Information theory of quantum entanglement and measurement

Abstract

Abstract We present a quantum information theory that allows for a consistent description of entanglement. It parallels classical (Shannon) information theory but is based entirely on density matrices rather than probability distributions for the description of quantum ensembles. We find that quantum (von Neumann) conditional entropies can be negative for entangled systems, which leads to a violation of entropic Bell inequalities. Quantum inseparability can be related, in this theory, to the appearance of “unclassical” eigenvalues in the spectrum of a conditional “amplitude” matrix that underlies the quantum conditional entropy. Such a unified information-theoretic description of classical correlation and quantum entanglement clarifies the link between them: the latter can be viewed as “super-correlation” which can induce classical correlation when considering a tripartite or larger system. Furthermore, the characterization of entanglment with negative conditional entropies paves the way to a natural information-theoretic description of the measurement process. This model, while unitary and causal, implies the well-known probabilistic results of conventional quantum mechanics. It also results in a simple interpretation of the Levitin-Kholevo theorem limiting the accessible information in a quantum measurement.