Choice of the state of complete uncertainty in a generalized probabilistic theory

Determine how to choose the distinguished interior state representing the state of complete uncertainty in a finite-dimensional generalized probabilistic theory, given that such a state need not be unique.

Background

The Protourgleichung is formulated using a distinguished state ∣μ)|\mu) that represents complete uncertainty. In classical and complex quantum theory, natural candidates are supplied by symmetry: the uniform distribution is invariant under simplex permutations, while the maximally mixed state is invariant under unitary transformations. For a general generalized probabilistic theory, however, the state invariant under all reversible symmetries may fail to be unique, and the fixed-point eigenspace of the reference-measurement transition matrix may likewise contain multiple normalized states.

The construction that follows only requires selecting some interior normalized state and transforming coordinates so that it is represented by (1,0,…,0)†(1,0,\ldots,0)^\dagger. The unresolved issue is therefore to identify a principled rule for selecting this state in the general GPT setting, especially when symmetry, geometric center, and maximal-entropy criteria do not coincide or do not determine a unique state.

References

More generally, given a GPT state space, we may consider a state $|\mu)$ which is invariant under the reversible transformations which preserve the state space, and adopt this as the ``state of complete uncertainty'' . But in general, there may not be a single unique such state . Moreover, in classical and quantum theory, several ideas coincide: a state which is invariant under state space symmetries, the geometric center of the state space, the maximal entropy state. In terms of our reference measurement formalism, we might be inclined to pick a state corresponding to an eigenvector with eigenvalue 1 of $P(R|R)$: indeed, since $P(R|R)$ is column stochastic, it must have at least one such eigenvector which is a probability distribution : but in general, it will not have a unique such eigenvector. Thus the choice of $|\mu)$ depends essentially on the nature of the GPT. Of course, the simplest solution is to take the uniform average of all the extremal states. For our purposes, however, it suffices merely to distinguish some state $|\mu)$ in the interior of the state space, and we leave open the question of how to choose it.

— Redesigning quantum theory  (2609.03333 - Weiss, 3 Sep 2026) in Chapter “Die Urgleichungen,” Section “Bloch form”