Quantum mutual information statistics for detecting dependence-structure change points in time series

Abstract

Detecting when the dependence between two components of a multivariate time series changes, while the marginals drift freely, requires a dependence-specific statistic. We take the inferential object to be a density operator -- the trace-normalised second moment of unit-norm random Fourier features of ranks -- rather than a probability distribution. Partial traces recover the marginal operators exactly, so von Neumann entropies yield a quantum mutual information (QMI) statistic computed from prefix sums of small matrices, without density estimation, matrix inversion, or a tuned parameter. We develop the inference it needs: a segment-separable cost that drives penalised optimal partitioning, its split gain a Holevo information; finite-sample exact calibration by joint pair permutation, a block-permutation form for serially dependent series, and an exact, provably consistent exchangeability diagnostic that selects between them. We also prove a weighted chi-square boundary law, at the segment length and not its square root, for the rank-based statistic exactly as computed. In 500 replicates QMI detects nonlinear, correlation-free dependence changes with more power than the Hilbert-Schmidt independence criterion, distance correlation, Spearman, and empirical-copula statistics on the same ranks, by at least 15 percentage points wherever any statistic detects the change. Its false-alarm rate stays near nominal under marginal drift, where the empirical-copula statistic reaches 0.87. On eight years of hourly Korean weather observations, a two-stage segment-and-certify procedure finds dependence-change candidates above chance (five of 27 at p $\le$ 0.05 against 1.4 expected); stage two certifies one as a pure coupling change and reclassifies eight as marginal-driven.