Loading…
On variational definition of quantum entropy
Entropy of distribution P can be defined in at least three different ways: 1) as the expectation of the Kullback-Leibler (KL) divergence of P from elementary δ-measures (in this case, it is interpreted as expected surprise); 2) as a negative KL-divergence of some reference measure ν from the probabi...
Saved in:
Main Author: | |
---|---|
Format: | Conference Proceeding |
Language: | English |
Subjects: | |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
cited_by | |
---|---|
cites | |
container_end_page | 204 |
container_issue | 1 |
container_start_page | |
container_title | |
container_volume | 1641 |
creator | Belavkin, Roman V |
description | Entropy of distribution P can be defined in at least three different ways: 1) as the expectation of the Kullback-Leibler (KL) divergence of P from elementary δ-measures (in this case, it is interpreted as expected surprise); 2) as a negative KL-divergence of some reference measure ν from the probability measure P; 3) as the supremum of Shannon’s mutual information taken over all channels such that P is the output probability, in which case it is dual of some transportation problem. In classical (i.e. commutative) probability, all three definitions lead to the same quantity, providing only different interpretations of entropy. In non-commutative (i.e. quantum) probability, however, these definitions are not equivalent. In particular, the third definition, where the supremum is taken over all entanglements of two quantum systems with P being the output state, leads to the quantity that can be twice the von Neumann entropy. It was proposed originally by V. Belavkin and Ohya [1] and called the proper quantum entropy, because it allows one to define quantum conditional entropy that is always non-negative. Here we extend these ideas to define also quantum counterpart of proper cross-entropy and cross-information. We also show inequality for the values of classical and quantum information. |
doi_str_mv | 10.1063/1.4905979 |
format | conference_proceeding |
fullrecord | <record><control><sourceid>proquest_osti_</sourceid><recordid>TN_cdi_osti_scitechconnect_22390863</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2124946874</sourcerecordid><originalsourceid>FETCH-LOGICAL-o211t-d7d97284564b8ac4cce8c05f324c3ce049b7bc0c94472d520a54d00369ad22ea3</originalsourceid><addsrcrecordid>eNpFj81KxDAURoMoWEcXvkHBrR1vbm6SZimDfzAwGwV3JU1S7DAmM00q-PYqCq4-DhwOfIxdclhyUOKGL8mANNocsYpLyRutuDpmFYChBkm8nrKznLcAaLRuK3a9ifWHnUZbxhTtrvZhGOP4A3Ua6sNsY5nf6xDLlPaf5-xksLscLv52wV7u755Xj8168_C0ul03CTkvjdfeaGxJKupb68i50DqQg0BywgUg0-vegTNEGr1EsJI8gFDGesRgxYJd_XZTLmOX3ViCe3MpxuBKhygMtEr8W_spHeaQS7dN8_R9InfIkQypVpP4AgOITp0</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>conference_proceeding</recordtype><pqid>2124946874</pqid></control><display><type>conference_proceeding</type><title>On variational definition of quantum entropy</title><source>American Institute of Physics:Jisc Collections:Transitional Journals Agreement 2021-23 (Reading list)</source><creator>Belavkin, Roman V</creator><creatorcontrib>Belavkin, Roman V</creatorcontrib><description>Entropy of distribution P can be defined in at least three different ways: 1) as the expectation of the Kullback-Leibler (KL) divergence of P from elementary δ-measures (in this case, it is interpreted as expected surprise); 2) as a negative KL-divergence of some reference measure ν from the probability measure P; 3) as the supremum of Shannon’s mutual information taken over all channels such that P is the output probability, in which case it is dual of some transportation problem. In classical (i.e. commutative) probability, all three definitions lead to the same quantity, providing only different interpretations of entropy. In non-commutative (i.e. quantum) probability, however, these definitions are not equivalent. In particular, the third definition, where the supremum is taken over all entanglements of two quantum systems with P being the output state, leads to the quantity that can be twice the von Neumann entropy. It was proposed originally by V. Belavkin and Ohya [1] and called the proper quantum entropy, because it allows one to define quantum conditional entropy that is always non-negative. Here we extend these ideas to define also quantum counterpart of proper cross-entropy and cross-information. We also show inequality for the values of classical and quantum information.</description><identifier>ISSN: 0094-243X</identifier><identifier>EISSN: 1551-7616</identifier><identifier>DOI: 10.1063/1.4905979</identifier><language>eng</language><publisher>Melville: American Institute of Physics</publisher><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ; COMMUTATION RELATIONS ; Divergence ; ENTROPY ; Entropy (Information theory) ; PROBABILITY ; QUANTUM ENTANGLEMENT ; QUANTUM INFORMATION ; Quantum phenomena ; QUANTUM SYSTEMS ; Transportation problem ; VARIATIONAL METHODS</subject><ispartof>AIP Conference Proceedings, 2015, Vol.1641 (1), p.204</ispartof><rights>2015 AIP Publishing LLC.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,310,311,315,786,790,795,796,891,23958,23959,25170,27957,27958</link.rule.ids><backlink>$$Uhttps://www.osti.gov/biblio/22390863$$D View this record in Osti.gov$$Hfree_for_read</backlink></links><search><creatorcontrib>Belavkin, Roman V</creatorcontrib><title>On variational definition of quantum entropy</title><title>AIP Conference Proceedings</title><description>Entropy of distribution P can be defined in at least three different ways: 1) as the expectation of the Kullback-Leibler (KL) divergence of P from elementary δ-measures (in this case, it is interpreted as expected surprise); 2) as a negative KL-divergence of some reference measure ν from the probability measure P; 3) as the supremum of Shannon’s mutual information taken over all channels such that P is the output probability, in which case it is dual of some transportation problem. In classical (i.e. commutative) probability, all three definitions lead to the same quantity, providing only different interpretations of entropy. In non-commutative (i.e. quantum) probability, however, these definitions are not equivalent. In particular, the third definition, where the supremum is taken over all entanglements of two quantum systems with P being the output state, leads to the quantity that can be twice the von Neumann entropy. It was proposed originally by V. Belavkin and Ohya [1] and called the proper quantum entropy, because it allows one to define quantum conditional entropy that is always non-negative. Here we extend these ideas to define also quantum counterpart of proper cross-entropy and cross-information. We also show inequality for the values of classical and quantum information.</description><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</subject><subject>COMMUTATION RELATIONS</subject><subject>Divergence</subject><subject>ENTROPY</subject><subject>Entropy (Information theory)</subject><subject>PROBABILITY</subject><subject>QUANTUM ENTANGLEMENT</subject><subject>QUANTUM INFORMATION</subject><subject>Quantum phenomena</subject><subject>QUANTUM SYSTEMS</subject><subject>Transportation problem</subject><subject>VARIATIONAL METHODS</subject><issn>0094-243X</issn><issn>1551-7616</issn><fulltext>true</fulltext><rsrctype>conference_proceeding</rsrctype><creationdate>2015</creationdate><recordtype>conference_proceeding</recordtype><recordid>eNpFj81KxDAURoMoWEcXvkHBrR1vbm6SZimDfzAwGwV3JU1S7DAmM00q-PYqCq4-DhwOfIxdclhyUOKGL8mANNocsYpLyRutuDpmFYChBkm8nrKznLcAaLRuK3a9ifWHnUZbxhTtrvZhGOP4A3Ua6sNsY5nf6xDLlPaf5-xksLscLv52wV7u755Xj8168_C0ul03CTkvjdfeaGxJKupb68i50DqQg0BywgUg0-vegTNEGr1EsJI8gFDGesRgxYJd_XZTLmOX3ViCe3MpxuBKhygMtEr8W_spHeaQS7dN8_R9InfIkQypVpP4AgOITp0</recordid><startdate>20150113</startdate><enddate>20150113</enddate><creator>Belavkin, Roman V</creator><general>American Institute of Physics</general><scope>8FD</scope><scope>H8D</scope><scope>L7M</scope><scope>OTOTI</scope></search><sort><creationdate>20150113</creationdate><title>On variational definition of quantum entropy</title><author>Belavkin, Roman V</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-o211t-d7d97284564b8ac4cce8c05f324c3ce049b7bc0c94472d520a54d00369ad22ea3</frbrgroupid><rsrctype>conference_proceedings</rsrctype><prefilter>conference_proceedings</prefilter><language>eng</language><creationdate>2015</creationdate><topic>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</topic><topic>COMMUTATION RELATIONS</topic><topic>Divergence</topic><topic>ENTROPY</topic><topic>Entropy (Information theory)</topic><topic>PROBABILITY</topic><topic>QUANTUM ENTANGLEMENT</topic><topic>QUANTUM INFORMATION</topic><topic>Quantum phenomena</topic><topic>QUANTUM SYSTEMS</topic><topic>Transportation problem</topic><topic>VARIATIONAL METHODS</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Belavkin, Roman V</creatorcontrib><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>OSTI.GOV</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Belavkin, Roman V</au><format>book</format><genre>proceeding</genre><ristype>CONF</ristype><atitle>On variational definition of quantum entropy</atitle><btitle>AIP Conference Proceedings</btitle><date>2015-01-13</date><risdate>2015</risdate><volume>1641</volume><issue>1</issue><epage>204</epage><issn>0094-243X</issn><eissn>1551-7616</eissn><abstract>Entropy of distribution P can be defined in at least three different ways: 1) as the expectation of the Kullback-Leibler (KL) divergence of P from elementary δ-measures (in this case, it is interpreted as expected surprise); 2) as a negative KL-divergence of some reference measure ν from the probability measure P; 3) as the supremum of Shannon’s mutual information taken over all channels such that P is the output probability, in which case it is dual of some transportation problem. In classical (i.e. commutative) probability, all three definitions lead to the same quantity, providing only different interpretations of entropy. In non-commutative (i.e. quantum) probability, however, these definitions are not equivalent. In particular, the third definition, where the supremum is taken over all entanglements of two quantum systems with P being the output state, leads to the quantity that can be twice the von Neumann entropy. It was proposed originally by V. Belavkin and Ohya [1] and called the proper quantum entropy, because it allows one to define quantum conditional entropy that is always non-negative. Here we extend these ideas to define also quantum counterpart of proper cross-entropy and cross-information. We also show inequality for the values of classical and quantum information.</abstract><cop>Melville</cop><pub>American Institute of Physics</pub><doi>10.1063/1.4905979</doi></addata></record> |
fulltext | fulltext |
identifier | ISSN: 0094-243X |
ispartof | AIP Conference Proceedings, 2015, Vol.1641 (1), p.204 |
issn | 0094-243X 1551-7616 |
language | eng |
recordid | cdi_osti_scitechconnect_22390863 |
source | American Institute of Physics:Jisc Collections:Transitional Journals Agreement 2021-23 (Reading list) |
subjects | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS COMMUTATION RELATIONS Divergence ENTROPY Entropy (Information theory) PROBABILITY QUANTUM ENTANGLEMENT QUANTUM INFORMATION Quantum phenomena QUANTUM SYSTEMS Transportation problem VARIATIONAL METHODS |
title | On variational definition of quantum entropy |
url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-09-21T19%3A40%3A41IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_osti_&rft_val_fmt=info:ofi/fmt:kev:mtx:book&rft.genre=proceeding&rft.atitle=On%20variational%20definition%20of%20quantum%20entropy&rft.btitle=AIP%20Conference%20Proceedings&rft.au=Belavkin,%20Roman%20V&rft.date=2015-01-13&rft.volume=1641&rft.issue=1&rft.epage=204&rft.issn=0094-243X&rft.eissn=1551-7616&rft_id=info:doi/10.1063/1.4905979&rft_dat=%3Cproquest_osti_%3E2124946874%3C/proquest_osti_%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-o211t-d7d97284564b8ac4cce8c05f324c3ce049b7bc0c94472d520a54d00369ad22ea3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2124946874&rft_id=info:pmid/&rfr_iscdi=true |