On the reversible extraction of classical information from a quantum source

Consider a source epsilon of pure quantum states with von Neumann entropy S . By the quantum source coding theorem, arbitrarily long strings of signals may be encoded asymptotically into S qubits per signal (the Schumacher lim it) in such a way that entire strings may be recovered with arbitrarily hig h fidelity. Suppose that classical storage is flee while quantum storage is expensive and suppose that the states of epsilon do not fall into two or m ore orthogonal subspaces. We show that if epsilon can be compressed with ar bitrarily high fidelity into A qubits per signal plus any amount of auxilia ry classical storage, then A must still be at least as large as the Schumac her limit S of epsilon. Thus no part of the quantum information content of epsilon can be faithfully replaced by classical information. If the states do fall into orthogonal subspaces, then A may be less than S, but only by a n amount not exceeding the amount of classical information specifying the s ubspace for a signal from the source.