Marginal spectral distributions on regular bipartite unitary orbits
Abstract
Fix the spectrum of a bipartite density matrix and randomize its eigenbasis according to Haar measure. We study the probability distributions induced on the spectra of the two marginal states. For arbitrary subsystem dimensions m and n, the joint characteristic function of the reduced density matrices is expressed as a Harish-Chandra-Itzykson-Zuber integral whose external eigenvalues are the pairwise sums xi + yj. Repeated external eigenvalues are handled by confluent determinant limits. In the two-qubit case, we derive an explicit alternating-spline formula for the joint density of the two marginal Bloch radii. Its support is the Bravyi-Klyachko compatibility region. We also obtain a compact truncated-power formula for the Bloch-radius density of either individual qubit marginal. In the qubit-qutrit case, we derive a truncated-power formula for the qubit Bloch-radius density and a bivariate spline formula for the joint density of the largest and smallest eigenvalues of the qutrit marginal. The latter two variables determine the full qutrit spectrum because the trace is fixed. The derivations combine confluent HCIZ integrals, distributional Fourier inversion, orbital measures, and the SU(2) and SU(3) derivative principles. The resulting densities are piece-wise polynomial on chambers determined by subset sums of the fixed global eigenvalues, in agreement with the Duistermaat-Heckman description of projected coadjoint-orbit measures.
Related
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- Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes
- Perfect Partitions of Regular Bipartite Graphs
BibTeX
@misc{zhang2026marginal,
title = {{Marginal spectral distributions on regular bipartite unitary orbits}},
author = {Lin Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30370},
eprint = {2608.30370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30370},
}
RIS
TY - UNPB
TI - Marginal spectral distributions on regular bipartite unitary orbits
AU - Lin Zhang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30370
UR - https://arxiv.org/abs/2608.30370
AB - Fix the spectrum of a bipartite density matrix and randomize its eigenbasis according to Haar measure. We study the probability distributions induced on the spectra of the two marginal states. For arbitrary subsystem dimensions m and n, the joint characteristic function of the reduced density matrices is expressed as a Harish-Chandra-Itzykson-Zuber integral whose external eigenvalues are the pairwise sums x_i + y_j. Repeated external eigenvalues are handled by confluent determinant limits. In the two-qubit case, we derive an explicit alternating-spline formula for the joint density of the two marginal Bloch radii. Its support is the Bravyi-Klyachko compatibility region. We also obtain a compact truncated-power formula for the Bloch-radius density of either individual qubit marginal. In the qubit-qutrit case, we derive a truncated-power formula for the qubit Bloch-radius density and a bivariate spline formula for the joint density of the largest and smallest eigenvalues of the qutrit marginal. The latter two variables determine the full qutrit spectrum because the trace is fixed. The derivations combine confluent HCIZ integrals, distributional Fourier inversion, orbital measures, and the SU(2) and SU(3) derivative principles. The resulting densities are piece-wise polynomial on chambers determined by subset sums of the fixed global eigenvalues, in agreement with the Duistermaat-Heckman description of projected coadjoint-orbit measures.
ER -