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]]. The conditional entropy is written <math>S(\rho|\sigma)</math>, or <math>H(\rho|\sigma)</math>, depending on the notation being used for the [[von Neumann entropy]].

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>{{Cite journal|last1=Cerf|first1=N. J.|last2=Adami|first2=C.|date=1997|title=Negative Entropy and Information in Quantum Mechanics|journal=[[Physical Review Letters]]|volume=79|issue=26|pages=5194–5197|doi=10.1103/physrevlett.79.5194|arxiv=quant-ph/9512022|bibcode=1997PhRvL..79.5194C|s2cid=14834430}}</ref><ref>{{Cite journal|last1=Cerf|first1=N. J.|last2=Adami|first2=C.|date=1999-08-01|title=Quantum extension of conditional probability|journal=[[Physical Review A]]|volume=60|issue=2|pages=893–897|doi=10.1103/PhysRevA.60.893|arxiv=quant-ph/9710001|bibcode=1999PhRvA..60..893C|s2cid=119451904 }}</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]].

ForIn thewhat remainder of the articlefollows, we use the notation <math>S(\rhocdot)</math> for the [[von Neumann entropy]], which will simply be called "entropy".

== Definition ==

Given twoa bipartite quantum statesstate <math>\rho^{AB}</math>, andthe entropy of the joint system AB is <math>S(AB)_\sigmarho \ \stackrel{\mathrm{def}}{=}\ S(\rho^{AB})</math>, and the vonentropies Neumannof entropiesthe subsystems are <math>S(A)_\rho \ \stackrel{\mathrm{def}}{=}\ S(\rho^A) = S(\mathrm{tr}_B\rho^{AB})</math> and <math>S(\sigmaB)_\rho</math>. The von Neumann entropy measures howan uncertainobserver's we areuncertainty about the value of the state;, that is, how much the state is a [[mixed state (physics)|mixed state]]. The [[joint quantum entropy]] <math>S(\rho,\sigma)</math> measures our uncertainty about the [[joint system]] which contains both states.

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

Unlike the classical [[conditional entropy]], the conditional quantum entropy can be negative. This is true even though the (quantum) von Neumann entropy of single variable is never negative. The negative conditional entropy is also known as the [[coherent information]], and gives the additional number of bits above the classical limit that can be transmitted in a quantum dense coding protocol. Positive conditional entropy of a state thus means the state cannot reach even the classical limit, while the negative conditional entropy provides for additional information.

==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 }}

* Mark M. Wilde, [http://arxiv.org/abs/1106.1445 "From Classical to Quantum Shannon Theory", arXiv:1106.1445].

[[Category:Quantum mechanical entropy]]