Bosonic and fermionic Gaussian states from Kähler structures

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.

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

“…where z can be an arbitrary phase-space vector z ∈ V , while G is a positive-definite symmetric bilinear form such that J = G Ω −1 has the property that all the eigenvalues of −J 2 are larger than one. In particular, the Gaussian state σ G,z is a pure state if and only if J 2 = − , in which case (Ω, G, J) form a so-called Kähler structure [40][41][42]. For the sake of a simpler notation, we omit the displacement vector when it is zero, i.e., we define σ G = σ G,0 All the quantum states with finite average energy 6 , which include all the quantum states that can be generated in physical experiments, have a well-defined covariance matrix.…”

Second-order statistics of fermionic Gaussian states

“…Its dual space V * in R 2N and V are equipped with a positive-definite bilinear form G ab : V * × V * → R and its inverse G −1 ab : V × V → R. Observables in quantum theory are linear operators on a Hilbert space. Here, it is convenient to introduce an operator-valued vector ξa , i.e., a quantization map [21,22] { ξa , ξb } = G ab I…”

“…with { ξa , ξb } = ξa ξb + ξb ξa denoting the anti-commutation relation and I being an identity operator. The operators ξa can be represented via the bases of the Majorana operators or fermionic creation and annihilation operators as discussed in details in [22].…”

“…Namely, Ω A and Ω B contain the same information regarding the entanglement between the two subsystems. The matrix Ω A takes the block diagonal form [21,22]…”

“…for a fixed f = m/(n + m). This asymptotic result (22) can be obtained by using the asymptotic behavior of polygamma functions [27]…”