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 quantum entropy measures the von Neumann entropy of a quantum state $\rho$, if we have already measured the value of a second quantum state $\sigma$. This conditional entropy is written $S(\rho |\sigma )$, or $H(\rho |\sigma )$, depending on the notation being used for the von Neumann entropy.

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

Definition

Given two quantum states $\rho$ and $\sigma$, the von Neumann entropies are $S(\rho )$ and $S(\sigma )$. 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 $S(\rho ,\sigma )$ measures our uncertainty about the joint system which contains both states.

By analogy with the classical conditional entropy, we define the conditional quantum entropy as $S(\rho |\sigma )\equiv S(\rho ,\sigma )-S(\sigma )$.

References

Nielsen, Michael A. and Isaac L. Chuang (2000). Quantum Computation and Quantum Information. Cambridge University Press, ISBN 0-521-63505-9.