The Cumulants Expansion Approach: The Good, The Bad and The Ugly

TL;DR

This paper examines the cumulants expansion method in quantum physics, analyzing its convergence and applicability in different quantum systems, highlighting its strengths in some cases and limitations in others.

Contribution

It provides a detailed analysis of cumulants expansion in quantum systems, identifying conditions for its convergence and illustrating its limitations through specific quantum problems.

Findings

Convergence improves with higher orders in quantum dissipation problems.

Higher-order cumulants are ineffective in adiabatic quantum simulation.

Numerical challenges and non-physical solutions arise beyond mean field in some cases.

Abstract

The configuration space, i.e. the Hilbert space, of compound quantum systems grows exponentially with the number of its subsystems: its dimensionality is given by the product of the dimensions of its constituents. Therefore a full quantum treatment is rarely possible analytically and can be carried out numerically for fairly small systems only. Fortunately, in order to obtain interesting physics, approximations often very well suffice. One of these approximations is given by the cumulants expansion, where expectation values of products of operators are approximated by products of expectation values of said operators, neglecting higher-order correlations. The lowest order of this approximation is widely known as the mean field approximation and used routinely throughout quantum physics. Despite its ubiquitous presence, a general criterion for applicability and convergence properties of higher order cumulants expansions remains to be found. In this paper, we discuss two problems in quantum electrodynamics and quantum information, namely the collective radiative dissipation of a dipole-dipole interacting chain of atoms and the factorization of a bi-prime by annealing in an adiabatic quantum simulator. In the first case we find smooth, convergence behavior, where the approximation performs increasingly better with higher orders, while in the latter going beyond mean field turns out useless and, even for small system sizes, we are puzzled by numerically challenging and partly non-physical solutions.