Quantum coherence of two-qubit over quantum channels with memory

Abstract

Using the axiomatic definition of the quantum coherence measure, such as the $l_{1}$ norm and the relative entropy, we study the phenomena of two-qubit system quantum coherence through quantum channels where successive uses of the channels are memory. Different types of noisy channels with memory, such as amplitude damping, phase damping, and depolarizing channels effect on quantum coherence have been discussed in detail. The results show that quantum channels with memory can efficiently protect coherence from noisy channels. Particularly, as channels with perfect memory, quantum coherence is unaffected by the phase damping as well as depolarizing channels. Besides, we also investigate the cohering and decohering power of quantum channels with memory.

Appendix

In this appendix, we give the explicit analytic forms of coherence measure for more general family two-qubit states suffering from different kinds of quantum channels with memory to support the statement of fact that increasing the memory can protect the coherence. A general class of two-qubit state can be written in the Hilbert–Schmidt representation as

$$\begin{aligned} \rho =\frac{1}{4}\left( I\otimes I+\sum _{i}^{i=3}a_{i}\sigma _{i}\otimes I+I \otimes \sum _{i}^{i=3}b_{i}\sigma _{i}+\sum _{i,j}^{i,j=3}c_{ij}\sigma _{i}\otimes \sigma _{j}\right) , \end{aligned}$$

(A1)

here I stands for $2\otimes 2$ the identity operator, $\sigma _{i}$ are the standard Pauli matrices. For an arbitrary two-qubit state, one can use these parameters $(a_{i}, b_{i}, c_{ij})$ to represent them. However, the general state of two-qubit system described by Eq. (A1), under proper unitary rotations [38, 39], can be parameterized by nine real parameters $\vec {A}=(a_{1},a_{2},a_{3})$, $\vec {B}=(b_{1},b_{2},b_{3})$ and $\vec {C}=(c_{1},c_{2},c_{3})$. Taking into quantum channel with memory affect on coherence account, the $l_{1}$ norm of coherence for an arbitrary two-qubit state suffering from the amplitude damping channel with memory is obtained

(A2)

where $\xi _{0}=1-p(1-\mu )$ and $\mu _{0}=(1-\sqrt{1-p})\mu$. In particular, there exists long-live quantum coherence in the limited $p\rightarrow 1$, namely time $t\rightarrow \infty$, and Eq. (A2) reduces to

$$\begin{aligned} C_{l_1}(\varepsilon (\rho ))=\frac{1}{2}\left( \sqrt{a_{1}^2+a_{2}^2} +\sqrt{b_{1}^2+b_{2}^2}+|c_{1}+c_{2}|\right) \mu , \end{aligned}$$

(A3)

which is proportional to parameter $\mu$. This indicates with the amplitude damping channel with memory $\mu$ increasing, quantum coherence can be protected effectively. Similarly, we can compute the $l_{1}$ norm of coherence for an arbitrary two-qubit state suffering from the phase damping channel with memory

(A4)

in the limited $p\rightarrow 1$, Eq. (A4) reduces to

$$\begin{aligned} C_{l_1}(\varepsilon (\rho ))=\frac{1}{2}\left( |c_{1}-c_{2}|+|c_{1}+c_{2}|\right) \mu \end{aligned}$$

(A5)

which is only dependent of parameters $(c_{1},c_{2})$ and proportional to parameter $\mu$. Besides, the $l_{1}$ norm of coherence for an arbitrary two-qubit state subjected to the depolarizing damping channel with memory is also gained

(A6)

Particularly, long-live quantum coherence is presented in the limited $p\rightarrow 1$, namely

(A7)

which is related to parameter $\mu$, increasing the memory $\mu$ can protect the coherence. However, the explicit expression of relative entropy of coherence $C_{R}$ is too complicated to present in the text.