-PPM	$\qquad N_R= 2.27708 \;\;$ photons/bit
64-PPM	$\qquad N_R= 1.95665 \;\;$ photons/bit
128-PPM	$\qquad N_R= 1.72722 \;\;$ photons/bit
256-PPM	$\qquad N_R= 1.55486 \;\;$ photons/bit
512-PPM	$\qquad N_R= 1.42072 \;\;$ photons/bit
1024-PPM	$\qquad N_R= 1.31332\;\;$ photons/bit

14 Overview of Squeezed States

Up to now we have considered Quantum Communications based on coherent states. In these last two sections, we consider the promising possibility to use squeezed states.

We have seen that a coherent state $|\alpha \rangle$ is completely determined by a complex parameter $\alpha$. A squeezed state, and more precisely, a squeezed-displaced state, say $|z,\alpha \rangle$ may be seen as a generalization of a coherent state, because of the dependence on two complex parameters, the displacement $\alpha$ and the squeeze factor $z=r\mathrm{e}^{\mathrm{i}\theta }$. In particular, setting $z$ to zero, the squeezed-displaced state gives back the coherent state

$$\begin{aligned} |0,\alpha \rangle =|\alpha \rangle \in \mathcal{G}. \end{aligned}$$

(7.147)

This simple property allows us to say that, using squeezed states in quantum communications with appropriate parameters, the performance cannot be worse than with coherent states. As we shall see, the squeeze factor $z$ allows us to control the photon statistic in such a way that, by choosing $z$ in an appropriate range, we get a considerable improvement of the system performance. This opportunity has been long recognized [14].

The theory of squeezed-displaced states will be formulated in Sect. 11.15 in the context of continuous variables, where it is shown that they represent the most general form of Gaussian states. In this section, we give the essential properties of squeezed states that are needed for Quantum Communications.

We shall use the following notations

• $|z,\alpha \rangle$: squeezed-displaced state

• $|0,\alpha \rangle =|\alpha \rangle$: coherent state

• $|z,0\rangle$: squeezed state or squeezed vacuum state.

14.1 Definition and Properties of Squeezed-Displaced States

Squeezed states live in the same Hilbert space as coherent states, that is an infinite dimensional Hilbert space where the Fock basis has been introduced. Squeezed-displaced states are the result of two distinct operations applied to the vacuum state: A squeezing and a displacement. They may be specified by the Fock expansion, whose Fourier coefficients result in

$$\begin{aligned} |z,\alpha \rangle _{n}=\frac{\sqrt{n!}}{\mu }\;\left( \frac{\beta }{\mu }\right) ^n \mathcal{H}_n\left( \frac{\mu \nu }{\beta ^2}\right) \mathrm{exp}\left( -{1 \over 2}|\beta |^2-\beta ^2\frac{\nu ^*}{2\mu }\right) . \end{aligned}$$

(7.148)

where $\mathcal{H}_n(x)$ are the polynomials (of degree $\lfloor n/2\rfloor$)

$$\begin{aligned} \mathcal{H}_n(x):=\sum _{j=0}^{\lfloor n/2\rfloor }\frac{1}{(n-2j)!j!} \;x^j. \end{aligned}$$

(7.149)

and

$$\begin{aligned} \mu =\cosh r \,,\qquad \nu =\sinh r \,\mathrm{exp}(\mathrm{i}\,\theta )\,,\qquad \beta =\mu \alpha -\nu \alpha ^*. \end{aligned}$$

(7.149a)

The deduction of (7.148) from the operations of squeezing and displaciment is made in Sect. 11.15.⁶

The class of squeezed-displaced states has two special subclasses, which are obtained for $z=0$ (absence of squeezing) and for $\alpha =0$ (absence of displacement). In the first case we have coherent states with coefficients (see (7.2))

$$\begin{aligned} |0,\alpha \rangle _{n}=\mathrm{e}^{-{1 \over 2}|\alpha |^2}\; \frac{\alpha ^n}{\sqrt{n!}} \end{aligned}$$

(7.150)

In the second case we have squeezed vacuum states with coefficients

$$\begin{aligned} |z,0\rangle _{n}=\sqrt{\text {sech}r}\;\sum _{n=0}^\infty \frac{\sqrt{(2n)!}}{2^n n!}\lambda ^n|2n\rangle \qquad \lambda =\tanh r \,\mathrm{e}^{\mathrm{i}\theta } \end{aligned}$$

(7.151)

that is, the state $|z,0\rangle$ is given by a linear combination of even photon number states, which means that the probability that the state contains an odd number of photons is zero.

Fig. 7.42

The probability distribution of photon number in squeezed-displaced states, $\; p_n(i)=\mathrm{P}[n=i\bigm ||(\alpha ,r)\rangle ]$, for $\alpha =3$ and different values of $r$

Fig. 7.43

Mean photon number $\bar{n }$ and variance $\sigma _n^2$ in squeezed-dispaced states versus the squeeze factor $r$ for three values of $|\alpha |^2$

14.2 Statistics of Squeezed-Displaced States

The probability distribution of the number of photons in a squeezed state is obtained by squaring the Fourier coefficients (7.148), that is,

$$\begin{aligned} p_n(i):=\mathrm{P}[n=i]=\left| |z,\alpha \rangle _i\right| ^2\; \end{aligned}$$

(7.152)

This distribution is illustrated in Fig. 7.42 for $\alpha =3$ and four values of $r$ with $\theta =0$. Note that for $r=0$, absence of squeezing, $p_n(i)$ becomes a Poisson distribution, but in general it may be far form the Poisson shape, and sometimes this is classified as sub-Poissonian statistic. Under certain conditions, this statistics may be controlled acting on the squeeze factor [14].

The mean photon number in a squeezed-displaced state is given by [4]

$$\begin{aligned} \bar{n}_{|z,\alpha \rangle }=|\alpha |^2+\sinh ^2\,r \end{aligned}$$

(7.153)

and the variance of the photon number is given by [15]

$$\begin{aligned} \sigma ^2_{ n_{|z,\alpha \rangle }}=|\alpha |^2 \left[ \mathrm{e}^{-2r}\cos ^2\theta +\mathrm{e}^{2r}\sin ^2\theta \right] +\frac{1}{2}\sinh ^2\,2r. \end{aligned}$$

(7.154)

Clearly these parameters, illustrated in Fig. 7.43, confirm the non-Poissonian statistic, because the mean and the variance are different.

14.3 Degree of Superposition of Squeezed-Displaced States

The most important parameter for Quantum Communications is given by the inner product, which must be studied in detail.

Proposition 7.4

The inner product of two squeezed-displaced states was evaluated by Yuen [4, Eq. 3.25] and reads

(7.155)

where

$$\begin{aligned}&z_i=r_i\mathrm{e}^{\mathrm{i}\theta _i}\,,\qquad i=0,1\nonumber \\&\mu _i=\cosh (r_i)\,,\qquad \nu _i=\sinh (r_i)\mathrm{e}^{\mathrm{i}\theta _i}\\&\beta _i=\mu _i\alpha _i-\nu _i\alpha _i^*\nonumber \\&A=\mu _0\mu _1^*-\nu _0\nu _1^*\,,\qquad B=\nu _0\mu _1-\mu _0\nu _1\nonumber \end{aligned}$$

(7.155a)

To study this complicate expression, we begin with remarking the dependence on the displacements $\alpha _i$ and on the squeeze factors $z_i$: The parameters $\beta _0$ and $\beta _1$ depend on both, while all the other parameters depend only on the squeeze factors. We can write (7.155) as

$$ \langle z_1,\alpha _1|z_0,\alpha _0\rangle =A^{-{1 \over 2}}\; \mathrm{exp}\left[ -\sum _{i=0}^1 \sum _{j=0}^1\left( a_{ij}\;\alpha _i\alpha _j +b_{ij}\;\alpha _i\alpha _j^* +d_{ij}\;\alpha _i^*\alpha _j^*\right) \right] $$

having at the exponent a bi-quadratic structure in $\alpha _0$, $\alpha _1$, $\alpha _0^*$, and $\alpha _1^*$, whose coefficients $a_{ij}$, $b_{ij}$, $d_{ij}$ depend only on the squeeze factors.

For $\alpha _0=\alpha _1=0$ (absence of displacement), (7.155) gives

$$\begin{aligned} \langle z_1,0|z_0,0\rangle =\,&A^{-{1 \over 2}}=(\mu _0\mu _1^*-\nu _0\nu _1^*)^{-{1 \over 2}}\nonumber \\ =\,&(\cosh r_0 \cosh r_1 - \sinh r_0 \,\sinh r_1 \mathrm{e}^{\mathrm{i}(\theta _0-\theta _1)})^{-{1 \over 2}} \end{aligned}$$

which is in agreement with the expression obtained in [16] for the inner product of two squeezed vacuum states

$$\begin{aligned} \langle z_1,0|z_0,0\rangle =\sqrt{\text {sech}\, r\;\text {sech}\, r_0}\bigm / \sqrt{1-\mathrm{e}^{\mathrm{i}(\theta _0-\theta )}\tanh r\,\tanh r_0} . \end{aligned}$$

(7.156)

For $z_0=z_1=0$ (absence of squeezing) we get $A=1$, $B=0$, $\beta _i=\alpha _i$, and then

$$ \langle 0,\alpha _1|0,\alpha _0\rangle = \mathrm{exp}\left[ -\frac{1}{2}\left( |\alpha _1|^2+|\alpha _0|^2- 2\alpha _1\alpha _0^*\right) \right] $$

which is the formula we got for the inner product of two coherent states (see (7.9)).

14.4 Squeezed-Displaced States as Gaussian States

As said above, squeezed-displaced states are the most general Gaussian states. The state $|z,\alpha \rangle$ depends on two complex parameters

$$\begin{aligned} z=r\,\mathrm{e}^{\mathrm{i}\theta }\,,\qquad \alpha =\varDelta \,\mathrm{e}^{\mathrm{i}\,\varepsilon }. \end{aligned}$$

(7.157)

We examine in detail the Wigner function $W(x,y)$ of the state (7.157), which is completely determined by the mean value and by the covariance matrix (see Sect. 7.2.5). Now, in $|z,\alpha \rangle$ the squeezing part does not give any contribution to the mean value, so we have

$$\begin{aligned} \begin{bmatrix} \overline{q}\\\overline{p} \end{bmatrix}= \begin{bmatrix}\mathfrak {R}\alpha \\\mathfrak {I}\alpha \end{bmatrix}= \begin{bmatrix} \varDelta \cos \varepsilon \\\varDelta \sin \varepsilon \end{bmatrix}. \end{aligned}$$

(7.158)

On the other hand the covariance matrix of the displacement component is the identity, so that the covariance matrix depends only on the squeeze factor as

$$\begin{aligned} V=&\left[ \begin{matrix} V_{11}&{}V_{12}\\ V_{12}&{}V_{22}\end{matrix}\right] \nonumber \\ =&\left[ \begin{array}{cc} \cosh ^2 r+ \sinh ^2 r+ \cos \theta \sinh 2 r &{} \sin \theta \sinh 2 r \\ \sin \theta \sinh 2 r &{} \cosh ^2 r+\sinh ^2 r- \cos \theta \sinh 2 r \end{array}\right] . \end{aligned}$$

(7.159)

Considering that $\det V=1$, the Wigner function results in

$$\begin{aligned} W(x,y)=\frac{1}{2\pi }\mathrm{exp}\left\{ -\frac{1}{2}\left[ V_{22} \big (x-\overline{q}\big )^2+ V_{11}\big (y-\overline{p}\big )^2- 2 V_{12} \big (x-\overline{q}\big ) \big (y-\overline{p}\big )\right] \right\} . \end{aligned}$$

(7.160)

A convenient representation of $W(x,y)$ in the $x,y$ plane is given by a contour level, which represents the curve given by the relation $W(x,y)=L$, with $L>0$ real. In general, these curves are tilted ellipses as shown in Fig. 7.44. The ellipses have the common center given by the displacement $\alpha$, and the main axis is tilted by the angle ${1 \over 2}\theta$. The lengths of the main axis and of the minor axis are proportional to $\mathrm{e}^{2r}$ and to $\mathrm{e}^{-2r}$, respectively, and so they are independent of the squeeze phase $\theta$.

Fig. 7.44

Contour level of the Wigner function W(x,y) (in red) of a squeezed-displaced state $|z,\alpha \rangle$ with $z=r\;\mathrm{e}^{\mathrm{i}\theta }$ and $\alpha =\varDelta \;\mathrm{e}^{\mathrm{i}\varepsilon }$. The mean vector $(\bar{q},\bar{p})= (\varDelta \cos \varepsilon ,\varDelta \sin \varepsilon )$ gives the displacement amount and determines the center of the elliptic countour. The main axis of the ellipse is tilted with respect to the $x$ axis of the angle ${1 \over 2}\theta _0$

14.5 Constellations of Squeezed-Displaced States with GUS

In Sect. 11.20, we will prove that the application of the rotation operator $R(\phi )$ to a squeezed-displaced state $|z,\alpha \rangle$, with $z=r\,\mathrm{e}^{\mathrm{i}\theta }$ and $\alpha =\varDelta \mathrm{e}^{\mathrm{i}\varepsilon }$, gives back the new squeezed-displaced state

$$ R(\phi )|z,\alpha \rangle =|z\mathrm{e}^{\,\mathrm{i}\, 2\phi },\alpha \mathrm{e}^{\,\mathrm{i}\, \phi }\rangle =|r\mathrm{e}^{\,\mathrm{i}\, (2\phi +\theta )},\varDelta \mathrm{e}^{\,\mathrm{i}\, (\phi +\varepsilon )}\rangle $$

that is, with the modification of squeeze factor $z\rightarrow z\mathrm{e}^{\,\mathrm{i}\, 2\phi }$ and of the displacement $\alpha \rightarrow \alpha \mathrm{e}^{\,\mathrm{i}\, \phi }$. In other words, the class of squeezed-displaced states is closed under rotations.

The above properties allow us to construct $K$-ary PSK constellations having the GUS, using as symmetry operator $S=R(2\pi /K)$. If $|z_0,\alpha _0\rangle$ is a reference squeezed-displaced state, the constellation has the form

$$\begin{aligned} \mathcal{S}=\{ S^k\,|z_0,\alpha _0\rangle = |z_0\mathrm{e}^{\,\mathrm{i}\, 2 k\,2\pi /K}\;,\;\alpha _0\mathrm{e}^{\,\mathrm{i}\, k \,2 \pi /K}\rangle \;,\;k=0,1,\ldots ,K-1\}. \end{aligned}$$

(7.161)

Figure 7.45 shows two 8-PSK constellations having the GUS with coherent states and squeezed-displaced states. The circles and the ellipses around the states represent the contour levels of the Wigner function of each state; the eccentricity of the ellipse depends only the squeeze factor $r$.

As seen for coherent states the GUS will allow us to find an optimal detection (minimum error probability) with the SRM approach.

Fig. 7.45

8-PSK constellations of coherent states (left) and of squeezed-displaced states (right) with the GUS. In both constellations the reference state has a real and positive displacement ($\alpha =\varDelta$, $\varepsilon =0$). The squeeze phase of the reference state is $\theta _0$ and the corresponding ellipse is tilted of ${1 \over 2}\theta _0$

15 Quantum Communications with Squeezed States

In this section, we evaluate the performance (error probability) in quantum communications systems where the information carrier is given by squeezed-displaced states instead of coherent states. We consider only PSK communications systems⁷; and therefore it is natural to choose constellations having the GUS. Then, for a given modulation order $K$, we have to choose a reference state of the constellation, $|z_0,\alpha _0\rangle$, because the other states are generated through the rotation operator, as indicated in (7.161).

In the choice of the reference state $|z_0,\alpha _0\rangle$ , where $z_0=r_0\,\mathrm{e}^{\mathrm{i}\theta _0}$, without restriction we can assume $\alpha _0$ real and positive. This parameter, together with $r_0$, determines the average number of photons contained in the state, which is given by

(7.162)

This number is very important because, in a PSK constellation with equally likely symbols, it also gives the average number of signal photons per symbol $N_s$. Now, choosing $r_0$ as a parameter that quantify the squeezing amount, it remains to choose the squeeze phase $\theta _0$ and this will be done by taking $\theta _0$ that minimizes the error probability.

Considering that the performance of PSK essentially depends on the quadratic superposition between the states of the constellation

$$ |X|^2=|\langle r_0\mathrm{e}^{\mathrm{i}\theta _0},\alpha _0|r_1\mathrm{e}^{\mathrm{i}\theta _1},\alpha _1\rangle |^2 $$

it is important to learn how $|X|^2$ depends on squeeze and displacement parameters. This is considered in Fig. 7.46 for the BPSK, where $r_1\mathrm{e}^{\mathrm{i}\theta _1}=r_0\mathrm{e}^{\mathrm{i}(\theta _0+2\pi )}$ and $\alpha _1=\alpha _0\mathrm{e}^{\mathrm{i}\pi }$. On the left of the figure $|X|^2$ is plotted versus $\alpha _0$ for coherent states (red curve) and also for squeezed-displaced states for a fixed value of $r_0$ and two values of $\theta _0$. It is remarkable the great improvement obtained with squeezed states, especially at the increase of the displacement amount $\alpha _0$. The right of the figure shows the strong dependence of $|X|^2$ on the squeeze phase $\theta _0$ and hence the importance of an appropriate choice of this parameter.

Fig. 7.46

Square of the inner product $X= \langle \alpha _0,r_0\mathrm{e}^{\mathrm{i}\theta _0}|\alpha _1,r_1\mathrm{e}^{\mathrm{i}\theta _1}\rangle$, represented by $-\log _{10} |X|^2$, for a BPSK constellation. On the left, the plot is versus the amplitude $\alpha _0$ of the displacement for coherent states (red) and for squeezed-displaced states with $r_0=0.9$ and two values of $\theta _0$. On the right, the plot is versus $\theta _0$ with $r_0=0.9$ and two values of $\alpha _0$

15.1 BPSK with Squeezed States

The BPSK constellation of squeezed-displaced states is obtained from (7.161) with $K=2$, namely

$$\begin{aligned} \mathcal{S}=\{ |r_0\mathrm{e}^{\mathrm{i}\theta _0},\alpha _0\rangle , |r_0\mathrm{e}^{\mathrm{i}\theta _0}\mathrm{e}^{\mathrm{i}2\pi },\alpha _0\mathrm{e}^{\mathrm{i}\pi }\} \rangle . \end{aligned}$$

(7.163)

For the evaluation of the error probability we can apply Helstrom’s theory (see (7.102)), which gives, with equally likely symbols,

$$\begin{aligned} P_e=\frac{1}{2}\left( 1-\sqrt{1-|X|^2}\right) . \end{aligned}$$

(7.164)

Thus, the only parameter needed is the quadratic superposition $|X|^2$ between the two states of the constellation. We have seen that $|X|^2$ is a function of $r_0$, $\alpha _0$, and $\theta _0$. Now, fixing $r_0$ and $\alpha _0$, we have the number of signal photons per symbol $N_s$ as

$$\begin{aligned} N_s= |\alpha _0|^2+\sinh ^2 r_0. \end{aligned}$$

(7.165)

and we choose the squeeze phase $\theta _0$ that achieves the minimum error probability.

In Fig. 7.47 the error probability is plotted versus $\theta _0$ for three values of $N_s$. Clearly, the minimum of $P_e$ is obtained for $\theta _0=\pi$, which means that in optimal BPSK constellation the ellipses appear to be vertically tilted, as shown at the right of the figure.

Fig. 7.47

BPSK system with squeezed-displaced states. On the left, the error probability $P_e$ versus the squeeze phase $\theta _0$ for three values of the number of signal photons per symbol $N_s$. All the curves have a minimum for $\theta _0=\pi$. On the right, the optimal BPSK constellation where the ellipse of the reference state is tilted of $\theta _0=\pi /2$ because the optimal squeeze phase is $\theta _0=\pi$

Fig. 7.48

Error probability $P_e$ versus the number of signal photons per symbol $N_s$ in 4-PSK. The dark curve refers to coherent states, while the colored curves refer to squeezed-displaced states with different values of the squeeze factor $r_0$ and optimal squeeze phase $\theta _0=-\pi /2$. The curve do not start at $N_s=0$, because $N_s=|\alpha _0|^2+\sinh ^2 r_0$ and for $N_s<\sinh ^2 r_0$ there is no room for the displacement $\alpha _0$

Finally, in Fig. 7.48 we compare the error probability $P_e$ versus $N_s$ obtained with coherent states and squeezed-displaced states. It is remarkable that the performance of the BPSK is highly improved with the presence of squeezing.

15.2 4-PSK with Squeezed States

The 4-PSK constellation of squeezed-displaced states is obtained from (7.161) with $K=4$, namely $\mathcal{S}=\{|z_i,\alpha _i\rangle \}$, where

$$\begin{aligned} z_i=z_0\mathrm{e}^{\,\mathrm{i}\, 2 k\,2\pi /4}\,,\qquad \alpha _i=\alpha _0\mathrm{e}^{\,\mathrm{i}\, k \,2\pi /2} \,,\qquad i=0,1,2,3 . \end{aligned}$$

(7.166)

Considering that the constellation has the GUS, for the evaluation of the error probability we apply the SRM approach, which turns out to be optimum. From Sect. 7.7 we recall that the evaluation of $P_e$ using the SRM is obtained as follows:

• (1) Evaluation of the inner products

  $$ G_{pq}=\langle z_p,\alpha _p|z_q,\alpha _q\rangle \,,\qquad p,q=0,1,2,3. $$

• (2) Evaluation of the eigenvalues $\lambda _i=\sum _{k=0}^3G_{0k}\,W_4^{-ki}$, and finally

  $$\begin{aligned} P_e=1-\left( \frac{1}{4}\sum _{i=0}^3 \sqrt{\lambda _i}\right) ^2 . \end{aligned}$$

  (7.167)

Fig. 7.49

4-PSK system with squeezed-displaced states. On the left, the error probability $P_e$ versus the squeeze phase $\theta _0$ for three values of the number of signal photons per symbol $N_s$ All the curves have a minimum for $\theta _0=-\pi /2$. On the right the optimal 4-PSK constellation; the ellipse of the reference state is tilted by ${1 \over 2}\theta _0=-\pi /4$ because the optimal squeeze phase is $\theta _0=-\pi /2$

Also in this case $P_e$ is a function of $r_0$, $\alpha _0$, and $\theta _0$ and the number of signal photons per symbol $N_s$ is still given by (7.165). Again, we choose $r_0$ and $\alpha _0$ as parameters and we search for the squeeze phase $\theta _0$ that gives the minimum error probability.

In Fig. 7.49 the error probability is plotted versus $\theta _0$ for three values of $N_s$. Clearly, the minimum of $P_e$ is obtained for $\theta _0=-\pi /2$, which means that in the optimal 4-PSK constellation the reference ellipse is tilted of ${1 \over 2}\theta _0=-\pi /4$, as shown on the right of the figure (see Fig. 7.45).

Finally, in Fig. 7.50 we compare the error probability $P_e$ versus. $N_s$ obtained with coherent states and with squeezed-displaced states. We realize that also the 4-PSK is highly improved with the presence of squeezing.

15.3 Conclusions

In this chapter, we have considered coherent states as the standard carrier for data transmission in quantum communications systems. On the other hand, in this last section, we have seen the possibility of a huge improvement using squeezed light, but we have limited the analysis only to the systems 2PSK and 4PSK for the reason that squeeze technique is not promising for the immediate future because of losses and excess noise present in this technique and because of the complexity and power required. However, quantum optics is making rapid progress so that quantum communications with squeezed states merits a special attention.

Fig. 7.50

Error probability $P_e$ versus the number of signal photons per symbol $N_s$ in 4-PSK. The dark curve refers to coherent states, while the colored curves refer to squeezed-displaced states with different values of the squeeze factor $r_0$ and optimal squeeze phase $\theta _0=-\pi /2$. The curve do not start at $N_s=0$, because $N_s=|\alpha _0|^2+\sinh ^2 r_0$ and for $N_s<\sinh ^2 r_0$ there is no room for the displacement $\alpha _0$