In this work, we make a step towards the extension of the Hudson's theorem to mixed 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, the purity of the states and the purity of Gaussian states determined by the same covariance matrix as of these states. Even though the bounds are not tight, they permit for a preliminary visualization of the space of states with positive Wigner functions and for the derivation of a bound on the purity of a state with strictly positive Wigner function and a given covariance matrix.
In this work, we make a step towards the extension of the Hudson's theorem to mixed 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, the purity of the states and the purity of Gaussian states determined by the same covariance matrix as of these states. Even though the bounds are not tight, they permit for a preliminary visualization of the space of states with positive Wigner functions and for the derivation of a bound on the purity of a state with strictly positive Wigner function and a given covariance matrix.
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.
In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.
We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value 1+lnπ. Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound πW≤1, we construct an explicit hierarchy of lower bounds Bn on S[W] by combining a truncated series lower bound for −lnx with moment identities of the Wigner function. This yields closed-form sufficient conditions, expressed in terms of the state purity μ:=Tr(ρ2)=2π∫dqdpW(q,p)2, ensuring S[W]≥1+lnπ. In particular, we first prove that all physical Wigner-non-negative states with μ≤4−23 satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, whose limiting sufficient condition is μ≤2/e. Finally, we show that the threshold 2/e is sharp under the relaxed constraints considered here, thereby identifying the need for additional quantum-realizability information in the remaining high-purity regime.
Hudson's theorem states that a pure state of a bosonic mode has a non-negative Wigner function if and only if it is Gaussian. It underwrites the reading of Wigner negativity as a faithful signature of pure-state non-classicality. We show the statement has no analogue on curved phase space. For the positive discrete series of SU(1,1), realised on the upper sheet of a two-sheeted hyperboloid, the Wigner-positive pure states form a strictly larger set than the Perelomov coherent orbit. Superpositions of the lowest weight state with the first excited state stay positive up to a mixing angle of 24.93∘ at Bargmann index k=1, and the admissible set has positive volume, with a maximal width that is not attained in the two-state direction. We show analytically that the quadratic form controlling positivity degenerates in the far field onto a single mixing angle. That degeneracy bounds the window by arctan(1/2k) and leaves the threshold itself fixed at intermediate hyperbolic distance.
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.