Dependence of exact uniformity on the initial state

Determine how exact uniformity of the return-conditioned positive sojourn-time distribution depends on the initial state, and characterize pairs consisting of a 2×2 unitary coin matrix and an initial state that yield an initial-state-independent uniform distribution.

Background

The main theorem fixes the initial coin state to φ∗=12(1,i)T\varphi_*=\frac{1}{\sqrt2}(1,i)^\mathsf{T}. Consequently, the rigidity result for the rotation-coin family does not establish whether the same uniformity phenomenon persists for other initial states.

The authors identify two unresolved aspects: how changing the initial state alters exact uniformity, and whether a condition can produce the uniform distribution independently of the initial state. They propose classifying coin–initial-state pairs to clarify the mechanism behind the Hadamard uniformity phenomenon.

References

In this paper, we fixed the initial state $\varphi_\ast$. It is also an important problem to study how the exact uniformity changes when the initial state is varied, and whether there exists a condition which gives the uniform distribution in a form independent of the initial state. By developing a classification for pairs of a coin matrix and an initial state, we expect to obtain a clearer understanding of the essence of the uniform distribution phenomenon for the Hadamard coin.

— Hadamard Rigidity of Positive Sojourn Time Distributions for Rotation Coins  (2609.05033 - Tamura et al., 4 Sep 2026) in Section “Conclusion and future problems”