Stability of Maximum-Entropy Inference in Finite Dimensions
James Tian
TL;DR
This work analyzes maximum-entropy inference for finite-dimensional quantum states under linear moment constraints, proving existence and uniqueness of the Gibbs-maximum-entropy state on the feasible set and establishing a convex-dual framework via the log-partition function. It delivers a convergence principle: if moment data converge to the target and the entropy matches the max-entropy value, the inferred state converges in trace norm, with stability preserved under unital CPTP post-processing. The authors derive explicit, dimension-free quantitative bounds linking moment deviations and entropy gaps to the trace distance and to observable deviations on the constraint subspace, enabling practical error control in finite-dimensional quantum inference. Collectively, the results provide a rigorous, self-contained foundation for robust maximum-entropy inference in quantum information and data-driven settings, applicable to experiment-driven state estimation and quantum learning tasks.
Abstract
We study maximum-entropy inference for finite-dimensional quantum states under linear moment constraints. Given expectation values of finitely many observables, the feasible set of states is convex but typically non-unique. The maximum-entropy principle selects the Gibbs state that agrees with the data while remaining maximally unbiased. We prove that convergence of moments and entropy implies convergence of states in trace norm, derive explicit quantitative bounds linking data and entropy deviations to state distance, and show that these results are stable under unital completely positive maps. The analysis is self-contained and relies on convex duality, relative entropy, and Pinsker-type inequalities, providing a rigorous and unified foundation for finite-dimensional maximum-entropy inference.