Abstract
We describe the class (semigroup) of quantum channels mapping states with finite entropy into states with finite entropy. We show, in particular, that this class is naturally decomposed into three convex subclasses, two of them are closed under concatenations and tensor products. We obtain asymptotically tight universal continuity bounds for the output entropy of two types of quantum channels: channels with finite output entropy and energy-constrained channels preserving finiteness of the entropy.
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Notes
It is easy to see that \(S\) is the homogeneous extension of the ‘‘classical’’ Shannon entropy defined on the set of probability distributions to the positive cone of \(\ell_{1}\).
The support \(\textrm{supp}\rho\) of a positive trace class operator \(\rho\) is the closed subspace spanned by the eigenvectors of \(\rho\) corresponding to its positive eigenvalues.
In terms of the sequence \(\{E_{k}\}\) of eigenvalues of \(H_{A}\) condition (9) means that \(\lim_{k\rightarrow\infty}E_{k}/\ln k=+\infty\), while (10) is valid if \(\liminf_{k\rightarrow\infty}E_{k}/\ln^{q}k>0\) for some \(q>2\) [7, Section 2.2].
Theorem 3 in [14] shows that \(F_{H_{A}}(E)=O(\ln E)\) as \(E\rightarrow+\infty\) provided that condition (15) holds.
Pure states are rank-one projectors—extreme points of the convex set \(\mathfrak{S}(\mathcal{H}_{A})\).
in the sense described at the begin of this subsection.
It is shown in [15] that \(\sigma\textrm{-}\textrm{co}f\neq\overline{\textrm{co}}f\) for a particular lower semicontinuous concave nonnegative unitarily invariant function \(f\) on \(\mathfrak{S}(\mathcal{H})\), so the above conjecture can not be proved by using only general entropy-type properties of the function \(H_{\Phi}\).
In general, the discrete convex roof construction applied to an entropy type function (in the role of \(H\)) may give a function which is not equal to zero at countably non-decomposable separable states [15, Remark 6].
By Proposition 1 in Section 2.2 this holds, in particular, if \(\hat{F}_{H_{A}}=\hat{F}^{*}_{H_{A}}\).
A continuity bound \(\sup_{x,y\in S_{a}}|f(x)-f(y)|\leq B_{a}(x,y)\) depending on a parameter \(a\) is called asymptotically tight for large \(a\) if \(\limsup_{a\rightarrow+\infty}\sup_{x,y\in S_{a}}\frac{|f(x)-f(y)|}{B_{a}(x,y)}=1\).
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ACKNOWLEDGMENTS
The authors are grateful to A.S. Holevo and to the participants of his seminar ‘‘Quantum probability, statistic, information’’ (the Steklov Mathematical Institute) for useful discussion.
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Shirokov, M.E., Bulinski, A.V. On Quantum Channels and Operations Preserving Finiteness of the von Neumann Entropy. Lobachevskii J Math 41, 2383–2396 (2020). https://doi.org/10.1134/S1995080220120392
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DOI: https://doi.org/10.1134/S1995080220120392