Maximum Entropy on the Complex Probability Framework

Abstract

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.