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Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique

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Abstract

We consider a quasi-classical version of the Alicki–Fannes–Winter technique widely used for quantitative continuity analysis of characteristics of quantum systems and channels. This version allows us to obtain continuity bounds under constraints of different types for quantum states belonging to subsets of a special form that can be called “quasi-classical”. Several applications of the proposed method are described. Among others, we obtain the universal continuity bound for the von Neumann entropy under the energy-type constraint which in the case of one-mode quantum oscillator is close to the specialized optimal continuity bound presented recently by Becker, Datta and Jabbour. We obtain semi-continuity bounds for the quantum conditional entropy of quantum-classical states and for the entanglement of formation in bipartite quantum systems with the rank/energy constraint imposed only on one state. Semi-continuity bounds for entropic characteristics of classical random variables and classical states of a multi-mode quantum oscillator are also obtained.

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Notes

  1. The support \(\textrm{supp}\rho \) of a state \(\rho \) is the closed subspace spanned by the eigenvectors of \(\rho \) corresponding to its positive eigenvalues.

  2. The necessity to consider the case \(\,\mathfrak {S}_0\ne \mathfrak {S}_m(\mathcal {H}_{A_{1}\ldots A_{n}})\) is shown in Section 4.

  3. This is a weakened form of the \(\Delta \)-invariance property [35, Section 3].

  4. \(\,[\rho -\varepsilon I_{A_1\ldots A_n}]_+\) is the positive part of the Hermitian operator \(\,\rho -\varepsilon I_{A_1\ldots A_n}\).

  5. In [43], the function \(F_{H}(E)\) is denoted by \(S(\gamma (E))\).

  6. \(\,[\rho -\varepsilon I_{\mathcal {H}}]_+\) is the positive part of the Hermitian operator \(\,\rho -\varepsilon I_{\mathcal {H}}\).

  7. We assume that \(\dim \mathcal {H}=+\infty \).

  8. Throughout this subsection, we assume that \(\rho \) and \(\sigma \) are q-c states having representation (58).

  9. The definitions of all the notions from the classical information theory used in this section can be found in [12].

  10. These inequalities are the classical versions of the inequalities in [33, formula (10)].

  11. The convex closure of a set S in a Banach space is the minimal closed convex set containing S.

  12. Another corollaries of the \(\mu \)-compactness of \(\mathfrak {S}(\mathcal {H})\) are presented in the first part of [34].

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Acknowledgements

I am grateful to A.S. Holevo and G.G. Amosov for valuable discussion. I am also grateful to L. Lami for the useful reference. Special thanks to A. Winter for the comment concerning Mirsky’s inequality. I am grateful to the unknown reviewers for valuable suggestions and typos found. This work was funded by Russian Federation represented by the Ministry of Science and Higher Education (Grant Number 075-15-2020-788).

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  1. Probability Department, Steklov Mathematical Institute, Moscow, Russia

    Maksim Shirokov

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I dedicate this article to the memory of Mary Beth Ruskai, an outstanding scientist and a wonderful person. .

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Shirokov, M. Close-to-optimal continuity bound for the von Neumann entropy and other quasi-classical applications of the Alicki–Fannes–Winter technique. Lett Math Phys 113, 121 (2023). https://doi.org/10.1007/s11005-023-01742-3

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