bibkey: utagi2024eternal
authors: Shrikant Utagi and Subhashish Banerjee and R. Srikanth
year: 2024
title: On the eternal non-Markovianity of non-unital quantum channels
doi: 10.1142/S0219749923500399
url: https://arxiv.org/abs/2203.10625
claim: "Section IV conjectures the impossibility of purely non-unital eternal non-Markovianity in arbitrary finite dimensions."
strata_touched:
  - D5/S3/QuantumChannels/PurelyNonUnitalEternalNonMarkovianity
license: citation-only
triage: anchor

Purely non-unital eternal non-Markovianity

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DOI: https://doi.org/10.1142/S0219749923500399

Source: https://arxiv.org/abs/2203.10625

The quoted text is from arXiv:2203.10625v2. The journal publication is International Journal of Quantum Information 22(1), 2350039 (2024).

Definition 1, p. 4:

When non-Markovianity originates solely from the non-unital part, i.e., from τ(t), then we call it non-unital non-Markovianity and such a channel will be called purely non-unital non-Markovian channel.

Section IV, p. 11:

These diverse examples examined in the Section suggest that a purely non-unital channel (i.e., one lacking any unital component) is impossible. Although we have only studied the qubit case, we conjecture that it is true for non-unital channels in arbitrary finite dimensions.

Section V, p. 11:

However, we conjecture that for a non-unital channel with no such unital component (in the above sense) to be ENM, is impossible.

Eqs. (9)–(11), pp. 3–4, split the Bloch action into its linear part and translation. For a qubit map the translation-free linear map on all matrices is $X\mapsto {\Phi }_t(X)-\mathrm{Tr}(X)({\Phi }_t(I)-I)/2$. The formal statement tests complete-positive divisibility of this unital part and failure of conditional complete positivity of the time-local generator at every positive time. This is the weak eternal non-Markovianity criterion; it does not assert negativity bounded away from zero at infinite time.

The source’s Theorems 1–2 concern generalized amplitude damping channels. The rotating replacement construction lies outside those restricted families.