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

TL;DR

This paper investigates the possibility of constructing non-unital quantum channels that are eternally non-Markovian, finding that such channels cannot exist in general for qudit GAD but can be quasi-eternally non-Markovian in qubits.

Contribution

The paper proves that a $d$-dimensional GAD channel cannot be eternally non-Markovian and constructs a quasi-eternally non-Markovian qubit GAD channel, highlighting the limitations of non-unital channels.

Findings

No $d$-dimensional GAD channel is eternally non-Markovian.

Constructed a quasi-eternally non-Markovian qubit GAD channel.

Impossibility of eternal non-Markovianity does not extend to all non-unital 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.