Aspects of Quantum Information Compression for Pure States

Abstract

Consider a source of pure quantum states with density matrix ρ and von Neumann entropy S. It is known that the quantum information of the source may be faithfully compressed to 5 qubits/signal (asymptotically for long sequences) and that this limit is optimal if we demand high fidelity for long strings of signals. The usual method for achieving this compression, based on the typical subspace of the source, requires knowledge of the density matrix. We describe a new “universal” compression scheme which does not even require knowledge of ρ. More precisely, suppose that the source is known to have von Neumann entropy less than some given bound S ₀ but is otherwise completely unspecified. Our method will faithfully compress the quantum information of any such source to S ₀ qubits/signal.

It is known that S is always less than or equal to the Shannon entropy H of the prior distribution of the source. According to Shannon’s theorem, H provides the limit for compression of classical signals, so one may ask the question: what is the physical origin of the extra compression available in the quantum case compared to the classical case (where the prior probabilities are the same)? One may suggest that this effect is a consequence of possible non-orthogonality of signal states. However we demonstrate that it is possible to make each pair of signal states of a source more parallel (keeping the prior probabilities the same) while increasing the von Neumann entropy. This casts doubt on the conventional wisdom that overlap of quantum states is a measure of their (non-)distinguishability.

The results reported here were obtained in collaboration with M. Horodecki, P. Horodecki, R. Horodecki⁶ and with J. Schlienz⁹.