Preprint

Inserted: 19 sep 2026
Last Updated: 19 sep 2026

Pages: 51
Year: 2026
Doi: 10.5281/zenodo.22699067
Notes:

Preprint.

MSC 2020: 49Q22 (primary); 46L57 (secondary); 47D07 (secondary); 49J45 (secondary); 81P17 (secondary).

Links: Author manuscript page, Zenodo record and version history

Abstract:

For every primitive GNS-symmetric quantum Markov generator on $M_d(\mathbb C)$, $d\ge2$, we construct reversible extensions on a countable direct sum with a common all-time diamond-norm semigroup limit and a common trace-norm $\Gamma$-limit of relative entropies. Their entropy-constrained logarithmic current actions nevertheless have a continuum of distinct $\Gamma$-limits. The additional cotangent form is $\chi(E-F_0(B))\lVert Q_0Ce_h\rVert^2$, where $\chi=\limsup_n b_n/\varphi_n^2$ is determined by the auxiliary rates. The joint action--Fisher functional has instead the original core limit. For canonically embedded faithful initial data, the corresponding energy--dissipation functionals $\Gamma$-converge, and their almost minimizers converge to the core semigroup. Every recovery family attaining a strict kinetic saving along a faithful $C^2$ curve below the entropy cap has integrated Fisher cost bounded below by a positive multiple of $\log^2(1/\delta)$. At fixed coupling, the current action need not be lower semicontinuous and can strictly exceed the metric energy of its induced distance. A second construction has a faithful stationary limit, vanishing generator and all-time semigroup differences in diamond norm, and entropy $\Gamma$-convergence, but collapsing distances between fixed faithful states. A bound in terms of the diamond norm of the entire auxiliary generator gives a complementary obstruction.

Keywords: Γ-convergence, Quantum Markov semigroups, noncommutative optimal transport, logarithmic transport metric, energy–dissipation principle, relative entropy

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