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arXiv:1111.1364 · Journal of Mathematical Physics

Quantal effects and MaxEnt

Federico Holik, Angelo Plastino

Published Jul 1, 2012• 7 citations
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Abstract

Convex operational models (COMs) are considered as great extrapolations to larger settings of any statistical theory. In this article, we generalize the maximum entropy principle (MaxEnt) of Jaynes’ to any COM. After expressing MaxEnt in a geometrical and lattice theoretical setting, we are able to cast it for any COM. This scope-amplification opens the door to a new systematization of the principle and sheds light into its geometrical structure.

DOI
10.1063/1.4731769

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Convex operational models (COMs) are considered as great extrapolations to larger settings of any statistical theory. In this article, we generalize the maximum entropy principle (MaxEnt) of Jaynes’ to any COM. After expressing MaxEnt in a geometrical and lattice theoretical setting, we are able to cast it for any COM. This scope-amplification opens the door to a new systematization of the principle and sheds light into its geometrical structure.
Jaynes' maximum entropy (MaxEnt) principle selects, from all probability distributions consistent with a set of constraints, the one that maximizes the Shannon entropy. The principle is well-defined for classical, real-valued probabilities, but extending it to complex-valued probability theories requires care. Youssef's complex probability framework, in which quantum mechanics is reformulated as a Bayesian theory with complex amplitudes and the observable probability is the squared modulus (Born rule), does not admit a natural MaxEnt extension: the L2 normalization of that framework is incompatible with the L1 normalization assumed by Shannon entropy, and the resulting complex-valued entropy functional cannot be maximized without additional structure absent from Youssef's axioms. The present paper proposes a different extension of Kolmogorov's axioms to the complex domain, replacing non-negativity with the condition that the real part of every probability be non-negative. This framework, which retains L1 normalization and recovers classical probability when the imaginary part vanishes, admits a natural MaxEnt principle: the real part of the complex Shannon entropy is a well-posed real functional, its Euler-Lagrange equations produce an explicit complex Gibbs distribution with complex Lagrange multipliers, and the classical Gibbs distribution is recovered as the special case of real multipliers. The die problem (Jaynes' Brandeis dice example) is worked out in detail for both the fair and the loaded cases, illustrating how imaginary constraints redistribute the observable probability mass in a way that has no counterpart in classical MaxEnt.
The MAXENT (maximum entropy) principle gives a natural correspondance between (classical and quantum) statistical mechanics and information theory. Its general formulation leverages on log-concavity (or Klein's inequality) and the celebrated information inequality, which is equivalent to Gibbs' inequality. The resulting convex optimization allows one to relate entropy to the log-canonical partition function by Legendre-Fenchel duality. This can be further generalized to dual variational characterization of relative entropy or Kullback-Leibler divergence, relating Gibbs variationnal principle to Donsker-Vardhan variational representations.From an information theoretical point of view, the Gibbs principle and its associated evidence lower bound serves as the basis of modern machine learning algorithms, such as the expectation-maximization (EM) algorithm and its restriction to parameters output by a neural network. Interestingly, a large class of alternating optimization algorithms including EM and Blahut-Arimoto for computing channel capacity and source rate-distortion functions can be cast in a generic information geometric alternating projections on convex sets of measures. On the other hand, Donsker-Vardhan representations, when restricted to the output of a neural network, allows one to efficiently estimate information measures, and when applied to parametric estimators with quadratic risk, provide HCR-like and CRB-like statistical lower bounds. An alternative derivation on such lower bounds in the Bayesian setting makes use of the Weyl-Heisenberg uncertainty principle, which opens up new perspectives.In this keynote, I will select and review several of the aforementioned topics, and discuss the interactions between classical and quantum interpretations.
The maximum-entropy principle has recently been discussed in this journal by R. Rosa. The present note stresses a few points made or touched upon there. First it is shown how to derive, by means of the principle under discussion, the quantal statistics, along with the intermediate ones. Then the point is made that the correct application of the principle and its ultimate acceptance by the scientific community require that one uses an appropriate representation for the entropy. Thus great importance is attributed to the characterizations of the various entropies.