Bosonic and fermionic Gaussian states from K\"ahler structures

TL;DR

This paper introduces a unified framework using K"ahler structures to characterize bosonic and fermionic Gaussian states, simplifying computations and revealing new identities in quantum state analysis.

Contribution

It extends Gaussian state characterization beyond covariance matrices to include K"ahler structures, unifying bosonic and fermionic cases and enabling algebraic computation methods.

Findings

Unified treatment of bosonic and fermionic Gaussian states.

Reduction of Gaussian state computations to algebraic operations.

Compilation of identities and formulas for state analysis.

Abstract

We show that bosonic and fermionic Gaussian states (also known as "squeezed coherent states") can be uniquely characterized by their linear complex structure $J$ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple $(G,\Omega,J)$ of compatible K\"ahler structures, consisting of a positive definite metric $G$, a symplectic form $\Omega$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian states can also be identified with such a triple, but with $J^2\neq -1\!\!1$. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.