Exact probability laws of random inner products: Dirichlet splines, overlaps, uncertainty, and Bargmann invariants
Abstract
Inner products of random matrices and random quantum states---such as orbital inner products, expectation values, overlaps, and cyclic Bargmann invariants---are shown to follow from two underlying mechanisms: the diagonal of an induced random state is Dirichlet distributed, and the expectation value of a random pure state in a rank-one or two-by-two operator is a numerical shadow on an elliptical disk. For states induced by an environment of dimension K, the expectation value of any observable is a Curry--Schoenberg spline with each eigenvalue repeated K times, and the vector of moments has an explicit density on a simplex; for a qutrit, the variance law is obtained in finite form. The overlap of two random qubit states attains its mode at its mean for every rotation-invariant ensemble, but for Hilbert--Schmidt qutrits the density at the mean has a strictly negative derivative, so mode and mean separate; for large environments, the centred overlap converges to a variance-gamma law rather than a Gaussian. For two noncommuting qubit observables, the joint law of the two standard deviations is derived for all K, and the probability of coming within a small distance of the optimal sum of variances scales as a power of that distance, with a different exponent when the Bloch axes are orthogonal. For three Haar-random qubit states, the exact density of the Bargmann invariant is given in terms of a complete elliptic integral, together with its local behaviour and elementary marginals. In every dimension, the probability that its real part is negative---certifying basis-independent coherence---is computed, along with all mixed moments. For cycles of arbitrary length, the invariant is shown to be driven by an exact three-parameter Markov chain, and the logarithm of its modulus obeys a law of large numbers and a central limit theorem.
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BibTeX
@misc{zhang2026exact,
title = {{Exact probability laws of random inner products: Dirichlet splines, overlaps, uncertainty, and Bargmann invariants}},
author = {Lin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25196},
eprint = {2610.25196},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25196},
}
RIS
TY - UNPB
TI - Exact probability laws of random inner products: Dirichlet splines, overlaps, uncertainty, and Bargmann invariants
AU - Lin Zhang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25196
UR - https://arxiv.org/abs/2610.25196
AB - Inner products of random matrices and random quantum states---such as orbital inner products, expectation values, overlaps, and cyclic Bargmann invariants---are shown to follow from two underlying mechanisms: the diagonal of an induced random state is Dirichlet distributed, and the expectation value of a random pure state in a rank-one or two-by-two operator is a numerical shadow on an elliptical disk. For states induced by an environment of dimension K, the expectation value of any observable is a Curry--Schoenberg spline with each eigenvalue repeated K times, and the vector of moments has an explicit density on a simplex; for a qutrit, the variance law is obtained in finite form. The overlap of two random qubit states attains its mode at its mean for every rotation-invariant ensemble, but for Hilbert--Schmidt qutrits the density at the mean has a strictly negative derivative, so mode and mean separate; for large environments, the centred overlap converges to a variance-gamma law rather than a Gaussian. For two noncommuting qubit observables, the joint law of the two standard deviations is derived for all K, and the probability of coming within a small distance of the optimal sum of variances scales as a power of that distance, with a different exponent when the Bloch axes are orthogonal. For three Haar-random qubit states, the exact density of the Bargmann invariant is given in terms of a complete elliptic integral, together with its local behaviour and elementary marginals. In every dimension, the probability that its real part is negative---certifying basis-independent coherence---is computed, along with all mixed moments. For cycles of arbitrary length, the invariant is shown to be driven by an exact three-parameter Markov chain, and the logarithm of its modulus obeys a law of large numbers and a central limit theorem.
ER -