Compatibility conditions on local and global spectra for $n$-mode Gaussian states

TL;DR

This paper establishes compatibility conditions linking the global and local spectra of n-mode Gaussian states, revealing that such states are uniquely determined by these spectra up to local transformations.

Contribution

It introduces new compatibility conditions for Gaussian states and shows that two-mode Gaussian states are uniquely characterized by their spectra, unlike qubits.

Findings

Compatibility conditions between global and local spectra are derived.

Every two-mode Gaussian state is uniquely determined by its spectra up to local transformations.

This property is not shared by qubit pairs.

Abstract

Compatibility conditions between the (global) spectrum of an $n$-mode Gaussian state and the spectra of the individual modes are presented, making optimal use of beam splitter and (two-mode) squeezing transformations. An unexpected bye-product of our elementary approach is the result that every two-mode Gaussian state is uniquely determined, modulo local transformations, by its global spectrum and local spectra -- a property shared not even by a pair of qubits.