The structure of general quantum Gaussian observable

TL;DR

This paper establishes a comprehensive structure theorem for multi-mode bosonic Gaussian observables, connecting their mathematical descriptions and providing minimal-dimension Naimark extensions, with implications for quantum optics measurements.

Contribution

It introduces a structure theorem that decomposes any multi-mode Gaussian observable into four fundamental types and links their characteristic function and POVM descriptions.

Findings

Any multi-mode Gaussian observable can be represented as a combination of four basic cases.

The paper provides a minimal-dimension Naimark extension construction for Gaussian observables.

Gaussian POVMs have bounded densities iff their noise covariance matrix is nondegenerate.

Abstract

The structure theorem is established which shows that an arbitrary multi-mode bosonic Gaussian observable can be represented as a combination of four basic cases, the physical prototypes of which are homodyne and heterodyne, noiseless or noisy, measurements in quantum optics. The proof establishes connection between the description of Gaussian observable in terms of the characteristic function and in terms of density of the probability operator-valued measure (POVM) and has remarkable parallels with treatment of bosonic Gaussian channels in terms of their Choi-Jamiolkowski form. Along the way we give the ``most economical'', in the sense of minimal dimensions of the quantum ancilla, construction of the Naimark extension of a general Gaussian observable. It is also shown that the Gaussian POVM has bounded operator-valued density with respect to the Lebesgue measure if and only if its noise covariance matrix is nondegenerate.