Bit symmetry entails the symmetry of the quantum transition probability

TL;DR

This paper demonstrates that in the transition probability framework, the bit symmetry postulate implies the symmetry of quantum transition probabilities, significantly constraining the models to classical and Euclidean Jordan algebra cases.

Contribution

It shows that bit symmetry entails transition probability symmetry within the transition probability framework, narrowing down the possible models to classical and Euclidean Jordan algebras.

Findings

Bit symmetry implies symmetry of transition probabilities between atoms.

Transition probability symmetry constrains models to classical and Euclidean Jordan algebras.

The results connect symmetry postulates with the structure of quantum and classical models.

Abstract

It is quite common to use the generalized probabilistic theories (GPTs) as generic models to reconstruct quantum theory from a few basic principles and to gain a better understanding of the probabilistic or information theoretic foundations of quantum physics and quantum computing. A variety of symmetry postulates was introduced and studied in this framework, including the transitivity of the automorphism group (1) on the pure states, (2) on the pairs of orthogonal pure states [these pairs are called 2-frames] and (3) on any frames of the same size. The second postulate is M\"uller and Ududec's bit symmetry, which they motivate by quantum computational needs. Here we explore these three postulates in the transition probability framework, which is more specific than the GPTs since the existence of the transition probabilities for the quantum logical atoms is presupposed either directly or indirectly via a certain geometric property of the state space. This property for compact convex sets was introduced by the author in a recent paper. We show that bit symmetry implicates the symmetry of the transition probabilities between the atoms. Using a result by Barnum and Hilgert, we can then conclude that the third rather strong symmetry postulate rules out all models but the classical cases and the simple Euclidean Jordan algebras.