Detecting quantum non-Gaussianity via the Wigner function

Abstract

We introduce a family of criteria to detect quantum non-Gaussian states of a harmonic oscillator, that is, quantum states that can not be expressed as a convex mixture of Gaussian states. In particular we prove that, for convex mixtures of Gaussian states, the value of the Wigner function at the origin of phase space is bounded from below by a non-zero positive quantity, which is a function only of the average number of excitations (photons) of the state. As a consequence, if this bound is violated then the quantum state must be quantum non-Gaussian. We show that this criterion can be further generalized by considering additional Gaussian operations on the state under examination. We then apply these criteria to various non-Gaussian states evolving in a noisy Gaussian channel, proving that the bounds are violated for high values of losses, and thus also for states characterized by a positive Wigner function.

Detecting quantum non-Gaussianity of noisy Schrödinger cat states

“…We here review QNG criteria based on the Wigner function which have been proposed in [9]. We will restrict here to single-mode systems, descrbed by bosonic operators satisfying the commutation relation [a, a † ] = 1.…”

“…In fact excluding the case of states with negative Wigner function, which are certainly quantum non-Gaussian, no general method is known to distinguish between the two sets. This state of knowledge triggered the development of sufficient methods to detect quantum non-Gaussianity in noisy setups, where no negativity of the Wigner function can be observed [8,9], allowing to witness the succesful implementation of QNL processes despite the high levels of noise [10,11]. In this paper we apply the method introduced in [9], to investigate QNG of Schrödinger cat states [12,13,14] undergoing severe optical loss, such that its Wigner function becomes everywhere positive.…”

“…This state of knowledge triggered the development of sufficient methods to detect quantum non-Gaussianity in noisy setups, where no negativity of the Wigner function can be observed [8,9], allowing to witness the succesful implementation of QNL processes despite the high levels of noise [10,11]. In this paper we apply the method introduced in [9], to investigate QNG of Schrödinger cat states [12,13,14] undergoing severe optical loss, such that its Wigner function becomes everywhere positive. Focusing on the socalled odd and even cat states, we find that QNG can be witnessed for any value of the model parameters in the former case, and for a significant but finite range of parameters in the latter.…”

“…In what follows, we start by briefly summarizing the results of Ref. [9] and the basics of the employed physical model, and then use these tools to carry out a detailed analysis of the problem of interest. ‡ In this manuscript, we use the term 'nonlinearity' to indicate any process that cannot be realized with Hamiltonians that are second degree polynomials in the bosonic operators.…”

Normal form decomposition for Gaussian-to-Gaussian superoperators

“…Determining whether a given stateρ belongs to the convex hull C of the Gaussian set is a difficult problem [15][16][17][18] . Then, there comes the need to find criteria to certify thatρ cannot be written in the form (21).…”