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 $x_i + y_j$. 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.