Quantum Entropy, Fidelity, and Hypothesis Testing

Corresponds to Lectures 13–14, opening the course’s information- theoretic half. The von Neumann entropy $S(\rho) = -\mathrm{Tr}(\rho \log\rho)$ generalizes Shannon entropy (entropy-mutual-information) to density matrices — it’s the Shannon entropy of $\rho$‘s eigenvalues, but behaves subtly differently on composite systems (e.g. it can be lower for a subsystem than for the whole, impossible classically, precisely when the whole system is entangled). Quantum hypothesis testing asks how well two candidate states $\rho$ and $\sigma$ can be distinguished by an optimal measurement, and fidelity $F(\rho,\sigma)$ is the standard similarity measure answering a related question (how close are two states), playing the role classical statistical distance/total variation distance plays for probability distributions — both are the quantum ingredients typicality-and-quantum-source-compression needs next.

Outline (TODO — flesh out each)

• Quantum entropies — von Neumann entropy and relatives
• Properties distinguishing it from Shannon entropy on composite systems
• Quantum hypothesis testing
• State fidelity