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

TL;DR

This work derives the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation from Euler–Poincaré reduction on an adjoint–coupled semidirect product (ACSP), showing that the Lindblad dissipator emerges as a curvature term produced by the adjoint torsion. By identifying the torquing action with a quadratic, $SU(n)$–equivariant dissipator, it proves uniqueness: the Lindblad double commutator is the only admissible dissipative term under natural symmetry and positivity constraints, and it characterizes orbital contraction toward the commutant of Lindblad operators. The framework unifies Hamiltonian (coadjoint) dynamics, metriplectic/double-bracket dissipation, and a contact–geometric view where the Lindblad term is the Reeb component, offering a geometric origin for decoherence. The results extend to finite-dimensional systems (qubits to qudits), provide explicit Bloch-ball descriptions for $SU(2)$ and $SU(3)$ channels, and suggest avenues for reservoir engineering, control, and quantum thermodynamics within a single Lie–geometric structure.

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.

Figures (9)

• Figure 1: Bloch representation of a qubit state $\rho = \frac{1}{2}(I+\mathbf{r}\cdot\boldsymbol{\sigma})$. Pure states lie on the surface ($\|\mathbf{r}\|=1$), mixed states in the interior.
• Figure 2: Hamiltonian evolution generated by $H = \tfrac{1}{2}\boldsymbol{\Omega}\cdot\boldsymbol{\sigma}$. The Bloch vector $\mathbf{r}(t)$ undergoes a rigid rotation about the axis $\boldsymbol{\Omega}$, corresponding to coadjoint motion on $\mathfrak{su}(2)^*$.
• Figure 3: Action of the double commutator $[L,[L,\rho]]$ for $L=\sigma_z$ on the Bloch sphere: the transverse component of the Bloch vector $\mathbf{r}(t)$ contracts exponentially toward the $z$-axis, while the longitudinal component is preserved. The contraction arises geometrically from the $\diamond$ coupling in the semidirect Euler--Poincaré equation.
• Figure 4: Diagram summarizing the geometric derivation of the GKSL equation from Euler--Poincaré reduction on the adjoint–coupled semidirect product $SU(2)\ltimes V$. The coadjoint term $\operatorname{ad}^*_{\xi}\mu$ produces, with a minus sign, the Hamiltonian commutator $-i[H,\rho]$, while the semidirect coupling $\frac{\delta\ell}{\delta\alpha}\diamond\alpha = [L,\alpha]$ generates the Lindblad double commutator via the choice $\alpha = \tfrac{\gamma}{2}[\rho,L]$.
• Figure 5: Pure dephasing along the $z$-axis produces an exponential contraction of the transverse components, driving the Bloch vector smoothly toward the $z$-axis while preserving $r_z$.

Theorems & Definitions (64)

• Definition 2.1
• Definition 2.2: Adjoint–coupled right action
• Definition 2.3: Adjoint torsion
• Remark 2.4: On the inevitability of the $\diamond$--induced nested commutator
• Remark 2.5
• Remark 2.6
• Lemma 3.1
• proof
• Proposition 3.2
• proof