1 Introduction

Quantum coherence is an essential ingredient in quantum optics, quantum information, solid state physics, and nanoscale thermodynamics [1, 2]. As a kind of quantum resource, some basic characterizations of coherence are studied such as quantification and manipulation. The quantum coherence manipulation is the conversion between quantum states under incoherent operations [3]. For pure states, one can be converted into another by incoherent operations if and only if their coherence vectors satisfy a majorization relation [4], which is a counterpart of the celebrated Nielson theorem in entanglement manipulation [5]. Then a necessary and sufficient condition for the transformation between two pure state ensembles is demonstrated [6]. The conversion from pure states to mixed states is studied and the necessary and sufficient condition has been provided in terms of a sequence of inequalities about a given coherence measure in Ref. [7]. The necessary and sufficient condition for the transformation from pure states into X state under incoherent operations is proved [8]. The transformation of mixed states under strictly incoherent operations and incoherent operations in qubit system is studied and the necessary and sufficient condition are presented in the Bloch sphere depiction [9, 10]. The probabilistic transformation between quantum states via strictly incoherent operations are also studied in Refs. [11,12,13].

The transformations between quantum states give rise to the equivalence of quantum states. The equivalence of two bipartite pure states under local incoherent operations assisted by classical communications (LICC) identifies to the equivalence under local operations assisted by classical communications (LOCC) [14]. Two multipartite pure states are LICC equivalent if and only if they are equivalent under local unitary incoherent operations [15].

In this paper, we study the transformations between quantum states under incoherent operations and unitary incoherent operations. In high dimensional systems, we show any two pure states are equivalent under incoherent operations if and only if they are equivalent under unitary incoherent operations. For mixed states, we show any two incoherent states are equivalent under incoherent operations. Furthermore, we show any incoherent states can be transformed by any mixed states under incoherent operations. In qubit systems, we prove the necessary and sufficient condition for the transformation between any two mixed states by incoherent operations alternatively and obtain the corresponding incoherent operations. Moreover we prove any two mixed qubit states with nonzero coherence are equivalent under incoherent operations if and only if they are equivalent under unitary incoherent operations.

2 The Transformation of Quantum States in Coherence Theory

The resource theory of coherence is composed of two basic elements: incoherent states and incoherent operations. Let H be a d dimensional Hilbert space. Take a set of basis \(\{|i\rangle \}_{i=1}^d\), we call the diagonal quantum state \(\rho =\sum _{i = 1}^d \lambda _i|i\rangle \langle i|\) under this set of basis as the incoherent state. Or else, the quantum state is called coherent. This set of incoherent states is labeled by \(\nabla \). \(\Phi =\sum _n K_n(\cdot )K_n^\dagger \) is an incoherent operation if it fulfills \(K_n\delta K_n^{\dagger }/\textrm{Tr}(K_n\delta K_n^{\dagger })\in \nabla \) for all \(\delta \in \nabla \) and for all n. In fact, the incoherent operators \(K_n\) can be represented as \(K_n=\sum _i a_i |\pi (i)\rangle \langle i|\) [16].

Before we study the transformations between quantum states under incoherent operations and unitary incoherent operations, we specify two equivalences between quantum states.

Definition 1

Two quantum states \(\rho \) and \(\sigma \) are called equivalent under incoherent operations if they can be transformed into each other by incoherent operations.

Definition 2

Two quantum states \(\rho \) and \(\sigma \) are called equivalent under unitary incoherent operations if they can be transformed into each other by unitary incoherent operations.

Since unitary operations do not change the eigenvalues of quantum states, so one necessary condition for the equivalence of two quantum states under unitary incoherent operations is that they have the same eigenvalues. It is obvious that unitary incoherent operations are incoherent, so the classification of quantum states under the unitary incoherent operations is finer than that under the incoherent operations. Next we analyze the transformation of quantum states and explore their equivalence under incoherent operations and unitary incoherent operations.

2.1 Pure State Case

Theorem 1

Two pure states \(|\psi \rangle =\sum _i \psi _i |i\rangle \) and \(|\phi \rangle =\sum _i \phi _i |i\rangle \) are equivalent under the incoherent operations if and only if they are equivalent under the unitary incoherent operations.

Proof

For two pure states \(|\psi \rangle =\sum _i \psi _i |i\rangle \) and \(|\phi \rangle =\sum _i \phi _i |i\rangle \), \(|\psi \rangle \) can be transformed into \(|\phi \rangle \) under incoherent operations if and only if \((|\psi _1|,|\psi _2|,\cdots , |\psi _{d}|)^{\downarrow T} \prec (|\phi _1|,|\phi _2|,\cdots , |\phi _{d}|)^{\downarrow T} \) with the superscript \(\downarrow \) denoting rearranging the elements in nonincreasing order and the superscript T denoting the transposition [17]. So two pure states \(|\psi \rangle \) and \(|\phi \rangle \) are equivalent under the incoherent operations if and only if \((|\psi _1|,|\psi _2|,\cdots , |\psi _{d}|)^{\downarrow T} \prec (|\phi _1|,|\phi _2|,\cdots , |\phi _{d}|)^{\downarrow T} \) and \((|\phi _1|,|\phi _2|,\cdots , |\phi _{d}|)^{\downarrow T} \prec (|\psi _1|,|\psi _2|,\cdots , |\psi _{d}|)^{\downarrow T} \), which means \((|\psi _1|,|\psi _2|,\cdots , |\psi _{d}|)^{\downarrow T} = (|\phi _1|,|\phi _2|,\cdots , |\phi _{d}|)^{\downarrow T}\). Therefore we suppose \(\phi _i= \psi _{\pi (i)} e^{\textrm{i} \theta _i}\) with \(\pi \) some permutation of \(\{1,2,\cdots , d\}\), with the roman letter i denoting the imaginary unit. Now let the incoherent unitary operator \(U=\sum _i e^{\mathrm{-i} \theta _i} |\pi (i)\rangle \langle i|\), then it has \(|\psi \rangle =U |\phi \rangle \). This implies two pure states are equivalent under the incoherent operations if and only if they are equivalent under the unitary incoherent operations. \(\square \)

2.2 Mixed State Case

Theorem 2

Any two incoherent states can be transformed into each other by incoherent operations.

Proof

Here we only need to prove that for any incoherent state \(\rho =\sum _{i} \rho _{ii} |i\rangle \langle i|\), \(\rho \) and \(\sigma =\frac{1}{d} I_d\) can be transformed into each other by incoherent operations.

First, suppose \(P=\sum _{i=1}^{d} |i\rangle \langle i+1 \mod d|\) is a permutation. For any incoherent state \(\rho =\sum _{i} \rho _{ii} |i\rangle \langle i|\), under the incoherent operation \(\Lambda (\rho )=\frac{1}{d} \sum _{s=1}^d P^s \rho (P^\dagger )^s \), it has \(\Lambda (\rho )= \frac{1}{d} I_d\). So any incoherent mixed state can be transformed into \(\sigma =\frac{1}{d} I_d\) by incoherent operations. Second, suppose \(Q=\sum _{i=1}^{d} \sqrt{\rho _{ii}}|i\rangle \langle i+1 \mod d|\), then under the incoherent operation \(\Lambda ^\prime (\rho )= \sum _{s=1}^d Q^s \rho (Q^\dagger )^s \), it has \(\Lambda ^\prime (\frac{1}{d} I_d)= \rho \). Therefore, any incoherent state \(\rho \) and \(\sigma =\frac{1}{d} I_d\) can be transformed into each other by incoherent operations. This completes the proof. \(\square \)

Theorem 2 shows any two incoherent states are equivalent under incoherent operations. However they are generally not equivalent under unitary incoherent operations, because two incoherent states probably have different eigenvalues. Therefore the classification of quantum states under incoherent operations is strictly more coarse than that under the unitary incoherent operations.

Theorem 3

For any given state \(\rho =\sum _{i,j} \rho _{ij} |i\rangle \langle j|\), it can be transformed into any incoherent state by incoherent operations.

Proof

For any given state \(\rho =\sum _{i,j} \rho _{ij} |i\rangle \langle j|\), under the projective measurement \(\Pi = \{|i\rangle \langle i|\}\), it is transformed into the incoherent state \(\rho ^\prime =\sum _{i} \rho _{ii} |i\rangle \langle i|\). Since incoherent states can be transformed into each other by incoherent operations, so any given state can be transformed into any incoherent state by incoherent operations. \(\square \)

2.3 Qubit Case

Now we consider the transformation between quantum states in qubit systems. We shall need the following lemma.

Lemma 1

For any qubit state \(\rho =\left( \begin{array}{cccccccc} \rho _{11} & \rho _{12}\\ \rho _{21} & \rho _{22}\end{array}\right) \) with \(\rho _{21}=\rho _{12}^*\), \(\rho _{11}+\rho _{22}=1\), \(\rho _{11},\rho _{22}\ge 0\), it can be transformed to \(\rho ^\prime =\left( \begin{array}{cccccccc} \rho _{11} & |\rho _{12}|\\ |\rho _{21}| & \rho _{22}\end{array}\right) \) under unitary incoherent operations.

Proof

For any qubit state \(\rho =\left( \begin{array}{cccccccc} \rho _{11} & \rho _{12}\\ \rho _{21} & \rho _{22}\end{array}\right) \) with \(\rho _{21}=\rho _{12}^*\), \(\rho _{11}+\rho _{22}=1\), \(\rho _{11},\rho _{22}\ge 0\), we suppose \(\rho _{12}=|\rho _{12}|e^{\textrm{i} \theta }\), then \(\rho _{21}=|\rho _{12}| e^{\mathrm{-i} \theta }\). Let \(U=\left( \begin{array}{cccccccc} e^{\mathrm{-i} \theta /2} & 0 \\ 0 & e^{\textrm{i} \theta /2} \end{array}\right) \), one can verify that U is incoherent and unitary. Furthermore we have \(U \rho U^\dagger =\left( \begin{array}{cccccccc} \rho _{11} & |\rho _{12}|\\ |\rho _{21}| & \rho _{22}\end{array}\right) \). This completes the proof. \(\square \)

Lemma 1 shows any qubit state \(\rho \) can be transformed into quantum state \(\rho ^\prime \) with real off-diagonals by incoherent and unitary operations. Conversely, any real quantum state \(\rho ^\prime \) can also be transformed into the quantum state \(\rho \) with complex off-diagonals having the same magnitude by incoherent and unitary operations. However it is not true for high dimensional systems. For example, for quantum state \(\rho =\sum _{i,j=1}^3 \rho _{ij} |i\rangle \langle j|\) with three nonzero off-diagonals, in order to eliminate the phases in the off-diagonals, we suppose the unitary incoherent operation \(U=\sum _{i=1}^3 e^{\textrm{i} \theta _i} |i\rangle \langle i|\). Then under the action of the unitary incoherent operation, \(\rho ^\prime =U \rho U^\dagger = \sum _{i,j=1}^3 \rho _{ij}e^{\textrm{i} (\theta _i-\theta _j)} |i\rangle \langle j|\). If \(\rho ^\prime \) is real, then \(\rho _{ij}e^{\textrm{i} (\theta _i-\theta _j)} \) is real for \(i\ne j\). Particularly, both \(\rho _{23}e^{\textrm{i} (\theta _2-\theta _3)} \) and \(\frac{\rho _{12}e^{\textrm{i} (\theta _1-\theta _2)}}{\rho _{13}e^{\textrm{i} (\theta _1-\theta _3)}} = \frac{\rho _{12}}{\rho _{13}} e^{\textrm{i} (\theta _2-\theta _3)}\) are real. This implies the complex numbers \(\rho _{23}\) and \( \frac{\rho _{12}}{\rho _{13}}\) have the same argument. However it is not true generally. so Lemma 1 is only valid in qubit systems. Now we prove the necessary and sufficient condition for the transformation of mixed states by incoherent operations in qubit systems.

Theorem 4

For any mixed qubit state \(\rho =\left( \begin{array}{cccccccc} \rho _{11} & \rho _{12}\\ \rho _{21} & \rho _{22}\end{array}\right) \) with \(\rho _{12}\ne 0\) and any mixed qubit state \(\sigma =\left( \begin{array}{cccccccc} \sigma _{11} & \sigma _{12}\\ \sigma _{21} & \sigma _{22}\end{array}\right) \), \(\rho \) can be transformed into \(\sigma \) by incoherent operations if and only if

  1. 1.

    \(|\rho _{12}|\ge |\sigma _{12}|\);

  2. 2.

    \(\frac{\sigma _{11}\sigma _{22}}{\rho _{11}\rho _{22}}\ge \frac{|\sigma _{12}|^2}{|\rho _{12}|^2}\).

Proof

By Lemma 1, we only need to consider the quantum states \(\rho \) and \(\sigma \) with real off-diagonals. Next we analyze the transformation between quantum state \(\rho =\left( \begin{array}{cccccccc} \rho _{11} & |\rho _{12}|\\ |\rho _{21}| & \rho _{22}\end{array}\right) \) and \(\sigma =\left( \begin{array}{cccccccc} \sigma _{11} & |\sigma _{12}|\\ |\sigma _{21}| & \sigma _{22}\end{array}\right) \). Without loss of generality, we suppose the diagonals of \(\rho \) and \(\sigma \) are arranged in nonincreasing order, \(\rho _{11}\ge \rho _{22}\) and \(\sigma _{11}\ge \sigma _{22}\).

Now we show if these two conditions are satisfied, then \(\rho \) can be transformed into \(\sigma \) by incoherent operations. First, if \(\rho _{11}\ge \sigma _{11}\), let the phase damping channel \(\Lambda _1\) be

$$\begin{aligned} K_1=\left( \begin{array}{cccccccc} 1 & 0 \\ 0 & \sqrt{1-q}\end{array}\right) ,\ \ K_2=\left( \begin{array}{cccccccc} 0 & 0 \\ 0 & \sqrt{q}\end{array}\right) \end{aligned}$$
(1)

with \(q=1-\frac{|\sigma _{12}|^{ 2}}{|\rho _{12}|^{2}}\), so we have \(\Lambda _1(\rho )=\left( \begin{array}{cccccccc} \rho _{11} & |\sigma _{12}| \\ |\sigma _{12}| & \rho _{22}\end{array}\right) \). Then let incoherent operation \(\Lambda _2\) be

$$\begin{aligned} \tilde{K}_1=\left( \begin{array}{cccccccc} \sqrt{p} & 0 \\ 0 & \sqrt{p}\end{array}\right) ,\ \ \tilde{K}_2=\left( \begin{array}{cccccccc} 0 & \sqrt{1-p} \\ \sqrt{1-p} & 0\end{array}\right) \end{aligned}$$
(2)

with \(p=\frac{\sigma _{11}-\rho _{22}}{\rho _{11}-\rho _{22}}\). After that, \(\rho \) is transformed into \(\sigma \) by incoherent operations \(\Lambda _2[\Lambda _1(\rho )]=\sigma \).

Second, if \(\rho _{11}<\sigma _{11}\), then the vector \((\rho _{11},\rho _{22})^{T}\) is majorized by \((\sigma _{11},\sigma _{22})^{T}\), \((\rho _{11},\rho _{22})^{T} \prec (\sigma _{11},\sigma _{22})^{T}\). So there is a doubly stochastic matrix such that \(A=\left( \begin{array}{cccccccc} a & 1-a \\ 1-a & a\end{array}\right) \) such that \(\left( \begin{array}{cccccccc} \rho _{11} \\ \rho _{22}\end{array}\right) =\left( \begin{array}{cccccccc} a & 1-a \\ 1-a & a\end{array}\right) \left( \begin{array}{cccccccc} \sigma _{11} \\ \sigma _{22}\end{array}\right) \). Now we define the incoherent operation \(\Lambda _1\) with Kraus operators

$$\begin{aligned} K_1=\left( \begin{array}{cccccccc} \sqrt{\frac{a\sigma _{11}}{\rho _{11}}} & 0 \\ 0 & \sqrt{\frac{a\sigma _{22}}{\rho _{22}}} \end{array}\right) ,\ \ K_2=\left( \begin{array}{cccccccc} 0 & \sqrt{\frac{(1-a)\sigma _{11}}{\rho _{22}}} \\ \sqrt{\frac{(1-a)\sigma _{22}}{\rho _{11}}} & 0\end{array}\right) , \end{aligned}$$
(3)

thus the output state is \(\Lambda _1(\rho )= \left( \begin{array}{cccccccc} \sigma _{11} & \sqrt{\frac{\sigma _{11}\sigma _{22}}{\rho _{11}\rho _{22}}} |\rho _{12}|\\ \sqrt{\frac{\sigma _{11}\sigma _{22}}{\rho _{11}\rho _{22}}} |\rho _{21}| & \sigma _{22}\end{array}\right) \). Next let the phase damping channel \(\Lambda _2\) be

$$\begin{aligned} \tilde{K}_1=\left( \begin{array}{cccccccc} 1 & 0 \\ 0 & \sqrt{1-q}\end{array}\right) ,\ \ \tilde{K}_2=\left( \begin{array}{cccccccc} 0 & 0 \\ 0 & \sqrt{q}\end{array}\right) \end{aligned}$$
(4)

with \(q=1-\frac{|\sigma _{12}|^{ 2}\rho _{11}\rho _{22}}{|\rho _{12}|^{2}\sigma _{11}\sigma _{22}}\), then we have \(\Lambda _2[\Lambda _1(\rho )]=\sigma \). Therefore we have proved the sufficient condition for the transformation of qubit states under incoherent operations.

Now we show if \(\rho \) can be transformed into \(\sigma \) by incoherent operations, then these two conditions must hold true. First, since \(l_1\) norm coherence is nonincreasing under incoherent operations and \(C_{l_1}(\rho )=|\rho _{12}|\), \(C_{l_1}(\sigma )=|\sigma _{12}|\) [16], so we have \(|\rho _{12}|\ge |\sigma _{12}|\). By Ref. [18], we know any incoherent qubit channel could be decomposed into four incoherent Kraus operators as

$$\begin{aligned} {K}_1=\left( \begin{array}{cccccccc} r\alpha _1 & \beta _1 \\ 0 & 0 \end{array}\right) ,\ \ {K}_2=\left( \begin{array}{cccccccc} 0 & 0 \\ \alpha _1 & -r\beta _1\end{array}\right) ,\ {K}_3=\left( \begin{array}{cccccccc} \alpha _2 & 0 \\ 0 & \beta _2\end{array}\right) ,\ {K}_4=\left( \begin{array}{cccccccc} 0 & \beta _3 \\ \alpha _3 & 0\end{array}\right) \end{aligned}$$
(5)

where \(r\ge 0\), \(\alpha _i\ge 0\), \(\beta _i\in \mathbb {C}\), and \(\alpha _2^2 +\alpha _3^2 + (r^2+1)\alpha _1^2= |\beta _2|^2 + |\beta _3|^2 + (r^2+1)|\beta _1|^2=1\). Suppose \(\Lambda \) is the incoherent operation with four Kraus operators in (5) such that \(\sigma = \Lambda (\rho )\), then we have

$$\begin{aligned} \sigma _{11}= & (r^2\alpha _1^2 +\alpha _2^2)\rho _{11} + (|\beta _1|^2 + |\beta _3|^2 )\rho _{22} + r \alpha _1 (\beta _1^*+\beta _1) |\rho _{12}|,\\ |\sigma _{12}|= & (\alpha _2\beta _2^*+\alpha _3\beta _3)|\rho _{12}|,\\ |\sigma _{21}|= & (\alpha _2\beta _2+\alpha _3\beta _3^*)|\rho _{21}|,\\ \sigma _{22}= & (\alpha _1^2 +\alpha _3^2)\rho _{11} + (r^2|\beta _1|^2 + |\beta _2|^2 )\rho _{22} - r \alpha _1 (\beta _1^*+\beta _1) |\rho _{12}|. \end{aligned}$$

By calculation, we get

$$\begin{aligned} \frac{\sigma _{11}\sigma _{22}}{\rho _{11}\rho _{22}}\ge & \frac{[\alpha _2^2\rho _{11} + |\beta _3|^2 \rho _{22}][\alpha _3^2 \rho _{11} + |\beta _2|^2 \rho _{22}]}{\rho _{11}\rho _{22}}\\= & \frac{\alpha _2^2 \alpha _3^2 \rho _{11}^2 + |\beta _2 \beta _3|^2\rho _{22}^2 + ( \alpha _2^2 |\beta _2|^2+ \alpha _3^2 |\beta _3|^2 )\rho _{11}\rho _{22} }{\rho _{11}\rho _{22}}\\\ge & \frac{ (2\alpha _2\alpha _3 |\beta _2 \beta _3| + \alpha _2^2 |\beta _2|^2+ \alpha _3^2 |\beta _3|^2 )\rho _{11}\rho _{22} }{\rho _{11}\rho _{22}}\\= & {( |\alpha _2\beta _2|+|\alpha _3\beta _3^*|)^2 }\\\ge & |\alpha _2\beta _2+\alpha _3\beta _3^*|^2\\= & \frac{|\sigma _{12}|^2}{|\rho _{12}|^2}, \end{aligned}$$

where we have used \(r^2\alpha _1^2 \rho _{11} + |\beta _1|^2 \rho _{22} \ge 2 r \alpha _1 |\beta _1| \sqrt{\rho _{11}\rho _{22}} \ge r \alpha _1 |\beta _1^*+\beta _1| |\rho _{12}|\) and \(\alpha _1^2 \rho _{11} + r^2 |\beta _1|^2 \rho _{22} \ge 2 r \alpha _1 |\beta _1| \sqrt{\rho _{11}\rho _{22}} \ge r \alpha _1 |\beta _1^*+\beta _1| |\rho _{12}|\) for the first inequality, \(\alpha _2^2 \alpha _3^2 \rho _{11}^2 + |\beta _2 \beta _3|^2\rho _{22}^2\ge 2\alpha _2\alpha _3 |\beta _2 \beta _3|\rho _{11}\rho _{22}\) for the second inequality, and \(|\alpha _2\beta _2|+|\alpha _3\beta _3^*|\ge |\alpha _2\beta _2+\alpha _3\beta _3^*|\) for the third inequality. Therefore we have proved the necessary condition for the transformation of qubit states under incoherent operations. \(\square \)

In fact, this necessary and sufficient condition has already been pointed out in Ref. [7, 10], which is expressed in Bloch representation. Here we prove it alternatively and construct the corresponding incoherent operations. In Ref. [10], they showed that for any quantum states \( \rho =\frac{1}{2}(I+\varvec{r} \cdot \varvec{\sigma })\) and \( \sigma =\frac{1}{2}(I+\varvec{s} \cdot \varvec{\sigma })\) with \( {\textbf {r}}=(r_x,r_y,r_z)\in \mathbb {R}^{3}\), \({\textbf {s}}=(s_x,s_y,s_z)\in \mathbb {R}^{3}\), \( \parallel {\textbf {r}}\parallel =\sqrt{r_x^{2}+ r_y^{2}+ r_z^{2}}\le 1 \), \( \parallel {\textbf {s}}\parallel =\sqrt{s_x^{2}+ s_y^{2}+ s_z^{2}}\le 1 \), and \(\varvec{\sigma }=(\sigma _{x},\sigma _{y},\sigma _{z})\) being the Pauli matrices, \(\sigma _{x}=|0\rangle \langle 1|+|1\rangle \langle 0|,\sigma _{y}=\textrm{i}(|0\rangle \langle 1|-|1\rangle \langle 0|),\sigma _{z}=|0\rangle \langle 0|-|1\rangle \langle 1|\). \(\rho \) can be transformed into \( \sigma \) if and only if \(s_x^{2}+ s_y^{2} \le r_x^{2}+ r_y^{2}\) and \(s_z^{2} \le 1-\frac{1-r_z^{2}}{r_x^{2}+ r_y^{2}} (s_x^{2}+ s_y^{2})\). To visualize the distribution of quantum state \(\rho \) and the transformed state \(\sigma \), we illustrate it by three cases with \({\textbf {r}}=(0.5,0.5,\sqrt{0.5})\), \({\textbf {r}}=(0.5,0.5,0.5)\), and \({\textbf {r}}=(0.5,0.5,0)\) respectively in Bloch sphere (See Fig. 1).

Fig. 1
figure 1

(Color Online) The distribution of quantum state \(\rho \) and the transformed state \(\sigma \) in Bloch sphere. The red points are the quantum state \(\rho \) and the blue surfaces are the quantum state \(\sigma \) transformed from \(\rho \). Here subfigure (a), subfigure (b), and subfigure (c) correspond to the Bloch vectors \({\textbf {r}}=(0.5,0.5,\sqrt{0.5})\), \({\textbf {r}}=(0.5,0.5,0.5)\), and \({\textbf {r}}=(0.5,0.5,0)\) for \(\rho \) respectively

Full size image

However, there does not exist necessary and sufficient conditions for the transformation between general mixed states in high dimensional systems yet. In d-dimensional systems with \(d\ge 4\), Ref. [7] has proved that a finite number of measure conditions are not sufficient to characterize coherence manipulation on general mixed states. By Theorem 4, we can get the transformation between pure states and mixed states.

Corollary 1

For every mixed qubit state \(\rho \), there exists a pure state \(|\psi \rangle \) that can be transformed into it by incoherent operations.

Proof

For any mixed qubit states \(\rho =\left( \begin{array}{cccccccc} \rho _{11} & \rho _{12}\\ \rho _{21} & \rho _{22}\end{array}\right) \), let \(|\psi \rangle =\sqrt{\rho _{11}} |1\rangle + \sqrt{\rho _{22}} |2\rangle \), then one can check that \(|\psi \rangle \) and \(\rho \) satisfy the two conditions in Theorem 4. Therefore \(|\psi \rangle \) can be transformed into \(\rho \) by incoherent operations. \(\square \)

In Corollary 1, we see mixed states can be derived by non maximally coherent pure states by incoherent operations. However in high-dimensional systems, any mixed states can be derived by maximally coherent pure states by incoherent operations, not necessarily non maximally coherent pure states [7]. By Theorem 4, one can get the equivalence of any two mixed qubit states under incoherent operations directly.

Theorem 5

For two mixed qubit states with nonzero coherence \(\rho =\left( \begin{array}{cccccccc} \rho _{11} & \rho _{12}\\ \rho _{21} & \rho _{22}\end{array}\right) \) and \(\sigma =\left( \begin{array}{cccccccc} \sigma _{11} & \sigma _{12}\\ \sigma _{21} & \sigma _{22}\end{array}\right) \), \(\rho \) and \(\sigma \) are equivalent under incoherent operations if and only if \(|\rho _{12}| = |\sigma _{12}|\) and \(\{\sigma _{11},\ \sigma _{22}\} = \{\rho _{11},\ \rho _{22}\}\).

Proof

By Theorem 4, we know \(\rho \) and \(\sigma \) can be transformed into each other under incoherent operations if and only if \(|\rho _{12}| = |\sigma _{12}|\) and \({\sigma _{11}\sigma _{22}} = {\rho _{11} \rho _{22}}\). Combined with the normalization \({\sigma _{11}+\sigma _{22}} = {\rho _{11} +\rho _{22}}=1\), we get \(\rho \) and \(\sigma \) are equivalent under incoherent operations if and only if \(|\rho _{12}| = |\sigma _{12}|\) and \(\{\sigma _{11},\ \sigma _{22}\} = \{\rho _{11},\ \rho _{22}\}\). This completes the proof. \(\square \)

Since Theorem 4 does not hold true in high dimensional systems, Theorem 5 is only valid in qubit systems correspondingly. Combined with Lemma 1, we get the equivalence of qubit states under unitary incoherent operations.

Corollary 2

Any two mixed qubit states with nonzero coherence are equivalent under incoherent operations if and only if they are equivalent under unitary incoherent operations.

3 Conclusions

To conclude, we have analyzed the transformations between quantum states under incoherent operations and unitary incoherent operations in high dimensional systems. Then we prove the necessary and sufficient condition for the transformations between quantum states under incoherent operations and unitary incoherent operations in qubit systems. We hope this study could helpful for the transformations between general quantum states and the classification of quantum states.