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arXiv:2511.21967

Lindblad Quantum Dynamics as Euler-Poincaré Reduction on Adjoint-Coupled Semidirect Products

Leonardo Jesus Colombo

Published Nov 26, 2025• 0 citations
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Abstract

We present a geometric and variational derivation of the Gorini--Kossakowski--Sudarshan--Lindblad equation from Euler--Poincar'e reduction on an adjoint--coupled semidirect product (ACSP). In this construction a Lie group $G$ acts on $V=\mathfrak{g}^{\oplus m}$ by the adjoint representation together with a second, adjointly compatible action whose failure to commute defines an adjoint torsion $K(ξ,v)$. This torsion generates a canonical quadratic curvature operator on $\mathfrak{g}^*$ that survives reduction and yields a metric double--bracket term. For $G=SU(n)$ the reduced Euler--Poincar'e equation reproduces exactly the GKSL generator: the Hamiltonian part arises from the coadjoint action, while the dissipator $-\tfracγ{2}[L,[L,ρ]]$ appears as the torsion--induced metric component of an ACSP bracket. We prove a characterization theorem showing that any quadratic $SU(n)$--equivariant operator generated by torsion factorizes into a Lindblad double commutator; a uniqueness theorem establishing that, under natural structural assumptions, the only admissible dissipator is the Lindblad form; and an orbit--contraction theorem showing strict contraction toward the commutant of the Lindblad operators. For $SU(2)$ and $SU(3)$ the ACSP geometry yields explicit Bloch equations for representative dissipative channels. We also show that the ACSP bracket fits into a metriplectic and contact--geometric framework in which the Lindblad term is the metric component and the Reeb part of a contact Hamiltonian flow. In this picture, decoherence is a curvature--induced contraction generated by Euler--Poincar'e reduction rather than a phenomenological input.

DOI
10.48550/arxiv.2511.21967

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We develop a Lie-Jordan geometric formulation of finite-dimensional open quantum dynamics, building on the algebraic framework introduced in our previous work (arXiv:2606.26477). The Hilbert-Schmidt operator space is endowed with an orthonormal Hermitian basis, in which the commutator and anticommutator are encoded by the structure tensors CμνλC_{μν}{}^λCμν​λ and BμνλB_{μν}{}^λBμν​λ. Within this formulation, the von Neumann and Gorini-Kossakowski-Lindblad-Sudarshan equations admit a direct component representation. Our central result is the identification of a basis-independent universal trilinear dissipative map, D(X,Y)Z=XZY−12{YX,Z}, \mathcal D(X,Y)Z=XZY-\frac12\{YX,Z\}, D(X,Y)Z=XZY−21​{YX,Z}, whose components define a universal operator-space tensor depending only on the Lie-Jordan structure tensors. The physical dissipator is obtained by contracting this tensor with the expansion coefficients of the Lindblad operators and the Kossakowski matrix, thereby separating the universal algebraic structure from the model-dependent physical information. We further show that the combinations (B+C)(B+C)(B+C) and (B−C)(B-C)(B−C) generate internal left and right transports, allowing the elementary dissipative map to be expressed as a left-right bimodule action corrected by an ordered Jordan contribution. The universal map satisfies basis-independent trace and Hermitian-conjugation identities, from which trace preservation, Hermiticity preservation of the complete dissipator, and the reality of its component representation in a Hermitian Hilbert-Schmidt basis follow. We also derive its Hilbert-Schmidt adjoint and illustrate the formalism for a qubit with pure dephasing and amplitude-damping channels. This construction provides a tensorial and affine-geometric interpretation of the universal superoperator structure derived in arXiv:2606.26477 while keeping the algebraic and model-dependent sectors explicitly separated.
This paper develops a bridge between bi-Hamiltonian structures of Poisson--Lie type, contact Hamiltonian dynamics, and the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) formalism for open quantum systems. On the classical side, we consider bi-Hamiltonian systems defined by a Poisson pencil with non-trivial invariants. Using an exact symplectic realization, these invariants are lifted and projected onto a contact manifold, yielding a completely integrable contact Hamiltonian system and a Jacobi-commutative algebra of observables. On the quantum side, we introduce a class of contact-compatible Lindblad generators: GKSL evolutions whose dissipative part preserves a commutative C∗C^\astC∗-subalgebra generated by the quantizations of the classical dissipated quantities, and whose Hamiltonian part admits an Egorov-type semiclassical limit to the contact dynamics. This construction provides a mathematical mechanism compatible with the semiclassical limit for pure dephasing, compatible with integrability and contact dissipation. An explicit Poisson--Lie pencil, inspired by deformed Euler top models, is developed as a fully worked-out example illustrating the resulting bi-Lindblad structure and its semiclassical behavior.
## Overview This preprint studies antisymmetric Kossakowski transport generated by compact Lie-algebraic jump operators in finite-dimensional Gorini--Kossakowski--Sudarshan--Lindblad (GKLS) dynamics. The central question is whether directed transport between representation sectors can itself select an operator-space geometry in which the transport becomes reciprocal or skew-adjoint. For a compact simple Lie algebra, the antisymmetric coefficient tensor admits an exact orthogonal bracket--kernel decomposition. The bracket projection carries all nonunitality and satisfies a Casimir--Jordan identity, whereas the bracket-kernel component is unital and Hilbert--Schmidt skew-adjoint. ## Main results Projection between Casimir eigensectors gives the universal scaled-adjoint law Fν→μ=−Δμν−CadΔμν+CadFμ→ν†,F_{\nu\to\mu}=-\frac{\Delta_{\mu\nu}-C_{\rm ad}}{\Delta_{\mu\nu}+C_{\rm ad}}F_{\mu\to\nu}^{\dagger},Fν→μ​=−Δμν​+Cad​Δμν​−Cad​​Fμ→ν†​, where Δμν=cν−cμ>0\Delta_{\mu\nu}=c_\nu-c_\mu>0Δμν​=cν​−cμ​>0 is the Casimir gap and CadC_{\rm ad}Cad​ is the adjoint Casimir. The critical gap Δμν=Cad\Delta_{\mu\nu}=C_{\rm ad}Δμν​=Cad​ produces a one-way transport edge, including the universal scalar-to-adjoint boundary. For the multiplicity-free symmetric-power hierarchy Vp=Sym⁡pCN,V_p=\operatorname{Sym}^p\mathbb C^N,Vp​=SympCN, every nonzero bracket direction activates every neighboring bond, and the forward and reverse neighboring blocks satisfy Rk=−kk+NFk†.R_k=-\frac{k}{k+N}F_k^\dagger.Rk​=−k+Nk​Fk†​. The resulting off-diagonal transport uniquely selects, up to an overall scale, the sector weights wk∝Γ(k)Γ(k+N).w_k\propto\frac{\Gamma(k)}{\Gamma(k+N)}.wk​∝Γ(k+N)Γ(k)​. Under the exact equivariant harmonic correspondence with CPN−1\mathbb CP^{N-1}CPN−1, this selected metric becomes a projective spectral multiplier. On nonconstant projective harmonics it is identified with the sign-corrected derivative, at the constant-kernel point, of a centered and rescaled projective cosine-transform intertwining family. Its inverse has the same scalar Gamma-ratio multiplier, when indexed by harmonic degree, as the corresponding order-NNN conformal spherical intertwiner. For N≥3N\ge3N≥3, a dabcd_{abc}dabc​-mediated same-sector Jordan block obstructs any positive SU(N)SU(N)SU(N)-invariant skew metric for the full traceless bracket transport. Consequently, within the symmetric-power family, SU(2)SU(2)SU(2) is exceptional: the bracket kernel disappears, the same-sector obstruction is absent, and the transport-selected metric reduces to inverse Casimir and extends over the entire traceless internal bracket dynamics. ## Interpretation and scope The Gamma-function sequence, projective harmonic correspondence, cosine-transform spectrum, and conformal spherical multipliers are established mathematical structures. The new mechanism developed here is their dynamical selection by Lie-algebraic Lindblad transport, together with the representation-theoretic obstruction that prevents the SU(2)SU(2)SU(2) inverse-Casimir geometry from extending generically to higher rank. The separate SU(2)SU(2)SU(2) prototype, including its dissipative decomposition, contraction theory, exact spherical Green geometry, and complete-positivity extremality, is developed in: **Lee Byoungwoo, “Inverse-Casimir Geometry of Nonunital Multipole Transport in Linear-Spin Lindbladians,” Zenodo Version v1.0 (2026), DOI 10.5281/zenodo.22266973.** ## Version status This record is **Version v1.0** of the public preprint. The manuscript is presented without internal research-version or freeze labels.
The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindblad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the correlations between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, among other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints that arise either because they are introduced by hand or because of singular Lagrangians. We derive an intriguing but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts such as quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.
Motivated by a geometric decomposition of the vector field associated with the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system.
Abstract In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. In this article we define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be traceless as demanded by the GKSL theorem). Moreover, we consider the converse construction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corresponding semigroup.