# Bosonic and fermionic Gaussian states from Kähler structures

### Submission summary

 As Contributors: Lucas Hackl Arxiv Link: https://arxiv.org/abs/2010.15518v3 (pdf) Date accepted: 2021-06-10 Date submitted: 2021-05-20 03:40 Submitted by: Hackl, Lucas Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Condensed Matter Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### 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.

Published as SciPost Phys. Core 4, 025 (2021)

### List of changes

As part of our reply to the referee reports, we included a revised manuscript with changes marked in red.