Mixed-permutation channel with its application to estimate quantum coherence

By this definition we see for any matrix $\boldsymbol{O}$ acting on $\mathbb{C}^{d}$, the action of the mixed-permutation channel is the average action over all possible permutation matrices. It is easily seen that $\boldsymbol{P}_{\pi}\boldsymbol{P}_{\sigma}=\boldsymbol{P}_{\pi\sigma}$ and $\boldsymbol{P}^{\dagger}_{\pi}=\boldsymbol{P}^{\scriptscriptstyle\mathsf{T}}_{\pi}=\boldsymbol{P}_{\pi^{-1}}$ for any $\pi$ and $\sigma$ in $S_{d}$. Moreover, $\Delta(\boldsymbol{O})=\frac{1}{d!}\sum_{\pi\in S_{d}}\boldsymbol{P}^{\dagger}_{\pi}\boldsymbol{O}\boldsymbol{P}_{\pi}$. The properties of the mixed-permutation channel $\Delta$ can be listed below:

1. (i)

   $\Delta\circ\Delta=\Delta$, that is $\Delta\circ\Delta(\boldsymbol{O})=\Delta(\boldsymbol{O})$ for any matrix $\boldsymbol{O}$ acting on $\mathbb{C}^{d}$.

2. (ii)

   $\Delta$ is unital, that is $\Delta(\mathbb{1}_{d})=\mathbb{1}_{d}$, where $\mathbb{1}_{d}$ is the identity matrix on $\mathbb{C}^{d}$.

3. (iii)

   $\Delta$ is self-adjoint with respect to Hilbert-Schmidt inner product, i.e., $\Delta=\Delta^{\dagger}$ in the sense of $\left\langle\Delta^{\dagger}(\boldsymbol{X}),\boldsymbol{Y}\right\rangle=\left\langle\boldsymbol{X},\Delta(\boldsymbol{Y})\right\rangle$ for all matrices $\boldsymbol{X},\boldsymbol{Y}$ acting on $\mathbb{C}^{d}$, that is, $\Delta^{\dagger}(\boldsymbol{O})=\frac{1}{d!}\sum_{\pi\in S_{d}}\boldsymbol{P}^{\dagger}_{\pi}\boldsymbol{O}\boldsymbol{P}_{\pi}=\Delta(\boldsymbol{O})$.

4. (iv)

   $[\Delta(\boldsymbol{O})]^{\scriptscriptstyle\mathsf{T}}=\Delta(\boldsymbol{O}^{{\scriptscriptstyle\mathsf{T}}})$, that is $[\Delta(\boldsymbol{O})]^{\scriptscriptstyle\mathsf{T}}=(\frac{1}{d!}\sum_{\pi\in S_{d}}\boldsymbol{P}_{\pi}\boldsymbol{O}\boldsymbol{P}^{\dagger}_{\pi})^{\scriptscriptstyle\mathsf{T}}=\frac{1}{d!}\sum_{\pi\in S_{d}}\boldsymbol{P}_{\pi}\boldsymbol{O}^{\scriptscriptstyle\mathsf{T}}\boldsymbol{P}^{\dagger}_{\pi}=\Delta(\boldsymbol{O}^{{\scriptscriptstyle\mathsf{T}}})$.

5. (v)

   $\Delta$ is invariant under any permutation matrices, that is, $\boldsymbol{P}_{\tau}\Delta(\boldsymbol{O})\boldsymbol{P}^{\dagger}_{\tau}=\Delta(\boldsymbol{O})$ for any permutation matrix $\boldsymbol{P}_{\tau}$, induced by the permutation $\tau\in S_{d}$.

Furthermore, the explicit form of the mixed-permutation channel on the $\mathbb{C}^{d}$ can be obtained analytically.

In Theorem 1, when $\boldsymbol{O}$ is taken as a quantum state $\rho\in\mathrm{D}\left(\mathbb{C}^{d}\right)$, i.e., if a quantum state $\rho$ goes through the mixed-permutation channel $\Delta$, then the output state $\Delta(\rho)$ can be characterized analytically.

In order to see clearly what is the image states $\Delta(\rho)$ of the channel $\Delta$ when input state is $\rho$, let us focus on qubit systems. In fact, any qubit state can be represented as

where $r_{i}\in\mathbb{R}$, $r:=\left\lvert\mspace{1.0mu}\boldsymbol{r}\mspace{1.0mu}\right\rvert=\sqrt{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}}\leqslant 1$, and $\boldsymbol{r}\cdot\boldsymbol{\sigma}:=\sum^{3}_{k=1}r_{k}\sigma_{k}$, $\boldsymbol{\sigma}:=(\sigma_{1},\sigma_{2},\sigma_{3})$ is the vector of the Pauli matrices, given by

It is easily seen that the action of $\Delta$ is

So the action of the mixed-permutation channel on the set of quantum states can be simulated by the transformation on the Bloch ball, i.e.,

From Figure 1, we see that the Bloch ball is mapped into a red line segment by the action of the mixed-permutation channel.

In order to estimate quantum coherence using the mixed-permutation channel, we need recall some notions about quantum coherence. Given the reference basis $\{|i\rangle\}_{i=1}^{d}$, a diagonal quantum state $\varrho=\sum_{i=1}^{d}\lambda_{i}|i\rangle\!\langle i|$ is called the incoherent state. The set of incoherent states is denoted as $\mathcal{I}$. The so-called incoherent operation is a completely positive linear mapping $\Phi$ such that its Kraus decomposition $\Phi(\rho)=\sum_{\nu}K_{\nu}\rho K_{\nu}^{\dagger}$ with $\sum_{\nu}K_{\nu}^{\dagger}K_{\nu}=\mathbb{1}$ fulfills $K_{\nu}\delta K_{\nu}^{\dagger}/\operatorname{Tr}\left(K_{\nu}\delta K_{\nu}^{\dagger}\right)\in\mathcal{I}$ for all $\delta\in\mathcal{I}$ and for all $\nu$.

The quantum coherence is quantified by a nonnegative function named the coherence measure. A coherence measure is required to satisfy the following conditions [22]:

1. (A1)

   Nonnegativity: $C(\rho)\geqslant 0$ for $\rho\in\mathrm{D}\left(\mathbb{C}^{d}\right)$, and moreover $C(\rho)=0$ for all $\rho\in\mathcal{I}$;

2. (A2)

   Monotonicity: $C(\Phi(\rho))\leqslant C(\rho)$ for any incoherent operation $\Phi$;

3. (A3)

   Strong monotonicity: $\sum_{\nu}p_{\nu}C(K_{\nu}\rho K_{\nu}^{\dagger}/p_{\nu})\leqslant C(\rho)$ for any incoherent operation $\Phi(\rho)=\sum_{\nu}K_{\nu}\rho K_{\nu}^{\dagger}$ with $p_{\nu}=\operatorname{Tr}\left(K_{\nu}\rho K_{\nu}^{\dagger}\right)$ and $\rho_{\nu}=K_{\nu}\rho K_{\nu}^{\dagger}/p_{\nu}$;

4. (A4)

   Convexity: $C(\rho)\leqslant\sum_{k}p_{k}C(\rho_{k})$ for any quantum state $\rho=\sum_{k}p_{k}\rho_{k}$ with $p_{k}\geqslant 0$ and $\sum_{k}p_{k}=1$.

Two commonly used coherence measures are the $l_{1}$-norm coherence and the relative entropy of coherence [22]. The $l_{1}$-norm coherence of the quantum state $\rho=\sum_{i,j}\rho_{ij}|i\rangle\langle j|$ is the sum of the magnitudes of all the off-diagonal entries

The relative entropy of coherence is the difference of von Neumann entropy between the density matrix and the diagonal matrix given by its diagonal entries,

where $\Pi(\rho)=\mathrm{diag}(\rho_{11},\rho_{22},\cdots,\rho_{dd})$ is the diagonal matrix obtained by the diagonal entries of $\rho$ and $S(\rho):=-\operatorname{Tr}\left(\rho\ln\rho\right)$ is the von Neumann entropy of the state $\rho$.

Now we come back to the mixed-permutation channel. It is obvious that $\boldsymbol{P}_{\pi}\rho\boldsymbol{P}^{\dagger}_{\pi}$ is incoherent for any incoherent state $\rho$. So the mixed-permutation channel is an incoherent operation. By the monotonicity of the coherence measures, we know the coherence of input state $\rho$ is nonincreasing under this channel. Since the output states of the mixed-permutation channel have an analytical form $\rho_{\star}$ in Eq. (8), we can utilize it to estimate the coherence of the input state $\rho$.

From the proof above, we see that Theorem 2 is universal in the sense that Eq. (30) is independent of the coherence measures. So Theorem 2 can be applied to all coherence measures. Additionally, the output states $\rho_{\star}$ is of one parameter, so its coherence is much easy to calculate compared with the input state. In fact, for any quantum state $\rho=\sum^{d}_{i,j=1}\rho_{ij}|i\rangle\langle j|$, the output state $\rho_{\star}$ in Corollary 1 can be decomposed as

where the weight is relevant to the quantum state as

and $|\Phi_{d}\rangle:=\frac{1}{\sqrt{d}}\sum^{d}_{i=1}|i\rangle$ is the maximally coherent state [29]. This state $\rho_{\star}$ is also called the maximally coherent mixed state in [30]. By this decomposition in Eq. (28), we get the eigenvalues of $\rho_{\star}$ is $\mathfrak{p}+\frac{1-\mathfrak{p}}{d}$ and $(1-\mathfrak{p})/d$ with multiplicity $d-1$. In view of this, we get the following result:

For $l_{1}$ norm coherence, the lower bound in Eq. (30) is tight for real quantum states, that is,

for any real quantum state $\rho$. Now we consider the $l_{1}$-norm coherence and the relative entropy of coherence in qubit system.

From another point of view, it is interesting to estimate quantum coherence from above. Let us turn to another quantity called the coherence of assistance, induced by any coherence measure $C$, which is defined in the form of concave bottom extension,

where the maximization is taken over all pure state decompositions of $\rho=\sum_{k}p_{k}|\psi_{k}\rangle\!\langle\psi_{k}|$. Coherence of assistance quantifies the coherence that can be extracted assisted by another party under local measurements and classical communication [31]. Suppose Alice holds a state $\rho^{A}=\rho=\sum_{k}p_{k}|\psi_{k}\rangle\!\langle\psi_{k}|$ with coherence $C(\rho)$. Bob holds another part of the purified state of $\rho$. The joint state between Alice and Bob is $\sum_{k}p_{k}|\psi_{k}\rangle_{A}\otimes|k\rangle_{B}$. Bob performs local projective measurements $\left\{|k\rangle\!\langle k|\right\}$ along the given basis and informs Alice the measurement outcomes by classical communication. Alice’s system will be in a pure state ensemble $\left\{p_{k},|\psi_{k}\rangle\!\langle\psi_{k}|\right\}$ with average coherence $\sum_{k}p_{k}C(|\psi_{k}\rangle\!\langle\psi_{k}|)$. The process is called assisted coherence distillation. The maximum average coherence is called the coherence of assistance which quantifies the one-way coherence distillation rate [31].

The coherence of assistance of any quantum state $\rho$ can be estimated by the output state $\rho_{\star}=\Delta(\rho)$.