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

Nifa, Anass

doi:10.5281/zenodo.22699067

Preprint · v1

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

Anass NifaInitial manuscript date: 10 June 2026Manuscript revision: 5 August 2026

Manuscript revision dated 5 August 2026, as confirmed by the author. Date metadata clarified on 21 September 2026. Mathematical content and references are unchanged.

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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.

MSC 2020

49Q22Primary: Optimal transportation
46L57Secondary: Derivations, dissipations and positive semigroups in C*-algebras
47D07Secondary: Markov semigroups and applications to diffusion processes
49J45Secondary: Methods involving semicontinuity and convergence; relaxation
81P17Secondary: Quantum entropies

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22699067
All versions: 10.5281/zenodo.22699066
Subjects: Quantum optimal transport; Metric geometry
Research index: OpenAlex record
Language: English

Cite this paper

Anass Nifa. Operational and free-energy convergence do not determine kinetic geometry in logarithmic quantum transport. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22699067

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References

As listed in the manuscript bibliography.

[1]
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, second edition, Birkhäuser, Basel, 2008.
[2]
M. Brooks and J. Maas, Characterisation of gradient flows for a given functional, Calc. Var. Partial Differential Equations 63 (2024), article 153.
[3]
E. A. Carlen and J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, J. Funct. Anal. 273 (2017), 1810–1869.
[4]
M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290.
[5]
Y. Deng, Y. Gao and L. Wang, Homogenization of time-discrete gradient flows, arXiv:2607.25206, 2026.
arXiv
[6]
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, New York, 2000.
[7]
P. Gladbach, E. Kopfer, J. Maas and L. Portinale, Homogenisation of dynamical optimal transport on periodic graphs, Calc. Var. Partial Differential Equations 62 (2023), article 143.
[8]
L. Lami and M. E. Shirokov, Attainability and lower semi-continuity of the relative entropy of entanglement and variations on the theme, Ann. Henri Poincaré 24 (2023), 4069–4137, doi:10.1007/s00023-023-01313-1.
DOI
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E. H. Lieb, Convex trace functions and the Wigner–Yanase–Dyson conjecture, Adv. Math. 11 (1973), 267–288.
[10]
M. Mittnenzweig and A. Mielke, An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models, J. Stat. Phys. 167 (2017), 205–233.
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A. Mielke, A. Montefusco and M. A. Peletier, Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence, Contin. Mech. Thermodyn. 33 (2021), 611–637.
[12]
A. Mielke and A. Stephan, Coarse-graining via EDP-convergence for linear fast–slow reaction systems, Math. Models Methods Appl. Sci. 30 (2020), no. 9, 1765–1807, doi:10.1142/S0218202520500360.
DOI
[13]
A. Müller-Hermes and D. Reeb, Monotonicity of the quantum relative entropy under positive maps, Ann. Henri Poincaré 18 (2017), 1777–1788.
[14]
M. A. Peletier and M. C. Schlottke, Gamma-convergence of a gradient-flow structure to a non-gradient-flow structure, Calc. Var. Partial Differential Equations 61 (2022), article 103.
[15]
A. Winter, Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints, Comm. Math. Phys. 347 (2016), 291–313.
[16]
M. Wirth, A dual formula for the noncommutative transport distance, J. Stat. Phys. 187 (2022), article 19, https://doi.org/10.1007/s10955-022-02911-9.
[17]
M. Wirth, A noncommutative transport metric and symmetric quantum Markov semigroups as gradient flows of the entropy, arXiv:1808.05419v2, 2021.
arXiv v2

[reference-1] [1]
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, second edition, Birkhäuser, Basel, 2008.

[reference-2] [2]
M. Brooks and J. Maas, Characterisation of gradient flows for a given functional, Calc. Var. Partial Differential Equations 63 (2024), article 153.

[reference-3] [3]
E. A. Carlen and J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, J. Funct. Anal. 273 (2017), 1810–1869.

[reference-4] [4]
M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285–290.

[reference-5] [5]
Y. Deng, Y. Gao and L. Wang, Homogenization of time-discrete gradient flows, arXiv:2607.25206, 2026.
arXiv

[reference-6] [6]
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics 194, Springer-Verlag, New York, 2000.

[reference-7] [7]
P. Gladbach, E. Kopfer, J. Maas and L. Portinale, Homogenisation of dynamical optimal transport on periodic graphs, Calc. Var. Partial Differential Equations 62 (2023), article 143.

[reference-8] [8]
L. Lami and M. E. Shirokov, Attainability and lower semi-continuity of the relative entropy of entanglement and variations on the theme, Ann. Henri Poincaré 24 (2023), 4069–4137, doi:10.1007/s00023-023-01313-1.
DOI

[reference-9] [9]
E. H. Lieb, Convex trace functions and the Wigner–Yanase–Dyson conjecture, Adv. Math. 11 (1973), 267–288.

[reference-10] [10]
M. Mittnenzweig and A. Mielke, An entropic gradient structure for Lindblad equations and couplings of quantum systems to macroscopic models, J. Stat. Phys. 167 (2017), 205–233.

[reference-11] [11]
A. Mielke, A. Montefusco and M. A. Peletier, Exploring families of energy-dissipation landscapes via tilting: three types of EDP convergence, Contin. Mech. Thermodyn. 33 (2021), 611–637.

[reference-12] [12]
A. Mielke and A. Stephan, Coarse-graining via EDP-convergence for linear fast–slow reaction systems, Math. Models Methods Appl. Sci. 30 (2020), no. 9, 1765–1807, doi:10.1142/S0218202520500360.
DOI

[reference-13] [13]
A. Müller-Hermes and D. Reeb, Monotonicity of the quantum relative entropy under positive maps, Ann. Henri Poincaré 18 (2017), 1777–1788.

[reference-14] [14]
M. A. Peletier and M. C. Schlottke, Gamma-convergence of a gradient-flow structure to a non-gradient-flow structure, Calc. Var. Partial Differential Equations 61 (2022), article 103.

[reference-15] [15]
A. Winter, Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints, Comm. Math. Phys. 347 (2016), 291–313.

[reference-16] [16]
M. Wirth, A dual formula for the noncommutative transport distance, J. Stat. Phys. 187 (2022), article 19, https://doi.org/10.1007/s10955-022-02911-9.

[reference-17] [17]
M. Wirth, A noncommutative transport metric and symmetric quantum Markov semigroups as gradient flows of the entropy, arXiv:1808.05419v2, 2021.
arXiv v2