Strong subadditivity of quantum entropy
In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems (or of one quantum system with three degrees of freedom). It is a basic theorem in modern quantum information theory. It was conjectured by D. W. Robinson and D. Ruelle in 1966 and O. E. Lanford III and D. W. Robinson in 1968 and proved in 1973 by E.H. Lieb and M.B. Ruskai, building on results obtained by Lieb in his proof of the Wigner-Yanase-Dyson conjecture. The classical version of SSA was long known and appreciated in classical probability theory and information theory. The proof of this relation in the classical case is quite easy, but the quantum case is difficult because of the non-commutativity of the reduced density matrices describing the quantum subsystems. Some useful references here include: "Quantum Computation and Quantum Information" "Quantum Entropy and Its Use" Trace Inequalities and Quantum Entropy: An Introductory Course Definitions We use the following notation throughout the following: A Hilbert space is denoted by , and denotes the bounded linear operators on . Tensor products are denoted by superscripts, e.g., .