Positivity Improving Quantum Markov Operators
Abstract
A Markov operator on a matrix algebra, that is, a unital completely positive map, is called positivity improving if it maps every nonzero positive element to an invertible one. This is the noncommutative counterpart of a stochastic matrix with strictly positive entries. We show that positivity improvement is self-dual and equivalent to a faithful quantum Markov--Dobrushin--Doeblin minorization. We also compare the scalar trace minorization with the optimal common-lower-bound construction of the quantum Markov--Dobrushin coefficient. We then characterize positivity improvement through the Kraus space of the map: a Markov operator is positivity improving if and only if the Hilbert--Schmidt orthogonal complement of its Kraus space contains no operator of rank one. As a consequence the Kraus rank of a positivity improving Markov operator on the algebra of d× d matrices is at least 2d−1, and an explicit Hankel-type channel shows that this bound is attained in every dimension. Although deciding positivity improvement is NP-hard in general, we obtain complete and effectively checkable classifications for qubits and qutrits, together with families of examples (Pauli, Weyl and Werner--Holevo channels) showing that all our estimates are optimal. Finally we show that, in infinite dimensions, positivity improvement no longer implies any contraction of the Markov operator.
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BibTeX
@misc{accardi2026positivity,
title = {{Positivity Improving Quantum Markov Operators}},
author = {Luigi Accardi and Abdessatar Souissi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.10042},
eprint = {2611.10042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.10042},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Positivity Improving Quantum Markov Operators
AU - Luigi Accardi
AU - Abdessatar Souissi
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.10042
UR - https://arxiv.org/abs/2611.10042
AB - A Markov operator on a matrix algebra, that is, a unital completely positive map, is called positivity improving if it maps every nonzero positive element to an invertible one. This is the noncommutative counterpart of a stochastic matrix with strictly positive entries. We show that positivity improvement is self-dual and equivalent to a faithful quantum Markov--Dobrushin--Doeblin minorization. We also compare the scalar trace minorization with the optimal common-lower-bound construction of the quantum Markov--Dobrushin coefficient. We then characterize positivity improvement through the Kraus space of the map: a Markov operator is positivity improving if and only if the Hilbert--Schmidt orthogonal complement of its Kraus space contains no operator of rank one. As a consequence the Kraus rank of a positivity improving Markov operator on the algebra of d× d matrices is at least 2d−1, and an explicit Hankel-type channel shows that this bound is attained in every dimension. Although deciding positivity improvement is NP-hard in general, we obtain complete and effectively checkable classifications for qubits and qutrits, together with families of examples (Pauli, Weyl and Werner--Holevo channels) showing that all our estimates are optimal. Finally we show that, in infinite dimensions, positivity improvement no longer implies any contraction of the Markov operator.
ER -