Loss of mixed unitarity after a positive time in a quantum dynamical semigroup
- Field
- Topic
Problem
Does there exist a norm-continuous semigroup \((\Phi_t)_{t\geq0}\) of unital completely positive trace-preserving maps on \(\mathcal L(\mathbb C^d)\), for some finite integer \(d\geq3\), such that \(\Phi_s\) is mixed unitary and \(\Phi_t\) is not mixed unitary for some \(0<s<t\)? Here \(\Phi_0=\operatorname{id}\) and \(\Phi_{u+v}=\Phi_u\circ\Phi_v\) for \(u,v\geq0\). A map \(\Psi\) is mixed unitary if it admits
for every \(X\in\mathcal L(\mathbb C^d)\), where \(N\) is finite and the \(U_j\) in Eq. (1) are unitary.
Source
Bhat and Devendra explicitly ask whether the first positive mixed-unitary time can precede the eventual threshold; see the unnumbered remark after Theorem 5.4, p. 21 of version 3 [BD26].
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Every such semigroup is mixed unitary at all sufficiently large times (Theorem 4.12). Unless it is mixed unitary at every time, Theorem 5.4 gives a first positive mixed-unitary time \(t_0\) and the least eventual threshold \(t_1\geq t_0\). The authors leave \(t_0<t_1\) open [BD26].
- Reported progress: Issue #75.
Comment
The question is equivalent to \(t_0<t_1\); eventual mixed unitarity alone does not settle it. Literature audit: 9 September 2026. The latest source version remains a preprint explicitly leaving this gap open. No later resolution was located in indexed literature; unindexed work cannot be excluded.