Non-equilibrium Quantum Many-Body Green Function Formalism in the light of Quantum Information Theory

TL;DR

This paper explores the quantum kinetic equations derived from Green functions, highlighting issues with positivity and entanglement in many-body systems, and connects these to quantum information theory concepts.

Contribution

It introduces a generalized Wigner distribution for quantum kinetic equations and analyzes the role of entanglement and positivity in approximate many-body Green function evaluations.

Findings

GWD is not positive everywhere, indicating quantum features.

Non-positivity issues relate to truncation of cumulant expansions.

Hartree-Fock approximation shows no entanglement in two-particle systems.

Abstract

The following issues are discussed inspired by the recent paper of Kadanoff (arXiv: 1403:6162): (a) Construction of a generalized one-particle Wigner distribution (GWD) function (analog of the classical distribution function) from which the quantum kinetic equation due to Kadanoff and Baym (KB) is derived, often called the Quantum Boltzmann Equation (QBE); (b) The equation obeyed by this has a collision contribution in the form of a two-particle Green function. This term is manipulated to have Kinetic Entropy in parallel to its counterpart in the classical Boltzmann kinetic equation for the classical distribution function. This proved to be problematic in that unlike in the classical Boltzmann kinetic equation, the contribution from the kinetic entropy term was non-positive; (3) Kadanoff surmised that this situation could perhaps be related to quantum entanglement that may not have been included in his theory. It is shown that GWD is not positive everywhere (indicating dynamical quantumness) just like the commonly recognized property of the Wigner function (negative property indicating quantumness of the state). The issue of non-positive feature appearing in approximate evaluation of patently positive entities in many particle systems is here pointed to an early discussion of this issue (Phys. Rev. A10, 1852 (1974)) in terms of a theorem on truncation of cumulant expansion of a probability distribution function due to Marcinkeiwicz. The last issue of presence or absence of entanglement in an approximate evaluation of a many particle correlation poses a new problem; it is considered here in terms of fermionic entanglement theory in the light of density matrix and Green function theory of many-fermion systems. The clue comes from the fact that the Hartree-Fock approximation exhbits no entantanglement in two-particle fermion density matrix and hence also in two-particle Green function.