Published August 25, 2026 | Version v1
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Quantum Wasserstein coarse-graining of detailed-balance Lindblad dynamics: Quotient Onsager geometry, section defects, and approximate basicity

Description

See also:  Covariant Coherent-Generator Geometry over Quantum Wasserstein State Space

This paper arises from the broader OT-GKSL architecture, in which detailed-balance GKSL dynamics are organized through a sequence of transport layers: a current representation, the induced state-space Onsager/quantum-Wasserstein geometry, and subsequent reduction to collective variables. The present work isolates and develops the native-to-collective coarse-graining layer of that architecture as a mathematically self-contained problem.

OT-GKSL Pedagogical Guide, Foundations Audit, Uniqueness & Substitutability Analysis, Literature Comparison, and Reading Guidance : A Pedagogical Guide to Understanding the OT–GKSL Architecture.

Abstract / Summary:

This work studies finite-dimensional coarse-graining of detailed-balance quantum Markov dynamics equipped with a specified Carlen–Maas noncommutative Wasserstein/Onsager geometry.

The starting point is a smooth manifold of faithful density matrices 𝓜₊ and a positive Onsager co-metric:

K_ρ : T*₊𝓜₊ → T_ρ𝓜₊
with inverse transport metric:

g_ρ(u, v) = ⟨K_ρ⁻¹ u, v⟩
For the declared detailed-balance sector, the dissipative drift is assumed to satisfy the Carlen–Maas gradient-flow identity:

b_db(ρ) = −K_ρ DF(ρ)
where F(ρ) = D(ρ || π) is the Umegaki relative entropy with respect to a faithful stationary state π. A specified current representation is written as:

K_ρ = Γ_ρ M_ρ Γ_ρ*
where Γ_ρ is the continuity/divergence map and M_ρ is a positive current mobility.

For a smooth collective map q : 𝓜₊ → Q, write P_ρ = dq_ρ. At a fixed microscopic representative ρ, the pointwise quotient Onsager co-metric is:

K_ρᵠ = P_ρ K_ρ P_ρ*
with inverse G_ρᵠ = (K_ρᵠ)⁻¹, and minimum-norm horizontal lift:

H_ρ = K_ρ P_ρ* G_ρᵠ
The current-to-state kinetic minimization and the minimum-norm Riemannian quotient construction are standard ingredients and are not claimed as new. Their typed composition gives the local collective quadratic contraction:

inf_{j : P_ρ[a(ρ) + Γ_ρ j] = dq/dt} 𝓛_cur(ρ, j) = (1/4) ⟨G_ρᵠ(dq/dt − ℓ_ρ), dq/dt − ℓ_ρ⟩
where ℓ_ρ = P_ρ b(ρ).

Key Findings & Core Structure:

The main purpose of the paper is to separate three issues that remain after this pointwise contraction:

1. Drift Basicity and Mobility Basicity are Distinct:

Exact autonomous deterministic closure requires:

P_ρ b(ρ) = Lᵇ(q(ρ))
for a vector field Lᵇ defined intrinsically on the collective space. By contrast, an intrinsic section-independent quadratic transport geometry requires:

P_ρ K_ρ P_ρ* = Kᵇ(q(ρ))
These are logically independent projectability conditions. In particular, an exact deterministic reduced equation may exist even when the reduced transport cost still depends on the microscopic representative.

A reversible three-state Markov-chain example makes this independence explicit. For q = p₁ + p₂, the projected deterministic dynamics closes exactly as:

dq/dt = 1 − 2q
whereas the logarithmic-mean quotient mobility is:

K_pᵇ = (1/4) Λ(4p₁, 2p₃) + (1/4) Λ(4p₂, 2p₃)
and therefore depends on the hidden split between p₁ and p₂. Here, Λ(a, b) = (a − b) / (log a − log b) for a ≠ b, with Λ(a, a) = a.

2. Exact Quotient-versus-Section Defect Identity:

Let s : U ⊂ Q → 𝓜₊ be a smooth representative section satisfying q ∘ s = Id. Define the section-dependent reduced drift Ls(q) = dq{s(q)} b(s(q)), the horizontal lift H_s, the vertical section tilt V_s = ds − H_s, and the native tangency residual R_s(q) = b(s(q)) − ds_q L_s(q). Both V_s and R_s are vertical.

The central exact decomposition is:

(1/4) || dsq (dq/dt) − b(s(q)) ||²{g_{s(q)}} = (1/4) ⟨G_sᵠ (dq/dt − L_s), dq/dt − L_s⟩ + (1/4) || V_s (dq/dt − Ls) − Rs ||²{g{s(q)}}
The first term is the pointwise quotient cost at the chosen representative. The second term is a nonnegative vertical square and is exactly the additional cost generated by imposing the reconstruction section instead of using the minimum-norm quotient lift.

The corresponding metric identity is:

s* g = G_sᵠ + V_s* g V_s ≥ G_sᵠ
Thus pullback along a parametrized reconstruction and quotient minimization over hidden tangent directions are distinct operations. Horizontal sections satisfy V_s = 0, but such sections need not exist locally: the transport-horizontal distribution may fail the Frobenius integrability condition because of nonzero Ehresmann curvature.

3. Approximate Basicity:

Exact fiber basicity is restrictive. The paper therefore introduces representative-dependent drift and mobility defects:

  • ε_b(ρ) = || ℓ_ρ − L̄(q) ||_h
  • ε_K(ρ) = || Kρᵠ − K̄(q) ||{op, h}
relative to a reference reduced pair (L̄, K̄) and an auxiliary Riemannian norm h.

Assuming uniform ellipticity with constant m > 0, and writing R = || dq/dt − L̄(q) ||_h, the local representative-dependent quadratic cost C_ρ obeys:

| C_ρ(q, dq/dt) − C̄(q, dq/dt) | ≤ (1/4) [ (ε_K(ρ) / m²) (R + ε_b(ρ))² + (ε_b(ρ) / m) (2R + ε_b(ρ)) ]
Hence the exact quotient theory is recovered as the zero-defect limit, while weak variation of projected drift and mobility along fibers yields an explicit quantitative coarse-graining estimate. Fiber oscillations can additionally be controlled by vertical regularity and the metric diameter of the fiber.

Quantum Consistency Check:

A two-level Davies generator is linearized around a faithful equilibrium state. In Hilbert–Schmidt tangent coordinates the entropy Hessian and Onsager operator take the diagonal form:

Cπ = diag(c⊥, c_⊥, c_z)
Kπ = diag(k⊥, k_⊥, k_z)
with k_⊥ = Γ₂ / c_⊥ and k_z = 2 Γ₁ p_g p_e. For a tilted local section s_η(z) = (η z, 0, z), the section-defect theorem gives:

(1/4) || dsη (dz/dt) − Adb s_η ||²{g_π} = (dz/dt + Γ₁ z)² / (4 k_z) + η² (dz/dt + Γ₂ z)² / (4 k⊥)
This example is intentionally local and linearized. It is a noncommutative tangent-space consistency check, not a nonlinear quantum coarse-graining theorem.

Novelty and Scope:

The paper does not claim novelty for:

  • The Carlen–Maas current-to-state kinetic-energy minimum;
  • Minimum-norm horizontal lifts or Riemannian quotient metrics;
  • The formula K_ρᵠ = dq_ρ K_ρ dq_ρ* in isolation;
  • Reduced Onsager operators as a general concept.
The contribution is the assembled reduction framework consisting of:

  1. A typed current-to-state-to-collective contraction;
  2. The explicit separation between drift projectability and mobility projectability;
  3. The exact arbitrary-section vertical-defect identity involving both section tilt and native tangency residual;
  4. An explicit perturbative stability theorem for approximate basicity;
  5. Geometric control of fiber defects through vertical regularity and fiber diameter.
The underlying quotient and section arguments are finite-dimensional Onsager/Riemannian statements once a positive co-metric K_ρ is supplied. Their quantum content in this work comes from their specialization to a declared Carlen–Maas noncommutative transport structure and detailed-balance quantum Markov dynamics.

Explicit Limitations:

  • The quadratic current functional introduced in the paper is not claimed to be a microscopic quantum path-space large-deviation rate function.
  • The paper does not prove that an arbitrary physical coarse-graining produces a reduced quantum Markov semigroup whose intrinsic Carlen–Maas Onsager operator is exactly dq_ρ K_ρ dq_ρ*.
Establishing such a functorial reduction theorem, for example for conditional expectations onto quantum subalgebras, is identified as a major open problem.
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Classification & Mathematics Subject Classification (2020):

  • Subjects: Mathematical Physics, Quantum Information / Open Quantum Systems, Optimal Transport, Differential Geometry, Probability / Markov Semigroups, Model Reduction
  • MSC 2020: 81S22, 47D07, 49Q22, 53C21, 60J27

Key References for Independent Validation:

  1. E. A. Carlen and J. Maas, Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balance, Journal of Functional Analysis 273 (2017), 1810–1869. DOI: 10.1016/j.jfa.2017.05.003
  2. E. A. Carlen and J. Maas, Non-commutative calculus, optimal transport and functional inequalities in dissipative quantum systems, Journal of Statistical Physics 178 (2020), 319–378. DOI: 10.1007/s10955-019-02434-w
  3. J. Maas, Gradient flows of the entropy for finite Markov chains, Journal of Functional Analysis 261 (2011), 2250–2292. DOI: 10.1016/j.jfa.2011.06.009
  4. J. Maas and A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics 181 (2020), 2257–2303. DOI: 10.1007/s10955-020-02663-4
  5. M. Liero, A. Mielke, O. Tse, and J.-J. Zhu, Evolution of Gaussians in the Hellinger–Kantorovich–Boltzmann gradient flow, Communications on Pure and Applied Analysis 31 (2026), 166–198. DOI: 10.3934/cpaa.2025105
  6. S. Becker and W. Li, Quantum statistical learning via quantum Wasserstein natural gradient, Journal of Statistical Physics 182 (2021), Article 7. DOI: 10.1007/s10955-020-02682-1
  7. D. F. Hornshaw, Quantum optimal transport for approximately finite-dimensional C-algebras*, arXiv:1910.03312.

 

 

        8.  OT-GKSL: Technical papers, companion papers, and supplementary materials:

 

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