Bosonic and fermionic Gaussian states from Kähler structures

Lucas Hackl,Eugenio Bianchi

doi:10.21468/scipostphyscore.4.3.025

Abstract

We show that bosonic and fermionic Gaussian states (also known as ``squeezed coherent states’’) can be uniquely characterized by their linear complex structure JJ 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)(G,Ω,J) of compatible Kähler structures, consisting of a positive definite metric GG, a symplectic form \OmegaΩ and a linear complex structure JJ with J^2=-\mathbb{1}J2=−1. Mixed Gaussian states can also be identified with such a triple, but with J^2\neq -\mathbb{1}J2≠−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.

Highlights

We show that bosonic and fermionic Gaussian states can be uniquely characterized by their linear complex structure J which is a linear map on the classical phase space
In applications to quantum information, Gaussian states are often described in a covariance matrix formalism [1–3]
We have adopted a language and selected aspects that are tailored to applications in quantum information and non-equilibrium physics

Summary

Introduction

Gaussian states play a distinguished role in quantum theory: they appear under various names (squeezed coherent states, squeezed vacua, quasi-free states, generalized Slater determinants, ground states of free Hamiltonians) and are used in vastly different research fields, from quantum information [1–3] to quantum field theory in curved spacetimes [4,5] They are often used as testing ground, as many concepts can be studied analytically when they are applied to Gaussian states (e.g., entanglement entropy, logarithmic negativity, circuit complexity). We highlight the fact that various expressions for information-theoretic quantities (e.g., the entanglement entropy) take the same form for bosons and fermions when written in terms of the complex structure J Due to their versatility, many properties of Gaussian states have been independently discovered in different research communities ranging from quantum optics and condensed matter physics to high energy theory and quantum gravity.

Bosons and fermions from Kähler structures

Phase space

Linear observables

Algebra of classical observables

Kähler structures

Groups and algebras

Gaussian states

Applications

Summary and discussion