Typicality and Quantum Source Compression

Corresponds to Lectures 15–16. Reviews the classical asymptotic equipartition property and Shannon’s source coding theorem (asymptotic-equipartition-property covers this in full) as the model to generalize: almost all the probability mass of a long i.i.d. sequence concentrates on a typical set of about $2^{nH}$ sequences, so compressing to $nH$ bits loses negligible fidelity. Quantum source compression (Schumacher compression) asks the same question when the source emits quantum states rather than classical symbols: there’s a typical subspace of the full $n$-fold tensor product Hilbert space, of dimension about $2^{nS(\rho)}$ (the von Neumann entropy from the previous note), that carries almost all of the source’s quantum information, so projecting onto it and re-encoding gives a compression scheme with fidelity approaching 1 as $n\to\infty$ — the direct quantum counterpart of Shannon’s theorem, with von Neumann entropy playing the role Shannon entropy played classically.

Outline (TODO — flesh out each)

• Typicality and Shannon source compression (classical review)
• Quantum source compression — the typical subspace
• Schumacher’s theorem