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
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].
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.