Non-symmetric transition probability in generalized qubit models

TL;DR

This paper introduces a class of binary models in quantum logic where the transition probability is not symmetric, contrasting with traditional quantum mechanics, and characterizes the conditions under which symmetry is restored.

Contribution

It presents a new class of models with non-symmetric transition probabilities using extreme points of the unit interval in an order unit space.

Findings

Transition probabilities are symmetric iff the state space is a Hilbert space unit ball.

State spaces are strictly convex smooth compact convex sets.

When symmetric, the quantum logic aligns with the projection lattice in a spin factor.

Abstract

The quantum mechanical transition probability is symmetric. A probabilistically motivated and more general quantum logical definition of the transition probability was introduced in two preceding papers without postulating its symmetry, but in all the examples considered there it remains symmetric. Here we present a class of binary models where the transition probability is not symmetric, using the extreme points of the unit interval in an order unit space as quantum logic. We show that their state spaces are strictly convex smooth compact convex sets and that each such set K gives rise to a quantum logic of this class with the state space K. The transition probabilities are symmetric iff K is the unit ball in a Hilbert space. In this case, the quantum logic becomes identical with the projection lattice in a spin factor which is a special type of formally real Jordan algebra.