Bosonic and fermionic Gaussian states from Kahler structures

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 Kahler structures, consisting of a positive definite metric G, a symplectic form Omega and a linear complex structure J with J(2) = -1. Mixed Gaussian states can also be identified with such a triple, but with J(2) not equal -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. Copyright L. Hackl and E. Bianchi.

Automated summary

Bosonic and fermionic Gaussian states can be uniquely characterized by a linear complex structure J, providing a unified framework to treat both types of particles simultaneously. Pure and mixed Gaussian states can be identified with compatible Kahler structures, with J(2) being a key parameter. By applying these methods, computations involving Gaussian states can be simplified to algebraic operations, leading to the discovery of new identities and facilitating the study of entanglement, system dynamics, and driven systems.