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arXiv:1501.00074

The introduction of symmetry constraints within MaxEnt Jaynes's methodology

Federico Holik, César Massri, Angelo Plastino

Published Dec 31, 2014• 1 citation
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Abstract

We provide a generalization of the approach to geometric probability advanced by the great mathematician Gian Carlo Rota, in order to apply it to generalized probabilistic physical theories. In particular, we use this generalization to provide an improvement of the Jaynes' MaxEnt method. The improvement consists in providing a framework for the introduction of symmetry constrains. This allows us to include group theory within MaxEnt. Some examples are provided.

DOI
10.48550/arxiv.1501.00074

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We provide a generalization of the approach to geometric probability advanced by the great mathematician Gian Carlo Rota, in order to apply it to generalized probabilistic physical theories. In particular, we use this generalization to provide an improvement of the Jaynes' MaxEnt method. The improvement consists in providing a framework for the introduction of symmetry constrains. This allows us to include group theory within MaxEnt. Some examples are provided.
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.
Symmetry shares an entwined history with the structure of physical theory. We propose a consequence of symmetry towards the axiomatic derivation of Hilbert space quantum theory. We introduce the notion of information symmetry (IS) and show that it constraints the state-space structure in any physical theory. To this end, we study the minimal error binary state discrimination problem in the framework of generalized probabilistic theories. A theory is said to satisfy IS if the probability of incorrectly identifying each of two randomly prepared states is the same for both the states. It is found that this simple principle rules out several classes of theories while being perfectly compatible with quantum theory.
We introduce a general framework for analysing general probabilistic theories, which emphasises the distinction between the dynamical and probabilistic structures of a system. The dynamical structure is the set of pure states together with the action of the reversible dynamics, whilst the probabilistic structure determines the measurements and the outcome probabilities. For transitive dynamical structures whose dynamical group and stabiliser subgroup form a Gelfand pair we show that all probabilistic structures are rigid (cannot be infinitesimally deformed) and are in one-to-one correspondence with the spherical representations of the dynamical group. We apply our methods to classify all probabilistic structures when the dynamical structure is that of complex Grassmann manifolds acted on by the unitary group. This is a generalisation of quantum theory where the pure states, instead of being represented by one-dimensional subspaces of a complex vector space, are represented by subspaces of a fixed dimension larger than one. We also show that systems with compact two-point homogeneous dynamical structures (i.e. every pair of pure states with a given distance can be reversibly transformed to any other pair of pure states with the same distance), which include systems corresponding to Euclidean Jordan Algebras, all have rigid probabilistic structures.