Revisiting Quantum Discord for Two-Qubit X States: The Error Bound to an Analytical Formula

Abstract

In this article, we investigate the error bound of quantum discord, obtained from the analytic formula of Ali et al. (Phys. Rev. A 81, 042105, 2010) for the case of general X states and from the analytic formula of Fanchini et al. (Phys. Rev. A 81, 052107, 2010) for the case of symmetric X states. For this purpose we consider 3-element POVM. We find that the results of Ali et al. and Fanchini et al. may have the worst-case errors of 0.004565 and 0.0009, respectively.

1 Introduction

A key ingredient in understanding quantum information is the quantum correlation. A well-known example of quantum correlation is entanglement. However,entanglement cannot be obtained by a local operation and classical communication(LOCC) [1, 2]. An entanglement is known to be very fragile to a local noisy channel. Furthermore, it has been shown that a quantum state without entanglement contains non-locality [4]. Henderson and Vedral [5] suggested a method to obtain the classical correlation between parties. Ollivier and Zurek [6] defined a quantum correlation called the quantum discord,, which is the total correlation minus the classical correlation in a bipartite quantum state. Quantum discord can contain a value even in a separable state [6]. The quantum discord is invariant under local unitary operation [5, 6]. In addition, quantum discord seems to have a relationship with noisy teleportation, entanglement distillation and quantum state merging [7]. Experimental efforts to investigate quantum discord are in progress [8, 9].

Quantum discord depends on the measurement setting. In [6] a projective measurement was used in obtaining the quantum discord. A measurement for the optimal quantum discord should produce the maximal classical correlation. It was shown in [10] that two element optimal POVM should be projective measurements. Here POVM stands for positive-operator valued measure. Quantum discord for a Bell-diagonal state was analytically obtained [11]. A great deal of research has focused on finding the quantum discord of the X state [14–17, 24–26].

The quantum discord of the two-qubit X state is significant for two reasons. The first reason is that X state does not change its form in almost every noisy environment [12]. The second one is that the general two qubit state can become a X state under a noisy channel [13]. Ali et al. [14] sought to develop an analytic formula of quantum discord for the X state; however, their results only hold for special X state [15]. Fanchini et al. [16] attempted to find an analytic formula for quantum discord for symmetric X state. Recently, by using von Neumann measurement, Huang showed that the result of [14] may be valid albeit with some worst-case error [17].

Researchers have investigated why the quantum discord of the X state can be important in physical processes. Recently, using an isomorphism between the X state and the Gibbs states with an XYZ Hamiltonian, Yurischev [18] showed that an optimal measurement angle can experience sudden change.. Also by generalization of qubit-qubit X state, Vinjanampathy and Rau [19] considered quantum discord of qubit-qudit states. Another interesting phenomemnon is a sudden change of quantum correlations. Pinto et al. [20] demonstrated that sudden change of quantum discord arises in Bell diagonal state but not in arbitrary close to Bell diagonal states. Karpat and Gedik [21, 22] analyzed quantum correlation including quantum discord of qubit-qutrit state interacting dephasing environment and showed that there exists a quantum correlation even in the region of entanglement sudden death. Li et al. showed that quantum discord in two qubit X state is more robust to decoherence than entanglement in two qubit X state [23].

In this paper we show that three element POVM can provide a better quantum discord. Shi et al. [24, 25] showed that there are some quantum states where 3 element POVM should be used for optimal quantum discord. The optimality for 3 element POVM can be found by a triangle formed by the direction vectors. By using 3 element POVM, we numerically obtained the quantum discord to the quantum state considered in [17, 26] and compared it with the result of [17].

2 Quantum Discord

The total correlation between the classical subsystem A and B can be defined by I(A : B) = H(p ^A) + H(p ^B) − H(p ^AB). Here H(p ^X)(X can be A or B) is the Shannon entropy of subsystem X. If the probability distribution of the subsystem becomes $p^{X}=\{{p_{1}^{X}},{p_{2}^{X}},\cdots ,{p_{n}^{X}}\}$, the Shannon entropy is found to be $H(p^{X})=-{\sum }_{i=1}^{n} {p_{i}^{X}} \log _{2} {p_{i}^{X}}$. H(p ^A,p ^B) is the joint entropy of the total system composed of subsystem A and B. When the probability distribution of the total system is known as $\{p_{ij}^{AB}\}(i=1,2,\cdots ,n,j=1,2,\cdots ,m)$, joint entropy is found to be ${H(p^{AB})}=-{\sum }_{i,j=1}^{n,m}p_{ij}^{AB} \log _{2} p_{ij}^{AB}$.

Let us consider the quantum case. In quantum information one may consider the information of the quantum system as the quantum state corresponding to the system. Let ρ ^AB denote the quantum state to total system. Then the quantum states of subsystem A and B can be found by ρ ^A=Tr _B ρ ^AB and ρ ^B=Tr _A ρ ^AB. The Von Neumann entropy of X and the total subsystem are given by S(ρ ^X)=−Tr{ρ ^X log2ρ ^X}(X becomes A or B) and S(ρ ^AB)=−Tr{ρ ^AB log2ρ ^AB} respectively. Therefore the total correlation of the quantum case is expressed by

$$ I(A:B)=S(\rho^{A})+S(\rho^{B})-S(\rho^{AB}). $$

(1)

Total correlation given by (1) contains the classical and quantum correlation. Therefore in order to extract the quantum correlation, one has to subtract the classical correlation from the total correlation. When one considers the positive operator valued measurement(POVM) $\{{M_{k}^{B}}\}$ on subsystem B, the classical correlation can be defined by (2) [5]

$$ J(A|\{{M_{k}^{B}}\})=S(\rho^{A})-\min\limits_{\{{M_{k}^{B}}\}}\sum\limits_{k} p_{k} S ({\rho_{k}^{A}}). $$

(2)

Here ${\rho _{k}^{A}}$ is the state of subsystem A, given as ${\rho _{k}^{A}} =\text {Tr}_{B} \{(1 \otimes {M_{k}^{B}} )\rho ^{AB}\}. S(A|\{{M_{k}^{B}}\})={\sum }_{k} p_{k} S({\rho _{k}^{A}})$ is the conditional entropy after measurement of subsystem B. Therefore the quantum correlation between subsystem A and B becomes [6]

$$\begin{array}{@{}rcl@{}} \delta_{\{{M_{k}^{B}}\}}(A:B)&=&I(A:B)-J(A|\{{M_{k}^{B}}\})\\ &=&S(\rho^{B})-S(\rho^{AB})+\min\limits_{\{{M_{k}^{B}}\}} \sum\limits_{k} p_{k} S({\rho_{k}^{A}}). \end{array} $$

(3)

This implies that optimizing (3) is identical to find a measurement to minimize the conditional entropy $S(A|\{{M_{k}^{B}}\})$. $S(A|\{{M_{k}^{B}}\})$ is invariant under the local unitary transformation [5].

The X state which appears in various physical cases [27, 28] is known to persist under local noisy channel [12]. The X state contains the Bell diagonal state and the Werner state [29]. The general form of subsystem ρ ^AB in two qubit states becomes

$$ \rho^{AB}=\frac{1}{4} \left\{\mathcal{I}\otimes\mathcal{I}+\sum\limits_{i} (A_{i} \mathcal{I}\otimes\sigma_{i} +B_{i} \sigma_{i}\otimes\mathcal{I})+\sum\limits_{i,j}t_{ij}\sigma_{i}\otimes\sigma_{j}\right\}. $$

(4)

Here σ _i (i = 1, 2, 3) are the Pauli’ spin matrices and $\vec {A}=(A_{1},A_{2},A_{3}), \vec {B}=(B_{1},B_{2},B_{3})$ and t _{i j} can be found by

$$\begin{array}{@{}rcl@{}} A_{i}=\text{Tr}\{(\mathcal{I}\otimes\sigma_{i})\rho^{AB}\} ,\\B_{i}=\text{Tr}\{(\sigma_{i}\otimes\mathcal{I})\rho^{AB}\} ,\\t_{ij}=\text{Tr}\{\sigma_{i}\otimes\sigma_{j}\}\rho^{AB}\}. \end{array} $$

(5)

X state $\rho _{X}^{AB}$ is described as (6).

$$ \rho_{X}^{AB}=\left(\begin{array}{cccc}a&0&0&\epsilon\\ 0&b&\delta&0\\0&\delta&c&0\\ \epsilon&0&0&d \end{array} \right). $$

(6)

Here, every element of $\rho _{X}^{AB}$ can be considered as a real number, since there exists a local unitary operation which can make every element of $\rho _{X}^{AB}$ real [3]. a,b,c and d satisfies a+b+c+d = 1. Equation (6) can be expressed as

$$ \rho_{X}^{AB}=\frac{1}{4}\left(\mathcal{I}\otimes\mathcal{I}+A\mathcal{I}\otimes\sigma_{3}+B\sigma_{3}\otimes\mathcal{I}+\sum\limits_{i} t_{i}\sigma_{i}\otimes\sigma_{i}\right). $$

(7)

Here A,B,t ₁, t ₂ and t ₃ in (7) become

$$\begin{array}{@{}rcl@{}} A&=&a-b+c-d,\\ B&=&a+b-c-d,\\ t_{1}&=&2(\delta+\epsilon),\\ t_{2}&=&2(\delta-\epsilon),\\ t_{3}&=&a-b-c+d. \end{array} $$

(8)

3 Optimization Strategy

Whether there is an optimal POVM for conditional entropy compared to projective measurement remains in question. In [10] it was shown that projective measurement is the optimal condition for the 2 element POVM. Therefore, POVM with more than two elements should be considered. In [25], it was found that there is an X state where 3-element POVM can be optimal. As is well known, it is very difficult to handle the optimal 3- element POVM analytically. The general form of 3-element POVM is expressed as [15]

$$ {M_{k}^{B}} = \mu_{k} (\mathcal{I}+\vec{n}^{(k)}\cdot\vec{\sigma}), k=1,2,3, \mu_{k}>0. $$

(9)

Here, $\vec {n}^{(k)}$ is the direction vector to ${M_{k}^{B}}$. Since, $|\vec {n}^{(k)}|=1$, the positivity of ${M_{k}^{B}}$ holds. When subsystem B is measured by $\{{M_{k}^{B}}\}$, the post-measurement state ${\rho _{k}^{A}}$ of subsystem A becomes ${\rho _{k}^{A}}=\left [1+\{t_{1} m_{x}^{(k)}\sigma _{1}+t_{2} m_{y}^{(k)}\sigma _{2} +(t_{3} m_{z}^{(k)}+B)\sigma _{3}\}/(1+Am_{z}^{(k)})\right ]/2$, where $m_{i}^{(k)}(i=x,y,z)$ is a component of the kth element in the direction vector $\vec {m}^{(k)}$. The eigenvalues of ${\rho _{k}^{A}}$ are $\{1\pm E(m_{x}^{(k)},m_{y}^{(k)},m_{z}^{(k)})\}/2$. Here $E(m_{x}^{(k)},m_{y}^{(k)},m_{z}^{(k)})$ is defined by

$$ E\left(m_{x}^{(k)},m_{y}^{(k)},m_{z}^{(k)}\right)=\frac{\sqrt{(t_{1}m_{x}^{(k)})^{2}+(t_{2}m_{y}^{(k)})^{2} +(t_{3}m_{z}^{(k)}+B)^{2}}}{1+Am_{z}^{(k)}}. $$

(10)

The probability to obtain outcome k is determined to be $p_{k}=\mu _{k}(1+m_{z}^{(k)}A)$. Therefore when 3-element POVM is used for measurement, the conditional entropy $S(A|\{{M_{k}^{B}}\})$ can be found as

$$ S(A|\{{M_{k}^{B}}\})=\sum\limits_{k=1}^{3}\mu_{k} (1+Am_{z}^{(k)})h\left(E\left(m_{x}^{(k)},m_{y}^{(k)},m_{z}^{(k)}\right)\right). $$

(11)

Here, h(x) is a function defined as $h(x)=-\frac {1+x}{2}\log _{2}\frac {1+x}{2}-\frac {1-x}{2}\log _{2} \frac {1-x}{2}$. The complete condition to $\{{M_{k}^{B}}\}$ becomes [15]

$$ \mu_{1}+\mu_{2}+\mu_{3}=1, $$

(12)

$$ \mu_{1}\vec{n}^{(1)}+\mu_{2}\vec{n}^{(2)}+\mu_{3}\vec{n}^{(3)}=0. $$

(13)

In [15], 3-element POVM of (14) can provide more optimal quantum discord than that of the projective measurement.

$$\begin{array}{@{}rcl@{}} \mu_{1}&=&\frac{\cos{\theta_{\text{opt}}}}{1+\cos{\theta_{\text{opt}}}},\ \ \vec{n}^{(1)}=(0,0,-1) \\ \mu_{2}&=&\frac{1}{2(1+\cos{\theta_{\text{opt}}})},\ \ \vec{n}^{(2)}=(\sin{\theta_{\text{opt}}},0,\cos{\theta_{\text{opt}}}) \\ \mu_{3}&=&\frac{1}{2(1+\cos{\theta_{\text{opt}}})},\ \ \vec{n}^{(3)}=(-\sin{\theta_{\text{opt}}},0,\cos{\theta_{\text{opt}}}) \end{array} $$

(14)

Here 𝜃 _opt is the POVM parameter minimizing (11).

In this paper we generalize the structure of 3-element POVM, using completeness and positivity. When there are three POVM elements, the direction vector for each element forms a triangle. The shape of this triangle depends on μ ₁, μ ₂ and μ ₃. Figure 1 shows a triangle made by $\vec {n}^{(1)},\vec {n}^{(2)}$ and $\vec {n}^{(3)}$ in the XY plane, and 𝜃 _{i j} denotes the angle between the direction vectors $\vec {n}^{(i)}$ and $\vec {n}^{(j)}$. From (12)–(13) one can obtain three equations to these angles

$$\begin{array}{@{}rcl@{}} \mu_{1}+\mu_{2}\cos\theta_{12}+\mu_{3}\cos\theta_{13}=0, \\ \mu_{1}\cos\theta_{12}+\mu_{2}+\mu_{3}\cos\theta_{23}=0, \\ \mu_{1}\cos\theta_{13}+\mu_{2}\cos\theta_{23}+\mu_{3}=0. \end{array} $$

(15)

From (15) the relations between 𝜃 ₁₂, 𝜃 ₂₃, 𝜃 ₁₃ and μ ₁, μ ₂, μ ₃ can be given as

$$\begin{array}{@{}rcl@{}} \theta_{12}=\cos^{-1} \frac{{\mu_{3}^{2}}-{\mu_{1}^{2}}-{\mu_{2}^{2}}}{2\mu_{1}\mu_{2}}, \\ \theta_{23}=\cos^{-1} \frac{{\mu_{1}^{2}}-{\mu_{2}^{2}}-{\mu_{3}^{2}}}{2\mu_{2}\mu_{3}}, \\ \theta_{13}=\cos^{-1} \frac{{\mu_{2}^{2}}-{\mu_{1}^{2}}-{\mu_{3}^{2}}}{2\mu_{1}\mu_{3}}. \end{array} $$

(16)

Fig. 1

Triangle composed of the direction vector $\vec {n}^{(1)}, \vec {n}^{(2)}, \vec {n}^{(3)}$

The condition where 𝜃 ₁₂, 𝜃 ₂₃, and 𝜃 ₁₃ are real can be found from −1< cos𝜃 ₁₂, cos𝜃 ₂₃, cos𝜃 ₁₃ < 1, which becomes (17). Figure 2 displays the region for (μ ₁, μ ₂) where 𝜃 ₁₂, 𝜃 ₂₃, and 𝜃 ₁₃ are real.

$$\begin{array}{@{}rcl@{}} |\mu_{2}-\mu_{3}|<\mu_{1}<\mu_{2}+\mu_{3} \\ |\mu_{1}-\mu_{3}|<\mu_{2}<\mu_{1}+\mu_{3} \\ |\mu_{1}-\mu_{2}|<\mu_{3}<\mu_{1}+\mu_{2} \end{array} $$

(17)

However, the measurement illustrated in Fig. 1 does not describe the most general POVM. In order to indicate the most general POVM one has to consider not the XY plane but an arbitrary plane. Therefore to find the direction vectors in an arbitrary plane, one can rotate them in Fig. 1 using the Euler angle. The completeness holds under the rotation of the direction vectors. There are three rotation matrices in (18)

$$ R(\psi,\theta,\phi)=R_{\psi}R_{\theta}R_{\phi}, $$

(18)

Where each rotation matrices are

$$\begin{array}{@{}rcl@{}} R_{\psi}&=&\left(\begin{array}{ccc}\cos\psi&0&\sin\psi\\ 0&1&0\\-\sin\psi&0&\cos\psi \end{array} \right), \\ R_{\theta}&=&\left(\begin{array}{ccc}1&0&0\\ 0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta \end{array} \right), \\ R_{\phi}&=&\left(\begin{array}{ccc}\cos\phi&-\sin\phi&0\\ \sin\phi&\cos\phi&0\\0&0&1 \end{array} \right). \end{array} $$

(19)

The direction vector of Fig. 1 are $\vec {n}^{(1)}=(1,0,0), \vec {n}^{(2)}=(\cos \theta _{12},\sin \theta _{12},0), \vec {n}^{(3)}=(\cos \theta _{13},-\sin \theta _{13},0)$. Through the rotation matrix R(ψ,𝜃,ϕ), one can obtain new direction vectors such as $\vec {m}^{(i)}=R(\psi ,\theta ,\phi )\vec {n}^{(i)}.$ However it is difficult to optimize (11) analytically. Therefore we use a Monte-Carlo simulation for optimizing (11). Our strategy is as follows.

• 1. In order to find (μ ₁, μ ₂) to minimize the conditional entropy $S\left (A|\{{M_{k}^{B}}\}\right )$ in the region of Fig. 2, we choose (μ ₁, μ ₂) randomly.

• 2. From the completeness condition (12), we decide μ ₃.

• 3. From (μ ₁, μ ₂, μ ₃), we decide the direction vector $\vec {n}^{(1)},\vec {n}^{(2)},\vec {n}^{(3)}$ of POVM element. When (μ ₁, μ ₂, μ ₃) is determined, by (16), 𝜃 ₁₂, 𝜃 ₂₃, 𝜃 ₁₃ between direction vectors are obtained.

• 4. By rotating $\vec {n}^{(1)},\vec {n}^{(2)},\vec {n}^{(3)}$ to minimize the conditional entropy $S\left (A|\{{M_{k}^{B}}\}\right )$, we obtain the direction vector $\vec {m}^{(1)},\vec {m}^{(2)},\vec {m}^{(3)}$. In fact 3-element POVM corresponding to $\vec {n}^{(1)},\vec {n}^{(2)},\vec {n}^{(3)}$ is not in a general form. The transformation of rotation of direction vector, which is R(ψ,𝜃,ϕ), is defined in (18) and (19).

• 5. Substituting the optimized conditional entropy $S\left (A|\{{M_{k}^{B}}\}\right )$, obtained in the previous step, into (3), we evaluate quantum discord.

Fig. 2

The permitted region of (μ ₁, μ ₂) for 3 element POVM. The edges are excluded

We examine a minimum conditional entropy in the region [0,2π] to the Euler angle ψ,𝜃,ϕ. It is found that a minimum conditional entropy does not depend on ϕ.

Y. Huang and Lu et al. considered quantum discord of special X states such as

$$ \rho_{1}^{AB}=\left(\begin{array}{cccc}0.027180&0&0&0.141651\\ 0&0.000224&0&0\\0&0&0.027327&0\\ 0.141651&0&0&0.945269 \end{array} \right) $$

(20)

$$ \rho_{2}^{AB}=\left(\begin{array}{cccc}0.021726&0&0&0.128057\\ 0&0.010288&0&0\\0&0&0.010288&0\\ 0.128057&0&0&0.957698 \end{array} \right) $$

(21)

and

$$ \rho_{3}^{AB}=\left(\begin{array}{cccc}0.0783&0&0&0\\ 0&0.1250&0.1000&0\\0&0.1000&0.1250&0\\ 0&0&0&0.6717 \end{array} \right). $$

(22)

In ref. [17] they treated the quantum discord of the X state using projective measurement. The condition for maximum discord was 𝜃 = π/2. Quantum discords for the X state of (21), (22) [17] and (23) [26] does not change dramatically according to the measurement setting.

Table 1 shows the quantum discord of 3 element POVM, that of the two projective measurements and that of measurement obtained by Ali et al. (δ ₂). As we can see, δ _3,min for $\rho _{1}^{AB}$ becomes 0.123010 which is 0.001613 less than the minimum value of the quantum discord obtained from two projective measurement in [17] (The quantum discord obtained here is a little different from the result in [17] because the quantum discord obtained in ref. [17] was expressed in terms of the natural logarithm(l o g _e ). In this paper all the results of the quantum discord are obtained in terms of l o g ₂. The results to quantum discord in terms of the natural logarithm(l o g _e ) can be found in Appendix. In addition, δ _3,min for $\rho _{2}^{AB}$ becomes 0.107873 which is 0.000075 less than the minimum value of the quantum discord obtained from two projective measurement in [17]. Furthermore δ _3,min for $\rho _{3}^{AB}$ becomes 0.132730 which is 0.000011 less than the minimum value of the quantum discord obtained from two projective measurement. For 3- element POVM, the optimized values to (μ ₁, μ ₂, μ ₃) are (0.4209,0.2938,0.2853) for $\rho _{1}^{AB}$, (0.4663,0.2489,0.2848) for $\rho _{2}^{AB}$ and (0.2748,0,2853,0.4349) for $\rho _{3}^{AB}$.

Table 1 Quantum discord for the states shown in (20)–(22) when 3 element POVM(δ _{3,m i n} ), projective measurement(δ _{2,m i n} ) and measurement obtained by Ali et al. (δ ₂) are used respectively

A better bound for $\rho _{1}^{AB}$, $\rho _{2}^{AB}$ and $\rho _{3}^{AB}$ can be obtained from 3-element POVM. Fanchini et al. provide an analytic formula for the symmetric X state [16]; however, the formula may not be optimal. Compared to Fanchini et al., the quantum discord to $\rho _{2}^{AB}$ is 0.000075 lower. Tables 1 and 2 clearly show that for the quantum states $\rho _{1}^{AB}$, $\rho _{2}^{AB}$ and $\rho _{3}^{AB}$, the 3- element POVM provide a better value to quantum discord.

Table 2 Difference between the quantum discord of 3 element POVM(2 projective measurement) and that of Ali et al., which is denoted by Δ₃(Δ₂)

4 Discussion

In this article we investigated the quantum discord to X states considered by Huang and Lu et al. We investigated the worst error to quantum discord from the analytic formula obtained by Ali et al. in case of general X states and by the analytic formula of Fanchini et al. in case of symmetric X states. By using projective measurement, Huang found the worst error to the quantum discord obtained by Ali et al. to be 0.002952. In this paper we extend the worst case error to 0.004565, by using 3-element POVM. Furthermore, for symmetric two-qubit X states using a 3-element POVM, the analytical formula derived by F. F. Fanchini et al. is valid with a worst-case error of 0.0009. In addition, 3-element POVM was found to supply a better quantum discord for the state considered in Lu et al. We numerically simulated the lower bound to the quantum states considered by Huang and Lu et al. However there is a need to provide an analytic optimal bound for these states, which is in progress.

Appendix A: Quantum Discord Expressed in Terms of the Natural Logarithm(loge)

In the appendix we supply results to quantum discord in terms of the natural logarithm(l o g _e ).

Table 3 Revisited quantum discord for the states shown in (20)–(22) when 3 element POVM(δ _{3,m i n} ), projective measurement (δ _{2,m i n} ) and measurement obtained by Ali et al. (δ ₂) are used respectively

Table 4 Difference between the quantum discord of 3 element POVM (2 projective measurement) and that of Ali et al., which is denoted by Δ₃(Δ₂)