The Independence of Complex Probabilities

Michael Strevens

Carlos Riveros Berger

We present an axiomatic construction of quaternionic probability, extending Kolmogorov’s classical framework to the noncommutative algebra of quaternions. The theory introduces quaternionic probability spaces, conditional probabilities, Bayes’ rules, independence, random variables, expectations, and transport equations, all formulated in a consistent manner. Classical probability is recovered through scalar projection, while restriction to complex subalgebras reproduces the standard quantum formalism. Uniquely quaternionic structures arise, including noncommutative conditional probabilities, inequivalent forms of independence, and quaternionic transport laws. The framework further develops quaternionic Markov chains, entropy, and divergence measures that separate scalar uncertainty from vectorial coherence. Several illustrative examples are provided to show how quaternionic probability captures order effects, hidden correlations, and orthogonal divergences—features invisible to both classical and complex approaches. These results establish quaternionic probability as a rigorous generalization of Kolmogorov’s axioms and as a potential foundation for future studies in noncommutative probability, integrable structures, and quaternionic extensions of mathematical physics.

Giorgio Parisi A

Luis Roa, Carla Hermann-Avigliano, Roberto Salazar +3 more

We find the allowed complex numbers associated with the inner product of N equally separated pure quantum states. The allowed areas on the unitary complex plane have the form of petals. A point inside the petal-shape represents a set of N linearly independent (LI) pure states, and a point on the edge of that area represents a set of N linearly dependent (LD) pure states. For each one of those LI sets we study the complete discrimination of its N equi-separated states combining sequentially the two known strategies: first the unambiguous identification protocol for LI states, followed, if necessary, by the error-minimizing measurement scheme for LD states. We find that the probabilities of success for both unambiguous and ambiguous discrimination procedures depend on both the module and the phase of the involved inner product complex number. We show that, with respect to the phase-parameter, the maximal probability of discriminating unambiguously the N non-orthogonal pure states holds just when there no longer be probability of obtaining ambiguously information about the prepared state by applying the second protocol if the first one was not successful.

Michail A. Zak