Continuity of the Maximum-Entropy Inference

An Erratum to this article was published on 29 July 2014


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 the image 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.

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Correspondence to Weis Stephan.

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Dedicated to Prof. Eyvind H. Wichmann on his 85th birthday

Communicated by A. Winter

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Stephan, W. Continuity of the Maximum-Entropy Inference. Commun. Math. Phys. 330, 1263–1292 (2014).

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  • Convex Body
  • Relative Entropy
  • Ranking Function
  • Exponential Family
  • Geodesic Closure