Criterion for Actual-Preparation TP2 Witnesses

Determine which diagonal observables with the independence property have the tree property of the second kind witnessed by actual preparations, and characterize this property in terms of bounded-coefficient indicators of an infinite independent sequence of nested clopen sets in the linear span of the responses in C(Kφ).

Background

The paper proves that mixed preparations cause every diagonal observable with the independence property to have the tree property of the second kind in its real-valued theory. However, the witnesses need not be actual preparations: for the Pauli observable, actual preparations witness the tree through syndrome sectors, whereas for the Rado observable the tree appears only after passing to idealized preparations.

The authors ask for a general criterion distinguishing these cases. They suggest examining whether the linear span of the responses in C(Kφ) contains indicators of an infinite independent sequence of nested clopen sets with uniformly bounded total coefficient norm. Establishing such a criterion would unify the Pauli and Rado examples.

References

Which diagonal observables with the independence property have the tree property of the second kind witnessed by actual preparations? cref{thm:pauli-tp2,thm:free} and cref{prop:sector} suggest a criterion in terms of the linear span of the responses in $C(K_\varphi)$: whether it contains, with coefficients of bounded total absolute value, the indicators of an infinite independent sequence of nested clopen sets.

— Classification of bipartite observables III: simplicity, mixtures and quantum error correction  (2610.22205 - Dueñez et al., 15 Oct 2026) in Section 10, Question 10.1, labeled ques:criterion

In classical model theory, a simple theory without the independence property is stable. Is a simple observable stable? By cref{thm:collapse}, a simple diagonal observable has no independence property, so the question is whether the unstable observables of the second paper, such as the comparison of photon numbers, have a tree property once preparations are mixed.

— Classification of bipartite observables III: simplicity, mixtures and quantum error correction  (2610.22205 - Dueñez et al., 15 Oct 2026) in Section 10, Question 10.3, labeled ques:nip-simple

Does cref{thm:collapse} hold for every observable whose responses have the independence property, not only for diagonal ones? The proof uses the values $0$ and $1$ of pure responses; for general observables they are replaced by two distinct levels, and the averages must be controlled with a gap.

— Classification of bipartite observables III: simplicity, mixtures and quantum error correction  (2610.22205 - Dueñez et al., 15 Oct 2026) in Section 10, Question 10.4, labeled ques:general