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 limit) in such a way that entire strings may be recovered with arbitrarily high 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 more orthogonal subspaces. We show that if epsilon can be compressed with arbitrarily high fidelity into A qubits per signal plus any amount of auxiliary classical storage, then A must still be at least as large as the Schumacher 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 an amount not exceeding the amount of classical information specifying the subspace for a signal from the source.