Definition
Given a bipartite quantum state, the entropy of the entire system is, and the entropies of the subsystems are and . The von Neumann entropy measures how uncertain we are about the value of the state; how much the state is a mixed state.
By analogy with the classical conditional entropy, one defines the conditional quantum entropy as .
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 in their paper "Quantum Information can be negative" .
Read more about this topic: Conditional Quantum Entropy
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