Structure of a General Quantum Gaussian Observable

A structure theorem is established which shows that an arbitrary multimode bosonic Gaussian observable can be represented as a combination of four basic cases whose physical prototypes are homodyne and heterodyne, noiseless or noisy, measurements in quantum optics. The proof establishes a connection between the descriptions of a Gaussian observable in terms of the characteristic function and in terms of the density of a probability operator-valued measure (POVM) and has remarkable parallels with the 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. We also show 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.