Operational and free-energy convergence do not determine kinetic geometry in logarithmic quantum transport

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 $\chi(E-F_0(B))\|Q_0Ce_h\|^2$, 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.