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., . The trace is denoted by . Density matrix A density matrix is a Hermitian, positive semi-definite matrix of trace one. It allows for the description of a quantum system in a mixed state. Density matrices on a tensor product are denoted by superscripts, e.g., is a density matrix on . Entropy The von Neumann quantum entropy of a density matrix is . Relative entropy Umegaki's quantum relative entropy of two density matrices and is . Joint concavity A function of two variables is said to be jointly concave if for any the following holds Subadditivity of entropy Ordinary subadditivity concerns only two spaces and a density matrix . It states that This inequality is true, of course, in classical probability theory, but the latter also contains the theorem that the conditional entropies and are both non-negative. In the quantum case, however, both can be negative, e.g. can be zero while . Nevertheless, the subadditivity upper bound on continues to hold. The closest thing one has to is the Araki–Lieb triangle inequality which is derived in from subadditivity by a mathematical technique known as purification. Strong subadditivity (SSA) Suppose that the Hilbert space of the system is a tensor product of three spaces: . Physically, these three spaces can be interpreted as the space of three different systems, or else as three parts or three degrees of freedom of one physical system. Given a density matrix on , we define a density matrix on as a partial trace: . Similarly, we can define density matrices: , , , , . Statement For any tri-partite state the following holds , where , for example. Equivalently, the statement can be recast in terms of conditional entropies to show that for tripartite state , . This can also be restated in terms of quantum