Quantum extension of conditional probability

Abstract

We analyze properties of the quantum conditional amplitude operator [Phys. Rev. Lett. 79, 5194 (1997)], which plays a role similar to that of the conditional probability in classical information theory. The spectrum of the conditional operator that characterizes a quantum bipartite system is shown to be invariant under local unitary transformations and reflects its inseparability. More specifically, it is proven that the conditional amplitude operator of a separable state cannot have an eigenvalue exceeding 1, which results in a necessary condition for separability. A related separability criterion based on the non-negativity of the von Neumann conditional entropy is also exhibited.