Bivariate probability densities with given margins

Ankit Kumar Jha, Ion Nechita

Abstract In this paper, we study random instances of the classical marginal problem. We encode the problem in a graph, where the vertices have assigned fixed binary probability distributions, and edges have assigned random bivariate distributions having the incident vertex distributions as marginals. We provide estimates on the probability that a joint distribution on the graph exists, having the bivariate edge distributions as marginals. Our study is motivated by Fine’s theorem in quantum mechanics. We study in great detail the graphs corresponding to CHSH and Bell–Wigner scenarios providing rations of volumes between the local and non-signaling polytopes.

Leon W. Cohen

Fan Hong-Yi, Lou Sen‐Yue

We introduce bivariate normal distribution operator for state vector |ψ〉 and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability | λ,ν 〈x|·ψ〉| 2 , where |x〉 λ,ν is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.

Lin Zhang

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.

Watanabe, Shinzo, Prokhorov, Jurii

Nguyen Tang