A qutrit semigroup can lose mixed unitarity after a positive time

An explicit analytic counterexample candidate, with exact membership at one positive time and a separating witness at a later time.

Full version (PDF, 6 pages) · LaTeX source and supplementary check (ZIP)

Status: a full counterexample candidate, internally reviewed by AI agents. External expert review and literature novelty remain unconfirmed. Both endpoint claims have analytic proofs; the supplementary numerical check is not a proof certificate.

The question

A mixed-unitary quantum channel is a finite probabilistic mixture of unitary conjugations. Bhat and Devendra prove that every continuous semigroup of unital quantum channels eventually becomes mixed unitary [1, Theorem 4.12]. The question recorded in QIQCOP [2] concerns the journey to that eventual regime: can a semigroup be mixed unitary at a positive time, then cease to be mixed unitary at a later time?

The manuscript gives an explicit qutrit construction with exactly that behavior. It is consistent with eventual mixed unitarity: membership at an earlier time need not persist continuously from that time onward.

The construction

Let

\[R=\begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad B=\operatorname{diag}(1,R),\qquad H=\operatorname{diag}(30,0,0),\]

and define a fixed generator on complex \(3\times3\) matrices by

\[\mathcal L(X)=\frac{\operatorname{Tr}(X)I_3-BX^T B^*}{2}-X-i[H,X].\]

Here \(T\) is transpose in the displayed basis and \(*\) is adjoint. This is a unital Lindblad generator, so \(\Phi_u=e^{u\mathcal L}\) is a norm-continuous unital CPTP semigroup. The two times are

\[s=\frac{2\pi}{\sqrt{3601}},\qquad t=\frac{3\pi}{\sqrt{3601}}>s.\]

The conclusion is mixed unitary at \(s\) and not mixed unitary at \(t\), for the same generator.

The dissipative construction comes from Bhat–Devendra [1, Lemma 5.7 and Example 5.11], as does the twisted-swap separation method [1, Lemmas 5.5–5.7]. The additional ingredient here is the specified Hamiltonian and the exact refocusing calculation that supplies the two endpoints.

Why the earlier channel is mixed unitary

An explicit solution of the block evolution gives

\[\Phi_s=\operatorname{Ad}_Q\circ e^{s\mathcal D_0},\qquad Q=\operatorname{diag}(-1,1,1),\]

where \(\operatorname{Ad}_Q(X)=QXQ^*\) and \(\mathcal D_0\) is a sum of five Hermitian-jump Lindblad generators. Each individual exponential is a Gaussian average of unitary conjugations. The finite-dimensional convex hull of unitary conjugations is compact, so these averages, their products, and the Lie-product limit all belong to that hull. This proves existence of a finite mixed-unitary representation of \(\Phi_s\), without replacing finite mixtures by a larger class of infinite mixtures.

Why the later channel is not mixed unitary

Use the normalized Choi matrix \(J(\Psi)=(\operatorname{id}\otimes\Psi)(|\Omega\rangle\langle\Omega|)\), where \(|\Omega\rangle=(|00\rangle+|11\rangle+|22\rangle)/\sqrt3\). With \(F\) the swap operator, set

\[C=\operatorname{diag}(-1,R),\qquad W=(I_3\otimes C)F(I_3\otimes C^*)+I_9/3.\]

For every mixed-unitary channel \(\Psi\), one has \(\operatorname{Tr}(WJ(\Psi))\geq0\). The proof applies to all complex qutrit unitaries: the skew-symmetric part of a unitary has rank at most two and operator norm at most one. Convexity then extends the bound to mixtures.

Direct evaluation at the later time gives

\[\operatorname{Tr}(WJ(\Phi_t)) =\frac{2+e^{-3t/2}-3e^{-t/2}-4e^{-t}/\sqrt{3601}}{3} <-\frac{456}{38125}<0.\]

The final inequality follows from elementary rational bounds on the exponentials and on \(t\). It rules out every mixed-unitary decomposition, without optimizing over a sampled family of unitaries or relying on a numerical solver’s tolerance.

Verification

The proof was generated within the QIQC agent workflow and checked in separate internal AI reviews. The checks covered the Lindblad representation, the exact refocusing identity, finite convex-hull membership, the global witness inequality, and the strict rational negative bound. No mathematical gap was found in that review. This is not external refereeing or formal proof verification.

The source archive includes a supplementary numerical reproduction of the Hermitian-jump identity, the full channel identity at \(s\), and the normalized Choi witness at \(t\). It is a diagnostic using NumPy and SciPy; the six-page manuscript contains the analytic proof.

What remains open

The construction is intended to answer the existence question in the linked QIQCOP record in full. It does not determine the first positive mixed-unitary time, the eventual membership threshold, or the complete set of intermediate membership times.

External mathematical review and broader literature checking remain necessary. No priority claim is made, and the ingredients inherited from [1] should retain their attribution.

References

  1. B. V. Rajarama Bhat and R. Devendra, On regions of mixed unitarity for semigroups of unital quantum channels, arXiv:2512.23598v3, revised 9 September 2026. See Theorems 4.12 and 5.4, the open question following Theorem 5.4, Lemmas 5.5–5.7, and Example 5.11.
  2. QIQCOP, Loss of mixed unitarity after a positive time in a quantum dynamical semigroup, problem op_722706a9205dcff2.