Conditional quantum entropy: Difference between revisions

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{{Short description|Measure of relative information in quantum information theory}}

The '''conditional quantum entropy''' is an [[entropy measure]] used in [[quantum information theory]]. It is a generalization of the [[conditional entropy]] of [[classical information theory]]. For a bipartite state <math>\rho^{AB}</math>, the conditional entropy is written <math>S(A|B)_\rho</math>, or <math>H(A|B)_\rho</math>, depending on the notation being used for the [[von Neumann entropy]]. The quantum conditional entropy was defined in terms of a conditional density operator <math> \rho_{A|B} </math> by [[Nicolas Cerf]] and [[Chris Adami]] ,<ref name="negent">{{citeCite journal | urllast1=http://prlCerf|first1=N.aps J.org/abstract/PRL/v79/i26/p5194_1 | last2=Adami|first2=C.|date=1997|title = Negative entropyEntropy and informationInformation in quantumQuantum mechanicsMechanics| publisher journal= [[Physical Review Letters]]| year volume= 199779|issue=26|pages=5194–5197|doi=10.1103/physrevlett.79.5194|arxiv=quant-ph/9512022|bibcode=1997PhRvL..79.5194C|s2cid=14834430}}</ref><ref>{{citeCite journal | urllast1=http://praCerf|first1=N.aps.org/abstract/PRA/v60/i2/p893_1 J.|last2=Adami|first2=C.|date=1999-08-01| title = Quantum extension of conditional probability| publisher journal= [[Physical Review A]]| year volume=60|issue=2|pages=893–897|doi=10.1103/PhysRevA.60.893|arxiv=quant-ph/9710001|bibcode=1999PhRvA..60..893C|s2cid=119451904 1999}}</ref>, who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum [[Separable state|non-separability]].

In what follows, we use the notation <math>S(\cdot)</math> for the [[von Neumann entropy]], which will simply be called "entropy".

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== Definition ==

Given a bipartite quantum state <math>\rho^{AB}</math>, the entropy of the joint system AB is <math>S(AB)_\rho \ \stackrel{\mathrm{def}}{=}\ S(\rho^{AB})</math>, and the entropies of the subsystems are <math>S(A)_\rho \ \stackrel{\mathrm{def}}{=}\ S(\rho^A) = S(\mathrm{tr}_B\rho^{AB})</math> and <math>S(B)_\rho</math>. The von Neumann entropy measures an observer's uncertainty about the value of the state, that is, how much the state is a [[mixed state (physics)|mixed state]].

By analogy with the classical conditional entropy, one defines the conditional quantum entropy as <math>S(A|B)_\rho \ \stackrel{\mathrm{def}}{=}\ S(AB)_\rho - S(B)_\rho</math>.

An equivalent (and more intuitive) operational definition of the quantum conditional entropy (as a measure of the [[quantum communication]] cost or surplus when performing [[quantum state]] merging) was given by [[Michał Horodecki]], [[Jonathan Oppenheim]], and [[Andreas Winter]].<ref>{{Cite injournal|last1=Horodecki|first1=Michał|last2=Oppenheim|first2=Jonathan|last3=Winter|first3=Andreas|title=Partial theirquantum paper "Quantum Information can be negative" [http://information|journal=Nature|volume=436|issue=7051|pages=673–676|arxiv.org/abs/=quant-ph/0505062]|doi=10.1038/nature03909|bibcode=2005Natur.436..673H|year=2005|pmid=16079840|s2cid=4413693}}</ref>

==Properties==

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==References==

{{reflist}}

* {{Cite book|title=Quantum Computation and Quantum Information|last1=Nielsen|first=Michael A.|last2=Chuang|first2=Isaac L.|publisher=Cambridge University Press|year=2010|isbn=978-1-107-00217-3|edition=2nd|___location=Cambridge|oclc=844974180|author-link=Michael Nielsen|author-link2=Isaac Chuang|title-link=Quantum Computation and Quantum Information (book)}}

Nielsen, Michael A. and [[Isaac L. Chuang]] (2000). ''Quantum Computation and Quantum Information''. Cambridge University Press, ISBN 0-521-63503-9.

*{{citation|first=Mark M.|last=Wilde|arxiv=1106.1445|title=Quantum Information Theory|pages=xi-xii|year=2017|publisher=Cambridge University Press|bibcode = 2011arXiv1106.1445W |doi=10.1017/9781316809976.001|chapter=Preface to the Second Edition|isbn=9781316809976|s2cid=2515538 }}

*{{cite book

| first = Mark

| last = Wilde

|

| title = Quantum Information Theory

| url= http://www.amazon.com/Quantum-Information-Theory-Mark-Wilde/dp/1107034256

| publisher = Cambridge University Press

| year = 2013

}}

[[Category:Quantum mechanical entropy]]