Negative Entropy in Quantum Information Theory

We present a quantum information theory that allows for the consistent description of quantum 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, unlike in Shannon theory, conditional entropies can be negative when considering quantum entangled systems such as an Einstein-Podolsky-Rosen pair, which leads to a violation of well-known bounds of classical information theory. Negative quantum entropy can be traced back to “conditional” density matrices which admit eigenvalues larger than unity. A straightforward definition of mutual quantum entropy, or “mutual entanglement”, can also be constructed using a “mutual” density matrix. 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 ternary or larger system.