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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 ρ A B {\displaystyle \rho ^{AB}} , the conditional entropy is written S ρ {\displaystyle S_{\rho }} , or H ρ {\displaystyle H_{\rho }} , depending on the notation being used for the von Neumann entropy. The quantum conditional entropy was defined in terms of a conditional density operator ρ A | B {\displaystyle \rho _{A|B}} by Nicolas Cerf and Chris Adami, 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 non-separability.
In what follows, we use the notation S {\displaystyle S} for the von Neumann entropy, which will simply be called "entropy".
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