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arXiv:2608.15237

On the zeros of the stellar representation in the discrete cylinder

Nicolas Fabre

quant-phmath-phSubmitted Aug 15, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

The discrete cylinder phase space offers a natural setting for twisted photons, Josephson junctions, and certain bosonic codes. In that context, we build a number-of-zeros hierarchy for the stellar distribution of states on the discrete cylinder phase space, tied to usefulness for quantum computation. The orbital angular momentum (OAM) and phase eigenstate, having no zeros, sits at the bottom of the hierarchy and share mathematical properties with the coherent state on the plane phase space, both sharing zero rank and a positive Wigner distribution. This reveals a structural peculiarity of the cylinder: its coherent state---the wrapped coherent state---is instead the most quantum, carrying infinitely many zeros, so "coherent state" need not signal classicality across different phase-space geometries. We identify the operations that preserve or change the zero count, giving explicit access to every rank of the hierarchy. We then ask whether this hierarchy also governs localization. Using the $L^r$ norm, the inverse participation ratio, and the Wehrl entropy as complementary measures of spread in the Husimi density, we find that the two hierarchies need not track one another: the wrapped coherent state can be more localized than the OAM eigenstate even though it carries infinitely many zeros.

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The discrete cylinder phase space offers a natural setting for twisted photons, Josephson junctions, and certain bosonic codes. In that context, we build a number-of-zeros hierarchy for the stellar distribution of states on the discrete cylinder phase space, tied to usefulness for quantum computation. The orbital angular momentum (OAM) and phase eigenstate, having no zeros, sits at the bottom of the hierarchy and share mathematical properties with the coherent state on the plane phase space, both sharing zero rank and a positive Wigner distribution. This reveals a structural peculiarity of the cylinder: its coherent state---the wrapped coherent state---is instead the most quantum, carrying infinitely many zeros, so "coherent state" need not signal classicality across different phase-space geometries. We identify the operations that preserve or change the zero count, giving explicit access to every rank of the hierarchy. We then ask whether this hierarchy also governs localization. Using the LrL^rLr norm, the inverse participation ratio, and the Wehrl entropy as complementary measures of spread in the Husimi density, we find that the two hierarchies need not track one another: the wrapped coherent state can be more localized than the OAM eigenstate even though it carries infinitely many zeros.
The so-called stellar formalism allows us to represent the non-Gaussian properties of single-mode quantum states by the distribution of the zeros of their Husimi Q function in phase space. We use this representation in order to derive an infinite hierarchy of single-mode states based on the number of zeros of the Husimi Q function: the stellar hierarchy. We give an operational characterization of the states in this hierarchy with the minimal number of single-photon additions needed to engineer them, and derive equivalence classes under Gaussian unitary operations. We study in detail the topological properties of this hierarchy with respect to the trace norm, and discuss implications for non-Gaussian state engineering, and continuous variable quantum computing.
Covariant integral quantization is developed on the double discrete cylinder phase space, with the torus as configuration space, and yields the Wigner distribution and the stellar representation. This phase space is particularly well suited to translationally invariant states, such as the Gottesman Kitaev Preskill (GKP) states. These states are central to fault-tolerant quantum computing architectures, yet their quantum resources are still under investigation. The stellar representations of ideal GKP logical states, which behave as phase eigenstates of the double discrete cylinder, have no zeros on this phase space, whereas those of their superpositions have infinitely many. The vacuum, although Gaussian, has zeros on this phase space, and the zeros of finitely squeezed GKP states move with the widths of the peaks and of the envelope, and reach this phase space for fine-tuned widths. These features echo the role of the vacuum as the resource that makes GKP-based architectures hard to simulate.
Under the contraction of SU(1,1)SU(1,1)SU(1,1) to the Heisenberg-Weyl group, the zeros of the Husimi function of a state redistribute according to a single scale, and we determine which of them survive. The stellar rank of a single-mode bosonic pure state, the number of these zeros, characterises the Gaussian states on the plane through the vanishing of the rank. On the Poincaré disk, where the same construction applies to the discrete series of SU(1,1)SU(1,1)SU(1,1), it does not, because the zero set leaves a zero-free factor undetermined. We therefore ask how the zero structure behaves in the contraction limit rather than how it should be classified at fixed Bargmann index. The entire difference between the two Hilbert spaces reduces to one weight sequence wk(m)=(2k)m/(2k)mw_k(m)=(2k)^m/(2k)_mwk​(m)=(2k)m/(2k)m​, whose monotonicity in kkk drives every estimate below. The rescaled states form a normal family, every nonzero limit point lies in the Segal-Bargmann space, and those with finitely many zeros are exactly the finite-rank Fock states. The rank is upper semicontinuous, with equality precisely when all zeros scale as O(1/2k)O(1/\sqrt{2k})O(1/2k​), the scale of the coherent-state labels. A two-sided Harnack estimate pins the profile on compact sets. Separately, a family of rank rrr degenerates if and only if the amplitudes on the lowest r+1r+1r+1 weight vectors tend to zero. Mass escapes at the edge of the rescaled disk instead, by a first-order transition of the semiclassical Husimi potential at an explicit critical coupling. Finally, the contraction limit retains exactly the lowest 2r+32r+32r+3 amplitudes, and this number cannot be lowered.
Phase-space representations reveal the geometry of quantum mechanics and distinguish quantum features from classical ones. However, the positivity of a density operator is among the properties of quantum states that phase-space representations capture least directly: deciding whether a phase-space function describes a physical quantum state has so far required global conditions coupling distant points in phase space. Here we show that these conditions can in fact be made local, by considering the Husimi representation in phase space. We establish a complete set of conditions which captures when a phase-space function fff is the Husimi function of a quantum state, with each condition involving only the derivatives of the function fff evaluated at a single phase-space point. We further show that each condition has an equivalent algebraic characterization, which takes the form of the nonnegativity of a determinant of the underlying operator in a displaced Fock basis. As a result, these conditions obey a recurrence relation organizing them into a differential hierarchy, which terminates based on the rank of the operator. Our results thus show that operator rank and positivity are local properties in phase space, reinforcing the role of phase-space representations as a geometric description of quantum mechanics.
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.