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MiS Preprint
6/2012

Continuity of the Maximum-Entropy Inference

Stephan Weis

Abstract

We study the inverse problem of inferring the state of a finite-level quantum system from expected values of a fixed set of observables, by maximizing a continuous ranking function. We have proved earlier that the maximum-entropy inference can be a discontinuous map from the convex set of expected values to the convex set of states because theimage contains states of reduced support, while this map restricts to a smooth parametrization of a Gibbsian family of fully supported states. Here we prove for arbitrary ranking functions that the inference is continuous up to boundary points. This follows from a continuity condition in terms of the openness of the restricted linear map from states to their expected values. The openness condition shows also that ranking functions with a discontinuous inference are typical. Moreover it shows that the inference is continuous in the restriction to any polytope which implies that a discontinuity belongs to the quantum domain of non-commutative observables and that a geodesic closure of a Gibbsian family equals the set of maximum-entropy states. We discuss eight descriptions of the set of maximum-entropy states with proofs of accuracy and an analysis of deviations.

Received:
Feb 23, 2012
Published:
Mar 1, 2012
MSC Codes:
62F30, 54C10, 52A05, 81P16, 94A17, 54A10
Keywords:
inference under constraints, continuous, open, maximum-entropy inference, exponential family, information topology, information projection

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