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Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences

We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum me...

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Autores principales: Bussandri, Diego G., Osán, Tristán M.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10296848/
https://www.ncbi.nlm.nih.gov/pubmed/37372256
http://dx.doi.org/10.3390/e25060912
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author Bussandri, Diego G.
Osán, Tristán M.
author_facet Bussandri, Diego G.
Osán, Tristán M.
author_sort Bussandri, Diego G.
collection PubMed
description We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension.
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spelling pubmed-102968482023-06-28 Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences Bussandri, Diego G. Osán, Tristán M. Entropy (Basel) Article We introduce a new family of quantum distances based on symmetric Csiszár divergences, a class of distinguishability measures that encompass the main dissimilarity measures between probability distributions. We prove that these quantum distances can be obtained by optimizing over a set of quantum measurements followed by a purification process. Specifically, we address in the first place the case of distinguishing pure quantum states, solving an optimization of the symmetric Csiszár divergences over von Neumann measurements. In the second place, by making use of the concept of purification of quantum states, we arrive at a new set of distinguishability measures, which we call extended quantum Csiszár distances. In addition, as it has been demonstrated that a purification process can be physically implemented, the proposed distinguishability measures for quantum states could be endowed with an operational interpretation. Finally, by taking advantage of a well-known result for classical Csiszár divergences, we show how to build quantum Csiszár true distances. Thus, our main contribution is the development and analysis of a method for obtaining quantum distances satisfying the triangle inequality in the space of quantum states for Hilbert spaces of arbitrary dimension. MDPI 2023-06-08 /pmc/articles/PMC10296848/ /pubmed/37372256 http://dx.doi.org/10.3390/e25060912 Text en © 2023 by the authors. https://creativecommons.org/licenses/by/4.0/Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Bussandri, Diego G.
Osán, Tristán M.
Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences
title Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences
title_full Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences
title_fullStr Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences
title_full_unstemmed Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences
title_short Quantum Distance Measures Based upon Classical Symmetric Csiszár Divergences
title_sort quantum distance measures based upon classical symmetric csiszár divergences
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10296848/
https://www.ncbi.nlm.nih.gov/pubmed/37372256
http://dx.doi.org/10.3390/e25060912
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