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arXiv:2206.00195 · Journal of Physics A Mathematical and Theoretical

Stellar representation of extremal Wigner-negative spin states

J. Davis, Robie A. Hennigar, Robert B. Mann, Shohini Ghose

Published Jun 8, 2023• 3 citations
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Abstract

Abstract The Majorana stellar representation is used to characterize spin states that have a maximally negative Wigner quasiprobability distribution on a spherical phase space. These maximally Wigner-negative spin states generally exhibit a partial but not high degree of symmetry within their star configurations. In particular, for spin j > 2, maximal constellations do not correspond to a Platonic solid when available and do not follow an obvious geometric pattern as dimension increases. In addition, they are generally different from spin states that maximize other measures of nonclassicality such as anticoherence or geometric entanglement. Random states display on average a relatively high amount of negativity, but the extremal states and those with similar negativity are statistically rare in Hilbert space. We also prove that all spin coherent states of arbitrary dimension have non-zero Wigner negativity. This offers evidence that all pure spin states also have non-zero Wigner negativity. The results can be applied to qubit ensembles exhibiting permutation invariance.

DOI
10.1088/1751-8121/acd918

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Abstract The Majorana stellar representation is used to characterize spin states that have a maximally negative Wigner quasiprobability distribution on a spherical phase space. These maximally Wigner-negative spin states generally exhibit a partial but not high degree of symmetry within their star configurations. In particular, for spin j > 2, maximal constellations do not correspond to a Platonic solid when available and do not follow an obvious geometric pattern as dimension increases. In addition, they are generally different from spin states that maximize other measures of nonclassicality such as anticoherence or geometric entanglement. Random states display on average a relatively high amount of negativity, but the extremal states and those with similar negativity are statistically rare in Hilbert space. We also prove that all spin coherent states of arbitrary dimension have non-zero Wigner negativity. This offers evidence that all pure spin states also have non-zero Wigner negativity. The results can be applied to qubit ensembles exhibiting permutation invariance.
The nonclassicality of simple spin systems as measured by Wigner negativity is studied on a spherical phase space. Several SU(2)-covariant states with common qubit representations are analyzed: spin coherent states, spin cat (Greenberger-Horne-Zeilinger or N00N) states, and Dicke (W\mathsf{\text{W}}W) states. We derive a bound on the Wigner negativity of spin cat states that rapidly approaches the true value as spin increases beyond $j\ensuremath{\gtrsim}5$. We find that spin cat states are not significantly Wigner negative relative to their Dicke state counterparts of equal dimension. We also find, in contrast to several entanglement measures, that the most Wigner-negative Dicke basis element is spin dependent, and not the equatorial state $|j,0\ensuremath{\rangle}$ (or $|j,\ifmmode\pm\else\textpm\fi{}1/2\ensuremath{\rangle}$ for half-integer spins). These results underscore the influence that dynamical symmetry has on nonclassicality and suggest a guiding perspective for finding novel quantum computational applications.
Abstract Majorana’s stellar representation provides an intuitive picture in which quantum states in high-dimensional Hilbert space can be observed using the trajectory of Majorana stars. We consider the Majorana’s stellar representation of the quantum geometric tensor for a spin state up to spin-3/2. The real and imaginary parts of the quantum geometric tensor, corresponding to the quantum metric tensor and Berry curvature, are therefore obtained in terms of the Majorana stars. Moreover, we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases. Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana’s stars.
The Majorana stellar representation maps a pure spin-SSS state to 2S2S2S points on a sphere. This review develops it with entanglement as the organising principle, and two objects recur throughout: the constellation, and the permanent of the Gram matrix of its stars. The degeneracy pattern of the constellation is invariant under stochastic local operations and classical communication, so the integer partitions of NNN label a finite set of families of symmetric NNN-qubit states. That labelling is a coarse-graining rather than a classification, since from four distinct stars onwards each family carries continuous Möbius moduli. The permanent supplies what the pattern omits. Normalised by it, inter-star chordal distances give the concurrence and the three-tangle in closed form, and the same permanent governs the anomalous contribution to the Berry phase acquired under adiabatic cyclic evolution, so that a single quantity links static correlations to dynamical holonomy. We also fix the computational reach of the geometry. Overlaps of symmetric states are permanents of matrices of rank at most two and are polynomially computable, whereas measures defined by an optimisation over the sphere are not reached by that argument. Interest in stellar representations has resurged, but the literature remains dispersed, and no existing treatment develops the link between constellation geometry, multipartite entanglement, and geometric phases within a single framework. The same two objects organise the applications reviewed here, from extremal states in metrology and permutation-invariant codes to collective spin models and photonic constellations, together with extensions to mixed states and to continuous-variable systems through the stellar rank. Whether the anomalous phase admits a bound in terms of any entanglement monotone remains open.
Spin anticoherent states acquired recently a lot of attention as the most "quantum" states. Some coherent and anticoherent spin states are known as optimal quantum rotosensors. In this work, we introduce a measure of quantumness for orthonormal bases of spin states, determined by the average anticoherence of individual vectors and the Wehrl entropy. In this way, we identify the most coherent and most quantum states, which lead to orthogonal measurements of extreme quantumness. Their symmetries can be revealed using the Majorana stellar representation, which provides an intuitive geometrical representation of a pure state by points on a sphere. Results obtained lead to maximally (minimally) entangled bases in the 2j+1 dimensional symmetric subspace of the 22j dimensional space of states of multipartite systems composed of 2j qubits. Some bases found are iso-coherent as they consist of all states of the same degree of spin-coherence.
Quasiprobability distributions for systems endowed with SU(1,1) dynamical symmetry have received surprisingly little attention, despite the central role of this symmetry in two-photon physics, squeezed states, and nonlinear interferometry. Here, we fill this gap by constructing a full covariant family of sss-ordered quasiprobability distributions defined on the two-sheeted hyperboloid, or equivalently, on the Poincaré unit disk via stereographic projection. A key result is that the Wigner function is strictly positive for all Perelomov SU(1,1) coherent states, in sharp contrast to the SU(2) case. This positivity endows Wigner negativity with an unambiguous operational meaning: any negative volume is a direct signature of genuinely quantum behavior. We further examine the stellar rank of SU(1,1) states, defined through the zeros of the Husimi QQQ-function, and show how it compares with Wigner negativity as a geometry-adapted witness of nonclassicality in this setting. We further introduce a hierarchy of generalized multipoles through a harmonic expansion of the density operator on the hyperboloid, providing a complementary framework for probing quantumness. This offers a comprehensive toolkit for characterizing and quantifying quantum resources in SU(1,1) systems.
A constellation of $N=d\ensuremath{-}1$ Majorana stars represents an arbitrary pure quantum state of dimension ddd or a permutation-symmetric state of a system consisting of nnn qubits. We generalize the latter construction to represent in a similar way an arbitrary symmetric pure state of kkk subsystems with ddd levels each. For $d\ensuremath{\ge}3$, such states are equivalent, as far as rotations are concerned, to a collection of various spin states, with definite relative complex weights. Following Majorana's lead, we introduce a multiconstellation, consisting of the Majorana constellations of the above spin states, augmented by an auxiliary, ``spectator'' constellation, encoding the complex weights. Examples of stellar representations of symmetric states of four qutrits, and two spin-3/2 systems, are presented. We revisit the Hermite and Murnaghan isomorphisms, which relate multipartite states of various spins, number of parties, and even symmetries. We show how the tools introduced can be used to analyze multipartite entanglement and to identify optimal quantum rotosensors, i.e., pure states which are maximally sensitive to rotations around a specified axis, or averaged over all axes.