III. Negative Probabilities

Li Jun

Coherent states of two dimensional asymmetrical harmonic oscillator,which prop- agate in additional octupole deformation potentials, are used to investigate the complexity of the density matrix and the sensitivity against the strength of disturbance. It was found that according to the regular, partly chaotic and overall chaotic characteristics of the phase space of the classical system,its quantum analogy bears out corresponding characters, which were associated with the existence of the negative curvature at the potential surface and the magni- tude of the negative curvature.

Andrei Yu. Khrennikov

Tayfun Ustun

Abstract. I prove a local uniqueness theorem for the Born rule in the setting of small categories equipped with complex morphism weights and path-amplitude probability functionals. Given (i) non-negativity, (ii) polynomiality of bounded total degree, (iii) global U(1) invariance, (iv) classical-limit additivity over mutually exclusive paths, and (v) normalization, I show that the probability assignment P : ℂ → ℝ≥₀ is uniquely determined to be P(z) = |z|^2. The notion of mutually exclusive paths is given a precise categorical formulation as the absence of a shared factorization through any common morphism. I relate the result to reconstructions of quantum probability due to Gleason, Hardy, and Chiribella–D’Ariano–Perinotti, and identify the extension to global coherence under morphism composition as an open problem connected to synthetic probability theory in Markov categories. The Born rule emerges as the unique locally consistent probability law on complex amplitudes, fixed solely by phase invariance and classical-limit behavior, independent of any Hilbert-space framework.

Tayfun Ustun

I prove a local uniqueness theorem for the Born rule in the setting of quiver-generated categories equipped with complex morphism weights and path-amplitude probability functionals. Given (i) non-negativity, (ii) invariance of bounded total degree, (iii) global U(1) invariance, (iv) classical-limit additivity over mutually exclusive paths, and (v) normalization, I show that the probability assignment P: C -> R>=0 is uniquely determined to be P(z) = |z|^2. The notion of mutually exclusive paths is given a precise categorical formulation as the absence of a shared factorization through any common morphism. I relate the result to reconstructions of quantum probability due to Gleason, Hardy, Chiribella-D'Ariano-Perinotti, and several further, more recent reconstructions, and identify the extension to global coherence under morphism composition as an open problem connected to synthetic probability theory in Markov categories. The Born rule emerges as the unique locally consistent probability law on complex amplitudes, fixed by phase invariance and classical-limit behavior together with polynomiality and normalization, independent of any Hilbert-space framework.

Lars M. Johansen

Christof Wetterich