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Some topological properties of the sets of non-negative Wigner functions

N. C. Dias

Published Nov 13, 2024• 0 citations
OpenAlex

Abstract

The set of non-negative Wigner functions is a convex set which, in the finite dimensional case, is compact and equal to the convex hull of its extreme points. Thus, in finite dimensions, these particular (extreme) Wigner functions can be used to generate the entire set of non-negative Wigner functions. We show that the extreme Wigner functions can be identified by the properties of their null sets, and discuss the problem of constructing these states explicitly. As a by-product, we also elaborate on the properties of the interior and the boundary of the set of non-negative Wigner functions. Finally, if time permits, we will discuss the difficulties of extending some of these results to the infinite dimensional case.

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The set of non-negative Wigner functions is a convex set which, in the finite dimensional case, is compact and equal to the convex hull of its extreme points. Thus, in finite dimensions, these particular (extreme) Wigner functions can be used to generate the entire set of non-negative Wigner functions. We show that the extreme Wigner functions can be identified by the properties of their null sets, and discuss the problem of constructing these states explicitly. As a by-product, we also elaborate on the properties of the interior and the boundary of the set of non-negative Wigner functions. Finally, if time permits, we will discuss the difficulties of extending some of these results to the infinite dimensional case.
For Hilbert spaces H⊆L2(R)\mathcal H\subseteq L^2(\mathbb R)H⊆L2(R) we consider the convex sets D+(H)\mathcal D_+(\mathcal H)D+​(H) of Wigner-positive states (WPS), i.e.~density matrices over H\mathcal HH with non-negative Wigner function. We investigate the topological structure of these sets, namely concerning closure, compactness, interior and boundary (in a relative topology induced by the trace norm). We also study their geometric structure and construct minimal sets of states that generate D+(H)\mathcal D_+(\mathcal H)D+​(H) through convex combinations. If H\mathcal HH is finite-dimensional, the existence of such sets follows from a central result in convex analysis, namely the Krein-Milman theorem. In the infinite-dimensional case H=L2(R)\mathcal H=L^2(\mathbb R)H=L2(R) this is not so, due to lack of compactness of the set D+(H)\mathcal D_+(\mathcal H)D+​(H). Nevertheless, we prove that a Krein-Milman theorem holds in this case, which allows us to extend most of the results concerning the sets of generators to the infinite-dimensional setting. Finally, we study the relation between the finite and infinite-dimensional sets of WPS, and prove that the former provide a hierarchy of closed subsets, which are also proper faces of the latter. These results provide a basis for an operational characterisation of the extreme points of the sets of WPS, which we undertake in a companion paper. Our work offers a unified perspective on the topological and geometric properties of the sets of WPS in finite and infinite dimensions, along with explicit constructions of minimal sets of generators.
Providing an operational characterization of the Wigner-positive states (WPS), i.e., the set of quantum states with non-negative Wigner function, is a longstanding open problem. For pure states, the only WPS are Gaussian states, but the situation is considerably more subtle for mixed states. Here, we approach the problem using convex geometry, reducing the question to the characterization of the extreme points of the set of WPS. We give a constructive method to generate a large class of such extreme WPS, which combines the following steps: (i) we characterize the phase-invariant extreme points of the superset of Wigner-positive quasi-states (WPQS); (ii) we introduce a new quantum map, named Vertigo map, which maps extreme WPQS to extreme WPS while preserving phase invariance; (iii) we identify families of extremality-preserving maps and use them to obtain extreme WPS while relaxing phase invariance. Our construction generates all extreme WPS of low dimension, starting from a specific kind of WPS known as beam-splitter states. Our results build upon new mathematical properties of the set of WPS derived in a companion paper and unveil the remarkable structure of mixed states with non-negative Wigner functions.
We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth by Van Herstraeten and Cerf [Phys. Rev. A 104, 042211 (2021)] and Hertz et al. [J. Phys. A: Math. Theor. 50, 385301 (2017)]. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to $1+ln\ensuremath{\pi}$. We prove this conjecture for the qubits formed by Fock states $|0\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}$ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for the conjecture's validity. Lastly, we elaborate on the states which are in accordance with our condition.
The bound states of a particle confined in a one-dimensional finite square well cannot be solved analytically, since the eigen-energies are determined by transcendental equations. Here, we numerically calculate the bound states and show their non-classical properties, using Wigner's quasi-probability distribution (also called the Wigner functions) in the phase space (x, p). In contrast to the infinite-well case, we find that the Wigner functions spread over the space dimension x, get squeezed along the momentum dimension p, and show negativity outside the well. Negativity in a Wigner function indicates non-classical properties of the bound states.
Abstract We consider the nonclassicality distance indicator of a state in finite-dimensional quantum systems which is evaluating a state nonclassicality by its remoteness from the set of “classical states”. The latter are identified with those states whose Wigner function is non-negative. The corresponding Wigner function’s positivity polytope in the simplex of qudit eigenvalues is introduced and the representation for the nonclassicality distance indicator as a piecewise function is derived. The results are exemplified by the qutrit case.
Quasiprobability has become an increasingly popular notion for characterising non-classicality in quantum information, thermodynamics, and metrology. Two important distributions with non-positive quasiprobability are the Wigner function and the Glauber-Sudarshan function. Here we study properties of the spin Wigner function for finite-dimensional quantum systems and draw comparisons with its infinite-dimensional analog, focusing in particular on the relation to the Glauber-Sudarshan function and the existence of absolutely Wigner-bounded states. More precisely, we investigate unitary orbits of mixed spin states that are characterized by Wigner functions lower-bounded by a specified value. To this end, we extend a characterization of the set of absolutely Wigner positive states as a set of linear eigenvalue constraints, which together define a polytope centred on the maximally mixed state in the simplex of spin- j states. The lower bound determines the relative size of such absolutely Wigner bounded (AWB) polytopes and we study their geometric characteristics. In each dimension a Hilbert-Schmidt ball representing a tight purity-based sufficient condition to be AWB is exactly determined, while another ball representing a necessary condition to be AWB is conjectured. Special attention is given to the case where the polytope separates orbits containing only positive Wigner functions from other orbits because of the use of Wigner negativity as a witness of non-classicality. Comparisons are made to absolute symmetric state separability and spin Glauber-Sudarshan positivity, with additional details given for low spin quantum numbers.