Conditional quantum entropy: Difference between revisions

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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">{{cite journal | url=http://prl.aps.org/abstract/PRL/v79/i26/p5194_1 | title = Negative entropy and information in quantum mechanics| publisher = [[Physical Review Letters]]| year = 1997}}</ref><ref>{{cite journal | url=http://pra.aps.org/abstract/PRA/v60/i2/p893_1 | title = Quantum extension of conditional probability| publisher = [[Physical Review]]| year = 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>.

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| first = Mark

| last = Wilde

|

| title = Quantum Information Theory

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