Probabilistic Q-function distributions in fermionic phase-space

TL;DR

This paper introduces a positive, probabilistic Q-function for fermionic systems using Gaussian operators, enabling sampling methods free of the sign problem and applicable to arbitrary states.

Contribution

It constructs a new fermionic Q-function valid for all density matrices, extending previous methods and providing a probabilistic phase-space framework for fermionic many-body systems.

Findings

Q-function is real and positive for any fermionic state

It provides a natural phase-space interpretation with SO(2M) symmetry

Enables sign-problem-free sampling methods

Abstract

We obtain a positive probability distribution or Q-function for an arbitrary fermionic many-body system. This is different to previous Q-function proposals, which were either restricted to a subspace of the overall Hilbert space, or used Grassmann methods that do not give probabilities. The fermionic Q-function obtained here is constructed using normally ordered Gaussian operators, which include both non-interacting thermal density matrices and BCS states. We prove that the Q-function exists for any density matrix, is real and positive, and has moments that correspond to Fermi operator moments. It is defined on a finite symmetric phase-space equivalent to the space of real, antisymmetric matrices. This has the natural SO(2M) symmetry expected for Majorana fermion operators. We show that there is a natural physical interpretation of the Q-function: it is the relative probability for observing a given Gaussian density matrix. The distribution has a uniform probability across the space at infinite temperature, while for pure states it has a maximum value on the phase-space boundary. The advantage of probabilistic representations is that they can be used for computational sampling without a sign problem.