Bit symmetry entails the symmetry of the quantum transition probability

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 Müller 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.

💡 Research Summary

The paper investigates three symmetry postulates that have been widely discussed in the framework of generalized probabilistic theories (GPTs) – weak symmetry (transitivity on pure states), bit symmetry (transitivity on pairs of orthogonal pure states, also called 2‑frames), and strong symmetry (transitivity on arbitrary frames of the same size). Rather than working directly in the abstract GPT setting, the author adopts the more concrete transition‑probability framework introduced in earlier work. In this framework a compact convex state space Ω is equipped with an order‑unit space AΩ of affine real‑valued functions, and a geometric condition (∗∗) is imposed: for every extreme point ω∈Ω the minimal non‑negative affine function e_ω that takes the value 1 at ω is itself affine and takes the value 1 only at ω. This condition guarantees that the extreme points of the interval