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

The eternally non-Markovian (ENM) 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 ENM 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-ENM qubit GAD channel, characterized by a time t* > 0, such that the channel is non-Markovian (NM) only and for all time t > t*. We further point out that our negative result for the qudit GAD channel, namely, the impossibility of the eternal NM property, does not hold for a general qubit or higher-dimensional non-unital channel.