On the eternal non-Markovianity of non-unital quantum channels

TL;DR

The paper analyzes whether a purely non-unital quantum channel can be eternally non-Markovian (ENM) under CP- and P-divisibility criteria, focusing on generalized amplitude damping (GAD) channels. Using the affine Bloch representation and canonical decay rates, it proves a no-go: for any finite dimension $d$, a non-unital GAD channel cannot be ENM when non-Markovianity originates solely from the non-unital part, and in the qubit case ENM is also ruled out at all times. Since a strict ENM is unattainable, the authors construct a quasi-eternal non-Markovian GAD channel with a finite threshold time $t^>0$ after which non-Markovianity persists, while ensuring the evolution remains P-divisible. They also show that the negative result does not extend to all non-unital channels: ENM can arise when mixing a unital ENM component with non-unital noise, and higher-dimensional examples can realize ENM without a purely non-unital origin. These findings refine the understanding of CP-/P-divisibility in non-unital dynamics and highlight the role of unital contributions in sustaining ENM within quantum channels.

Abstract

The eternally non-Markovian Pauli channel is an example of a unital channel characterized by a negative decay rate for all time $t>0$. Here we consider the problem of constructing an analogous non-unital channel, and show in particular that a $d$-dimensional generalized amplitude damping (GAD) channel cannot be eternally non-Markovian when the non-Markovianity originates solely from the non-unital part of the channel. We study specific ramifications of this result for qubit GAD. Specifically, we construct a quasi-eternally non-Markovian qubit GAD channel, characterized by a time $t^\ast > 0$, such that the channel is non-Markovian only and for all time $t > t^\ast$. We further point out that our negative result for the qudit GAD channel, namely the impossibility of the eternal non-Markovian property, does not hold for a general qubit or higher-dimensional non-unital channel.

Figures (2)

• Figure 1: (Color online) The decay rates for the quasi-eternal non-Markovian GAD channel described by the Lindblad rates Eq. (\ref{['eq:rate12']}), with values $m:=3, n:= 2$ and $\nu:=1$. Whilst rate $\gamma_2$ is always positive, rate $\gamma_1$ is positive until time $t^\ast$, Eq. (\ref{['eq:tast']}), before becoming negative, and thereafter remains negative throughout thereafter. It is a consequence of Eq. (\ref{['eq:tast']}) that $t^\ast$ can be made arbitrarily close to 0, but not 0 itself, by virtue of Theorem 1. The symmetry between the two decay rates is a consequence of the fact, which follows from Eqs. (\ref{['eq:rate1']}) and \ref{['eq:rate2']}), that $\gamma_1(t)+\gamma_2(t) = \nu$.
• Figure 2: (Color online) The HCLA measure of QENMGAD channel (\ref{['eq:kraus_operators']}) against $m$.

Theorems & Definitions (5)

• Definition 1
• Theorem 1
• Lemma 1
• Theorem 2
• Theorem 3