The $n$-th decay rate of coherence for Bell-diagonal states under quantum channels

TL;DR

This paper analyzes how the coherence of Bell-diagonal quantum states decays under various quantum channels over multiple iterations, providing explicit formulas and exploring conditions for coherence freezing.

Contribution

It introduces the concept of the $n$-th decay rate of coherence and derives explicit formulas for different coherence measures under multiple channels.

Findings

The $n$-th decay rate decreases faster with increasing $n$ as the state parameter $p$ increases.

Bell-diagonal states become fully incoherent under certain channels as $p$ increases, depending on the channel type.

Coherence freezing occurs under specific conditions, but is less likely for larger $n$ under some channels.

Abstract

We study the degree to which the coherence of quantum states is affected by noise. We give the definition of the $n$-th decay rate and investigate the coherence of Bell-diagonal states under $n$ iterations of channels. We derive explicit formulas of the $n$-th decay rates based on $l_1$ norm of coherence, relative entropy of coherence and skew information-based coherence. It is found that the larger $n$ is, the faster the $n$-th decay rate decreases as the parameter $p$ of Bell-diagonal states increases. Moreover, for any fixed $n$, with the increase of $p$, Bell-diagonal states can be completely incoherent under generalized amplitude damping (GAD) channels, depolarization (DEP) channels and phase flip (PF) channels, while this is not the case for bit flip (BF) channels and bit-phase flip (BPF) channels. We also investigate the geometry of the relative entropy of coherence and skew information-based coherence of Bell-diagonal states under different channels when the $n$-th decay rate is one, i.e., the coherence is frozen. It is shown that compared with BF and BPF channels, when $n$ is large enough, the coherence of Bell-diagonal states will not be frozen under GAD, DEP and PF channels. For skew information-based coherence, similar properties of coherence freezing are found.