Stellar rank under the contraction of SU(1,1) to the Heisenberg-Weyl group
Abstract
Under the contraction of 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), 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)m, whose monotonicity in k 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)), the scale of the coherent-state labels. A two-sided Harnack estimate pins the profile on compact sets. Separately, a family of rank r degenerates if and only if the amplitudes on the lowest r+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+3 amplitudes, and this number cannot be lowered.
Related
- Towards the Geometry of Quantum Optical Measurements: Kraus Trajectories in the Instrumental Weyl-Heisenberg Group Manifold
- Learning to Rank Tensor Network Contraction Plans for GPU-Accelerated Quantum Circuit Simulation
- Evidence of valence bond solid and quantum paramagnetism in the J1--J2--J3 Heisenberg model from pseudofermion functional renormalization group
- Witnessing and quantifying stellar rank with the Wigner function
- Optimal stellar rank approximation of squeezed cat states with photon catalysis
BibTeX
@misc{kam2026stellar,
title = {{Stellar rank under the contraction of SU(1,1) to the Heisenberg-Weyl group}},
author = {Chon-Fai Kam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00133},
eprint = {2610.00133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00133},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stellar rank under the contraction of SU(1,1) to the Heisenberg-Weyl group
AU - Chon-Fai Kam
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00133
UR - https://arxiv.org/abs/2610.00133
AB - Under the contraction of 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), 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 w_k(m)=(2k)^m/(2k)_m, whose monotonicity in k 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)), the scale of the coherent-state labels. A two-sided Harnack estimate pins the profile on compact sets. Separately, a family of rank r degenerates if and only if the amplitudes on the lowest r+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+3 amplitudes, and this number cannot be lowered.
ER -