Extending Hudson’s theorem to mixed quantum states

Abstract

According to Hudson's theorem, any pure quantum state with a positive Wigner function is necessarily a Gaussian state. Here, we make a step toward the extension of this theorem to mixed quantum states by finding upper and lower bounds on the degree of non-Gaussianity of states with positive Wigner functions. The bounds are expressed in the form of parametric functions relating the degree of non-Gaussianity of a state, its purity, and the purity of the Gaussian state characterized by the same covariance matrix. Although our bounds are not tight, they permit us to visualize the set of states with positive Wigner functions. The Wigner representation of quantum states ͓1͔, which is realized by joint quasiprobability distributions of canonically conjugate variables in phase space, has a specific property which differentiates it from a true probability distribution: it can attain negative values. Among pure states, it was proven by Hudson ͓2͔ ͑and later generalized to multimode quantum systems by Soto and Claverie ͓3͔͒ that the only states which have non-negative Wigner functions are Gaussian states ͓4͔. The question that naturally arises ͓2͔ is whether this theorem can be extended to mixed states, among which not only Gaussian states may possess a positive Wigner function. A logical extension of the theorem would be a complete characterization of the convex set of states with positive Wigner function. Although this question can be approached by using the notion of Wigner spectrum ͓5͔, a simple and operational extension of Hudson's theorem has not yet been achieved due to the mathematical complications which emerge when dealing with states with positive Wigner functions ͓5͔.Motivated by the increasing interest for non-Gaussian states in continuous-variable quantum information theory ͑see, e.g., ͓6͔͒ and the need for a better understanding of the de-Gaussification procedures for mixed states ͑see, e.g., ͓7͔͒, we attempt here an exploration of the set of states with positive Wigner functions using Gaussian states as a reference. More precisely, we consider the subset of such states that have the same covariance matrix as a reference Gaussian state. We obtain a partial solution to the problem by analytically deriving necessary conditions ͑bounds͒ on a measure of non-Gaussianity for a state to have a positive Wigner function. This set of conditions bounds a region in a threedimensional space with coordinates being the purity of the state, the purity of the corresponding Gaussian state, and the non-Gaussianity. As intuitively expected, the maximum degree of non-Gaussianity increases with a decrease in the purity of both the state and its Gaussian corresponding state.Before deriving the main results of this paper, let us recall a convenient representation of the trace of the product of two one-mode quantum states, and Ј, in terms of the Wigner representation ͓8͔,where W is the Wigner function of the state . For example, the purity of a state, ͓͔ =Tr͑ 2 ͒, may be calculated with the help of this formula. For a state with a G...

Quantifying non-Gaussianity for quantum information

“…The same conclusion is indirectly suggested by the results obtained in [83] and this leads to formulate the following Conjecture A5 For single-mode quantum states we have that…”

“…These measures have been used to assess the role of nG in different quantum information and communication tasks as teleportation [79], quantum estimation [31], experimental entanglement quantification [80] and entanglement transfer between CV states and qubits [81,82]. In [83] the relationship between nG and the Hudson's theorem [84] have been studied, obtaining at fixed purity an upper bound for non-Gaussian states having a positive Wigner function, while in [85] nG bounded uncertainty relations are derived. The entropic measure proposed in [77] has been used to quantify exactly the nG of experimentally produced photon-added coherent states and a lower bound has been evaluated experimentally in [86] for conditional states obtained via an inefficient photo-detection on classically correlated thermal beams.…”

Quantum uncertainty relation saturated by the eigenstates of the harmonic oscillator

“…Our method works by reduction to a constrained optimization problem (even for solving the unconstrained SR inequality), so it can be simply adapted to find the MS with an extra constraint on Gaussianity. Several measures of non-Gaussianity have been used in the literature [15,16,[33][34][35]], but we instead suggest using a parameter g capturing the degree of Gaussianity, inspired from our former work on non-Gaussian states with a positive Wigner function [36,37]. Denoting asρ G the Gaussian state that has the same covariance matrix γ (and same mean values x and p ) as stateρ, we define the Gaussianity ofρ as…”

Unconditional generation of bright coherent non-Gaussian light from exciton-polariton condensates

“…While simple, elementary quantum protocols such as teleportation [6] can be realized using Gaussian continuous-variable (CV) states and Gaussian operations, any more advanced application such as universal quantum computation would require some non-Gaussian elements [7][8][9]. Moreover, even non-Gaussian states may be insufficient as resources for quantum computation: trivially, those which are expressible as mixtures of Gaussian states; however, as well those which are not representable as mixtures of Gaussian states and yet possess a positive Wigner function [10,11] (for the discrete analogue, see [12]). As a consequence, the occurrence of a negative Wigner function can be seen as a prerequisite for a potential quantum mechanical speed-up [13] (for related results on the discrete case, see [14]).…”