A logic for probability in quantum systems

Abstract

Quantum computation deals with projective measurements and unitary transformations in finite dimensional Hilbert spaces. The paper presents a prepositional logic designed to describe quantum computation at an operational level by supporting reasoning about the probabilities associated to such measurements: measurement probabilities, and transition probabilities (a quantum analogue of conditional probabilities). We present two automatizations, one for the logic as a whole and one for the fragment dealing just with measurement probabilities. These axiomatizations are proved to be sound and complete. The logic is also shown to be decidable, and we provide results characterizing its complexity in a number of cases. © Springer-Verlag Berlin Heidelberg 2003.