 Quantum relative entropy

Quantum relative entropy compares two density operators bywhen the support of a positive operator $\rho$ is contained in that of $\sigma$. If that containment fails, the value is $+\infty$. Zero eigenvalues of $\rho$ use the continuous convention $0\log 0=0$ on its support.

For a qubit reference $\sigma =I/2$, base-two logarithms give $D(\rho \Vert I/2)=1-S(\rho )$. A pure qubit state is therefore one bit away from that reference in this relative-entropy sense, while the reference itself gives zero. Nonnegativity of quantum relative entropy holds generally, but the quantity is not symmetric and need not satisfy the triangle inequality, so it is not a metric. Specifying the reference and logarithm base is necessary when comparing values.

Quantum relative entropy is a concept from quantum information theory that quantifies the difference between two quantum states in terms of information theory. It is a generalization of the classical relative entropy (or Kullback-Leibler divergence) to the quantum domain.