Conditional quantum 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. The conditional entropy is written S(ρ | σ), or H(ρ | σ), depending on the notation being used for the von Neumann entropy.

For the remainder of the article, we use the notation S(ρ) for the von Neumann entropy.

Definition

Given two quantum states ρ and σ, the von Neumann entropies are S(ρ) and S(σ). 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 S(ρ,σ) 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 $S(\rho|\sigma) \ \stackrel{\mathrm{def}}{=}\ S(\rho,\sigma) - S(\sigma)$.

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" [1].

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.

References

Nielsen, Michael A. and Isaac L. Chuang (2000). Quantum Computation and Quantum Information. Cambridge University Press, ISBN 0-521-63503-9.

• Mark M. Wilde, "From Classical to Quantum Shannon Theory", arXiv:1106.1445.