Quantum non-Gaussianity witnesses in the phase space

We address detection of quantum non-Gaussian states, i.e. nonclassical states that cannot be expressed as a convex mixture of Gaussian states, and present a method to derive a new family of criteria based on generic linear functionals. We then specialise this method to derive witnesses based on $s$-parametrized quasiprobability functions, generalising previous criteria based on the Wigner function. In particular we discuss in detail and analyse the properties of Husimi Q-function based witnesses and prove that they are often more effective than previous criteria in detecting quantum non-Gaussianity of various kinds of non-Gaussian states evolving in a lossy channel.

From a physical point of view this is a particularly important distinction, as quantum non-Gaussian states can be only produced by means of highly non-linear processes, while states belonging to the Gaussian convex hull can be generated by means of Gaussian operations only and classical randomisation. In [34,35] the first attempt to detect quantum non-Gaussianity was pursued, by deriving criteria based respectively on photonnumber probabilities and on the Wigner function. The criterion [34] has been already used to detect quantum non-Gaussian states produced in different experimental settings [36][37][38].
We here present a framework to derive QNG criteria based on generic linear functionals. We apply these results to the case of s-parametrized quasi-probability distributions, generalising the criteria obtained in [35] for the Wigner function, to the Husimi Q-function (s = −1) and in general to any distribution characterised by a parameter s < 0.
The paper is structured as follows: in Section II we present the problem by defining QNG, while Section III illustrates how to derive bounds of linear functionals on the Gaussian convex hull, along with their most important properties. In Section IV we present QNG criteria based on quasi-probability distributions. In Section V the effectiveness of these criteria are investigated for Fock states, photon-added coherent states, and photonsubtracted squeezed states, focusing in particular on the performances corresponding to the different quasiprobability distributions considered. In Section VI we illustrate how the uncertainty on the measured average photon number, propagates to the derived bounds, for different values of the parameter s. Section VII concludes the paper with final discussions and remarks.

II. QUANTUM NON-GAUSSIANITY
We begin by recalling the definition of the Gaussian convex hull where p(λ) can be an arbitrary probability distribution, |ψ G (λ) are pure Gaussian states and B(H) is the set of bounded operators. In general, all pure singlemode Gaussian states can be parametrized as |ψ G (λ) = D(α)S(ξ)|0 where D(α) and S(ξ) are the displacement and squeezing operators, respectively, α, ξ are arbitrary complex numbers and λ = {α, ξ}. The set G includes mixed Gaussian states as they can always be decomposed in the form (1), but also non-Gaussian states, that is states having a non-Gaussian Wigner function, as mixtures of coherent and squeezed states.
In line with Refs. [34,35], a quantum state ρ is defined quantum non-Gaussian iff ρ / ∈ G. To understand the importance of QNG in Quantum Optics, consider a simple example: given a single-mode field initially prepared in the vacuum state, it is easy to verify that states belonging to G can be prepared by applying a combination of Gaussian operations and classical randomization. In contrast, the preparation of a quantum non-Gaussian state ρ / ∈ G starting from the vacuum field can only be achieved by means of some non-Gaussian operation, such as the application of a highly non-linear Hamiltonian (i.e. more than quadratic in the mode operators) or probabilistic non-Gaussian operations as photon addition/subtraction [39].

III. BOUNDING LINEAR FUNCTIONALS ON THE GAUSSIAN CONVEX HULL
Before proceeding to specialize our analysis to phasespace quasi-probability distributions, it is worthwhile to discuss the general approach we shall take in order to witness QNG. Suppose that a single-oscillator quantum state is the output of some experiment. Assume that the data of our experiment allows us to estimate a certain quantity Φ[ ], where Φ is a linear functional on the space of quantum states, and a bound n on the average photon number, that is Tr[ â †â ] ≤ n. Remarkably, it may be possible to gain some information on the QNG character of the state , solely based on those two quantities. To see this, let us consider which can be easily seen to be convex subsets of G for any n ≥ 0, and define the function In other words, B(n) is the lowest possible value that Φ[ ] could take compatible with the assumptions (i) ∈ G; (ii) Tr[ â †â ] ≤ n. Hence, if our state verifies (ii), but we find the quantity Φ[ ] to be below B(n), we must conclude that / ∈ G (conversely, finding Φ[ ] ≥ B(n) must be interpreted as an inconclusive result).
A key step in this procedure is the calculation of the function B(n) for a given Φ. In general, this can be seen as a problem of linear optimization over an infinitedimensional parameter space [see Eq. (3)]. Luckily, the optimization can be dramatically simplified by exploting the properties of B(n). It turns out that it is sufficient to look for the (constrained) minimum of Φ amongst the set of pure Gaussian states, and that of Rank-2 mixtures of Gaussian states. Therefore, for a fixed Φ and n, only a finite number of parameters needs to be optimized in order to find B(n). While one may be able to derive these results by applying standard techniques of convex analysis, we find it worthwhile to present their proof in our context in Appendix A.

IV. QUANTUM NON-GAUSSIANITY WITNESSES IN PHASE-SPACE
In the present work we show that that the structure of the states in Eq. (1) implies nontrivial constraints on their associated quasiprobability distributions. As a consequence, we will be able to certify QNG when those constraints are violated. We start by recalling the results of [35], where it was shown that the Wigner function of any quantum state belonging to the Gaussian convex hull satisfies the following inequality We aim at obtaining bounds for other s-parametrized quasiprobabilities. For a quantum state of density operator , Here There are three values of s for which the quasiprobability function is typically explored: s = 1 is the Glauber-Sudarshan P-function [2,3], s = 0 is the Wigner function [1], and s = −1 is the Husimi Q-function [41] For the purposes of this paper, the only necessary requirement on the parameter s is going to be s < 0, in order to avoid singularities in our quasi-probability distributions. Even though the function in Eq. (5) may lose some of the appealing properties of a quasi-probability distribution when s < −1, it still allows us to obtain useful and experimentally friendly QNG criteria as we will discuss in Sec. V.

A. General QNG criteria in Phase space
The general problem under investigation can be formulated in the general framework of Section III, by noting that Q s [ ](α) is a linear functional of the state at fixed α and s. Thus, having fixed a particular value of s < 0, and assuming Tr[ a † a] ≤ n, we ask ourselves whether the structure given in Eq. (1) implies a non-trivial lower bound on the possible values that Q s can take. More precisely, we define For every value of s < 0, B s (n) is positive and convex. Moreover, we show in Appendix B that B s (n) is strictly decreasing in n, B s (n) → 0 as n → ∞, and the minimizing state in G n has an average photon number exactly equal to n. The functions B s are therefore non-trivial and can be exploited in the formulation of QNG criteria as follows. Criterion 1: For a quantum state , define the QNG witness where Tr[ a † a] ≤n. Then, that is, is quantum non-Gaussian.
Criterion 2: Consider now a quantum state and a Gaussian map E G , or a convex mixture thereof. Define: where Before proceeding further, we note that the monotonicity of B s implies that the criteria become harder to satisfy as n andn E are increased (indeed, both ∆ s would correspondingly increase). Therefore, in the remainder of this paper we shall apply these criteria respectively for Tr[ a † a] =n and Tr[E G ( )a † a] =n E , which provide the highest chance of detecting QNG. On the other hand, experimentally it may be more practical to estimate an upper bound to the average photon number, rather than its actual value.

B. Near-optimality of pure states
As discussed in Section III, we can restrict the optimization in Eq (7) to Rank-1 and Rank-2 mixtures of Gaussian states. In all the considered examples, however, we found strong numerical evidence that the minimum was being reached by a pure Gaussian state. We have thus proven the near-optimality of pure Gaussian states for a number of s-values of interest through a semianalytical approach, whose details are provided in Appendix D. A pure state lower bound to each quasiprobability can be defined as where the |ψ G 's are pure Gaussian states. Clearly, the bound in Eq. (12) is in practice easier to calculate than the one in Eq. (7), however B P s (n) ≥ B s (n), since we can not exclude that the minimum may be reached by a Rank-2 state. Nevertheless, our numerical studies for the cases meaning that the pure state lower bound B P s is an excellent approximation to the true bound B s (n) in a wide range of average photon numbers [see Appendix D]. The level of approximation provided by Eq. (13) is sufficient to guarantee the validity of our findings in the following section.

V. DETECTING QUANTUM NON-GAUSSIANITY OF STATES EVOLVING IN A LOSSY CHANNEL
In this section we will test the effectiveness of the criteria introduced in Section IV A. Specifically, we shall investigate whether these criteria can be exploited to certify that pure non-Gaussian states evolving in a lossy channel remain quantum non-Gaussian. We will consider initially pure non-Gaussian states evolving in a lossy bosonic channel described by the following master equation:˙ The corresponding quantum channel E is Gaussian and can be characterised by a single parameter, = 1 − e −γt . We will look for the maximum values of such that the criteria are violated, in particular we define (a) Since for > 1 2 , no negativity of the Wigner function can be observed, we will be interested in larger values of s , so that our criteria will be able to detect quantum non-Gaussian states with positive Wigner function. The usefulness of the Wigner-function-based criterion has been extensively shown in [35,43]. We will start by comparing the witnesses ∆ (a) s for initial Fock states, while next we will discuss both the witnesses ∆  [44]. While the parameter characterising the lossy channel is supposed to be unknown, and our goal is to understand the maximum value of noise such that our criteria will be able to detect quantum non-Gaussian states, the inefficiency of the detector is known to the experimentalist as it is possible to determine its value by probing the detector with known states. In fact, as illustrated in Fig. 1, considering different values of s is equivalent to detecting QNG of unknown states evolved through a lossy channel, with a choice of detectors, one corresponding to each s. As regards the examination of ∆ s , we will focus on the special cases of s = 0 and s = −1. In both cases we will observe how, in particular in the low energy regime, the witnesses derived for lower values of s show a larger robustness against loss.
and the value of the corresponding s-parametrized quasiprobability distribution at the origin can be evaluated using the formula for a generic Fock state We have evaluated the corresponding values of the witnesses ∆  Perhaps surprisingly, the witnesses appear to be more sensitive as s decreases, that is, they provide a larger value of (a) s in the relevant range m ≤ 5. One can notice an interesting tradeoff in the behaviour of the witnesses ∆ (a) s for the Fock state |1 in Fig. 3(a). We note that the absolute value of ∆ (a) s is decreasing by considering more negative values of s. Even though such monotonous behaviour is lost for the Fock state |3 [see Fig. 3(b)], similar conclusions can be drawn for higher Fock states. Hence, while one can in principle detect QNG for larger values of the noise parameter by decreasing s, the amount of violation quantified by ∆ (a) s may be generally smaller. The impact of such tradeoff on the experimental detection of QNG, however, cannot be assessed without a thorough analysis of the propagation of experimental errors for the various witnesses. A first attempt towards this direction will be done in Sec. VI, while a complete analysis goes beyond the scope of our paper.

B. PAC states
A photon-added coherent state is defined as where N is the normalisation factor. Its average photon number isn for α ∈ R. The s-parametrized quasiprobability distributions are determined using the convolution expression presented in [42]: with the condition that this holds provided s < s. Using this expression, the values of the witnesses ∆ states, smaller values of s produce a more effective bound for the certification of QNG in noisy PAC, provided the parameter α is smaller or equal to about 10, as illustrated in Fig. 4. However, we observe again evidence that there is a compromise between a tighter bound and the amount of violation quantified by the criterion ∆ We can now consider the optimised witness defined in Eq. (10). For the PAC state, it is observed that the minima of the quasiprobability distributions are displaced from the origin of the phase space, and it is therefore possible to decrease the quantum non-Gaussianity indicator by re-displacing the minimum to the origin. Thus the quasiprobability function Q s [ ](−β) and the average photon number of the displaced state are computed, yielding: where A 0 = ψ 0 |A|ψ 0 , and for |ψ 0 = |ψ PAC , We then minimise ∆ (b) s (β) over the possible displacement parameters β. We find that the optimal value of β for large values of and α 1.5, which is nearly the same for the Wigner and Q functions, can be approximated as Taking β = β opt , we compare the values of the QNG witness based on the second criterion for s = 0 and s = −1. Both the plots and numerical investigations indicate that

C. PSS states
Taking the squeezing parameter r to be real, define the photon-subtracted squeezed state as |ψ PSS = NâS(r)|0 . The average photon number for the PSS state is The quasiprobability distributions for the PSS state are computed using the convolution of Eq. 22. The bounds were found, and again demonstrate the same characteristics found with the other states. In this case, more negative values of s allow for a larger value of the loss parameter Similarly to how PAC states inherit a displacement, PSS states inherit additional squeezing on the evolved state, and we can use the optimized witness (10) by using additional squeezing operations. In this case, there is a difference between how the Wigner function changes under this squeezing versus how the Q function changes: while the Wigner function at the origin is unchanged by the squeezing operation, the Q function is not invariant and as a result the following argument for the optimisation of the squeezing parameter is only valid for the Wigner function. This proves not to be a problem, and will be discussed below. First, we determine the value q opt that minimises the average photon number of S(q)ρS † (q),

FIG. 5. Loss parameters for PSS states
where µ t = cosh t, ν t = sinh t and for an initial PSS state |ψ PSS , â 2 0 = â †2 0 = 3µ r ν r . In this case the optimal squeezing value can be evaluated analytically: We again follow the format of the PAC state analysis, assigning the squeezing parameter its optimal value q opt and plotting the criterion as a function of . We plot both the Wigner and Q functions for q opt and illustrate that while the q opt used is only optimised for the Wigner function, the maximum noise (b) s for the Q function for this value of s is 1 for all values and therefore already giving the desired result. So even if this is not the optimal squeezing for the Q function, it is sufficient to detect quantum non-Gaussianity by means of the Q function based witness. This feature is illustrated in Fig. 5(b).

VI. ESTIMATION OF ERROR ON THE BOUNDS
While a full error propagation to evaluate the various witnesses is beyond the scope of this paper, it is straightforward to evaluate the bounds B s (n) for the different s-values, based on uncertainty in the mean photon number n. The method of determining the error on the bound is chosen to best approximate the experimental realities. To this end, we suppose that we have a photon number resolving detector with which we would like to measure different average values of n that we assign to a set n avg . These values are a discretised version of the range of n we would like to consider. In an experiment, we need to measure our state k times, for preferably large k. We then define n tot as the total number of photons measured over all k trials, that is: For n tot we assume a Poissonian distribution for simplicity. Before evaluating the means and variances of the bounds we divide by k as we wish to evaluate for a distribution about n avg . Normalising the results so all bounds evaluate to 1 at n avg = 0, we get the distributions in Fig.  6. As we can observe, the errors on the different bounds B s (n) are comparable. This shows how the errors coming from the measurement of the quasi-probability distributions values Q s [ ](α), will probably play a major role when the proposed witnesses will be used in an actual experiment.

VII. CONCLUSIONS
We have presented a general method to derive bounds of linear functionals on the Gaussian convex hull. After having presented the main properties of the bounds, we used it to define QNG witnesses based on s-parametrised quasiprobability distributions, with s < 0. The witnesses are based on bounding from above the average photon number of the quantum state, and measuring the value of the corresponding quasiprobability distribution in a particular point of phase space (typically the origin).
Following the determination of these witnesses, we consider three different states and test the criteria for three different values of s for each state. Motivation to consider the bound for different s-values comes from the freedom it provides to change the type of detection used in experiment. While it is known that s = 0, −1 correspond to the Wigner and Q functions respectively, s = −2 is comparable to measuring the Q function with an inefficient detector. As the inefficiency of the detector can be known from trials using known states, allowing for s < −1 provides a more general description less dependent on the type of detection. From the different states for which the bound was considered we see that there is a region for which a smaller s-value provides a witness for QNG and allows more channel loss than the originally considered Wigner function bound. There is, however, a tradeoff between the maximum amount of loss for which QNG may be witnessed and the amount of violation quantified by the criterion which is generally smaller, for smaller values of s. conclusive results have been obtained. π 1 + s(s − 2 − 4m) (B1) where n = |α| 2 +sinh 2 r is the average number of photons and m = sinh 2 r ≤ n is the squeezing fraction. The condition s < 0 ensures that the expression in Eq. (B1) is real. From this expression, we can prove some general properties of the functions B s (n). Firstly, we notice that B s (n) > 0 for any n ≥ 0. Also, since the only state in G 0 is the vacuum |0 , we have B s (0) = 2 One can also see that lim n→∞ B s (n) = 0 for any s < 0. Then, it is easy to show that B s (n) is strictly decreasing: suppose that n > n but B s ( n) = B s (n). Since B s tends to zero for large n, it is possible to find N > n > n such that B s (N ) < B s ( n), and q ∈ (0, 1) such that qn + (1 − q)N = n. Then one would obtain qB s (n) + (1 − q)B s (N ) ≥ B s ( n) = B s (n), on the other hand qB s (n) + (1 − q)B s (N ) < qB s (n) + (1 − q)B s ( n) = B s (n) thus reaching a contradiction. Finally, we show that the bound B s (n) is achieved by a state with n average photons, that is B s (n) = Q s [ρ n ](0) and Tr[ρ n a † a] = n (for brevity, in what follows we shall omit the phase space argument of Q s , assuming it to be always "(0)"). Assuming that this is not the case, we write B s (n) = Q s [ρ n ], s.t. Tr[ρ n a † a] = n < n. However, one has ρ n ∈ G n ⊂ G n , and as a consequence we reach the absurd conclusion B s (n) = Q s [ρ n ] ≥ B s ( n), which is in contradiction with the strict monotonicity of B s .
We remark that all the properties derived in this section hold for any linear functional whose bound satisfies the properties: (i) B(0) > 0 and (ii) lim n→∞ B(n) = 0.