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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]]. 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]].
For the remainder of the article, we use the notation <math>S(\rho)</math> for the [[von Neumann entropy]].
==Definition==
Given two quantum states <math>\rho</math> and <math>\sigma</math>, the von Neumann entropies are <math>S(\rho)</math> and <math>S(\sigma)</math>. The [[von Neumann entropy]] measures how uncertain we are about the value of the state; how much the state is a [[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(\rho|\sigma) \equiv S(\rho,\sigma) - S(\sigma)</math>.
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