Nonextensive statistical mechanics : implications to quantum information

Abstract

The interactions and correlations among the constituents of manybody systems are manifested in characteristic physical properties such as ferromagnetism, superconductivity, etc. A list of some that have been studied in the last century is given in the Box (see below). A parallel development in quantum information was initially slow but over the past two decades progress has been very rapid. Fundamentally this is another aspect of quantum correlations in composite systems arising from the twin features of the superposition principle and the tensor product structure of state space. These features are not utilized in the same manner in quantum many-body physics. In the Box, a corresponding parallel list with properties of many-body systems is given because there has been an interplay between the two research efforts and their understanding. The basic quantum mechanical principles apply to both cases except different aspects are utilized because the goals are different in each. Both areas of investigation are based on a probabilistic foundation with a variational underpinning founded on an “entropy” maximization, which may be called Quantum Statistical Mechanics [1]. The specific form of the entropy as a functional of the density matrix will be made explicit presently. See Table I for definitions of density matrix and associated quantities. The equilibrium properties of the many-body systems are then given by a maximization of von Neumann entropy subject to certain constraints such as the average value of the Hamiltonian of the system “energy”. This leads to the familiar exponential probabilities of the Boltzmann-Gibbs (BG) form. Any nonequilibrium properties are studied by a quantum time evolution equation. Non-equilibrium properties such as the anomalous relaxation in time are often analyzed with the quantum version of the Tsallis entropy with an entropic index, q, which leads to powerlaw probabilities in contrast to the BG type [2]. (See the Box in the Introduction of Tsallis and Boon). In quantum information, the maximum entropy scheme is not of use because its origins are elsewhere as will be made clear presently. The evolution is replaced by processing of information