Characterizing Schwarz maps by tracial inequalities

DOI10.1007/S11005-023-01636-4arXiv2203.03433MaRDI QIDQ6393028

Eric Carlen, Alexander Müller-Hermes

Publication date: 7 March 2022

Abstract: Let $phi$ be a linear map from the $nimesn$ matrices ${mathcalM}_n$ to the $mimesm$ matrices ${mathcalM}_m$. It is known that $phi$ is $2$-positive if and only if for all $Kin{mathcalM}_n$ and all strictly positive $Xin{mathcalM}_n$, $phi(K^{*}{X}^{-1}K)geqphi(K{)}^{*}phi(X{)}^{-1}phi(K)$. This inequality is not generally true if $phi$ is merely a Schwarz map. We show that the corresponding tracial inequality $mTr[phi(K^{*}{X}^{-1}K)]geqmTr[phi(K{)}^{*}phi(X{)}^{-1}phi(K)]$ holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results.

Mathematics Subject Classification ID

Positive matrices and their generalizations; cones of matrices (15B48) Miscellaneous inequalities involving matrices (15A45) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) Operator theory (47-XX) Quantum entropies (81P17)

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