Bargmann invariants of Gaussian states

TL;DR

This paper derives an explicit formula for the Bargmann invariant of multiple bosonic Gaussian states using their means and covariance matrices, enabling analysis of their quantum properties.

Contribution

It provides a novel expression for the Bargmann invariant of bosonic Gaussian states in terms of their means and covariance matrices.

Findings

Derived the expression of Bargmann invariant for multi-mode Gaussian states.

Analyzed the possible values of Bargmann invariants for these states.

Abstract

Given a set of ordered quantum states, described by density operators $% \{\rho _{j}\}_{j=1}^{n}$, the Bargmann invariant of $\{\rho _{j}\}_{j=1}^{n}$ is defined as tr($\rho _{1}\rho _{2}...\rho _{n}$). Bargmann invariant serves as a fundamental concept for quantum mechanics and has diverse applications in quantum information science. Bosonic Gaussian states are a class of quantum states on infinite-dimensional Hilbert space, widely used in quantum optics and quantum information science. Bosonic Gaussian states are conveniently and conventionally characterized by their means and covariance matrices. In this work, we provide the expression of Bargmann invariant tr($\rho _{1}\rho _{2}...\rho _{n}$) for any $m$-mode bosonic Gaussian states $\{\rho _{j}\}_{j=1}^{n}$ in terms of the means and covariance matrices of $\{\rho _{j}\}_{j=1}^{n}.$ We also use this expression to explore the permissible values of Bargmann invariants for bosonic Gaussian states.