Continuity of the Maximum-Entropy Inference
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Submission date: 23. Feb. 2012 (revised version: May 2014)
published as: Continuity of the maximum-entropy inference.
In: Communications in mathematical physics, 330 (2014) 3, p. 1263-1292
DOI number (of the published article): 10.1007/s00220-014-2090-1
published as: Erratum to: Continuity of the maximum-entropy inference.
In: Communications in mathematical physics, 331 (2014) 3, p. 1301
DOI number (of the published article): 10.1007/s00220-014-2125-7
MSC-Numbers: 62F30, 54C10, 52A05, 81P16, 94A17, 54A10
Keywords and phrases: inference under constraints, continuous, open, maximum-entropy inference, exponential family, information topology, information projection
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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.