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|urllast=http://prl.aps.org/abstract/PRL/v79/i26/p5194_1 Cerf|archive-urlfirst=https://archiveN.is/20130223112832/http://prl.aps.org/abstract/PRL/v79/i26/p5194_1 J.|dead-urllast2=yes Adami|first2=C.|archive-date=2013-02-23 1997|title=Negative entropyEntropy and informationInformation in quantum mechanicsQuantum Mechanics|publisherurl=https://link.aps.org/doi/10.1103/PhysRevLett.79.5194|journal=[[Physical Review Letters]]|yearvolume=79|issue=26|pages=5194–5197|doi=10.1103/physrevlett.79.5194|via=1997 }}</ref><ref>{{citeCite journal|urllast=http://praCerf|first=N.aps.org/abstract/PRA/v60/i2/p893_1 J.|last2=Adami|first2=C.|date=1999-08-01|title=Quantum extension of conditional probability|publisherurl=https://link.aps.org/doi/10.1103/PhysRevA.60.893|journal=[[Physical Review A]]|yearvolume=1999 }}{{dead link60|dateissue=November 2016 2|botpages=InternetArchiveBot 893–897|fix-attempteddoi=10.1103/PhysRevA.60.893|via=yes }}</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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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|last=Horodecki|first=Michał|last2=Oppenheim|first2=Jonathan|last3=Winter|first3=Andreas|date=|title=Partial theirquantum paper "Quantum Information can be negative" [information|url=http://arxivwww.orgnature.com/absdoifinder/10.1038/nature03909|journal=Nature|volume=436|issue=7051|pages=673–676|arxiv=quant-ph/0505062]|doi=10.1038/nature03909|via=}}</ref>
==Properties==
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==References==
{{reflist}}
Nielsen,* Michael A. and{{Cite book|title=[[IsaacQuantum L.Computation Chuang]]and (2000).Quantum Information ''(book)|Quantum Computation and Quantum Information'']]|last1=Nielsen|first=Michael A.|last2=Chuang|first2=Isaac L.|publisher=Cambridge University Press, {{ISBN|0year=2010|isbn=978-5211-63503107-900217-3|edition=2nd|___location=Cambridge|pages=|oclc=844974180|author-link=Michael Nielsen|author-link2=Isaac Chuang}}.
