Lindblad Quantum Dynamics as Euler-Poincar\'e Reduction on Adjoint-Coupled Semidirect Products

TL;DR

This paper derives the Lindblad quantum dynamics using geometric variational methods on semidirect product structures, revealing the underlying geometric origin of dissipative quantum evolution and providing explicit equations for specific groups.

Contribution

It presents a novel geometric derivation of Lindblad equations via Euler--Poincaré reduction on adjoint-coupled semidirect products, linking dissipation to curvature and torsion in the geometric framework.

Findings

Derivation of Lindblad equations from geometric reduction

Explicit Bloch equations for SU(2) and SU(3) channels

Establishment of a geometric framework connecting dissipation and curvature

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 $K(\xi,v)$. 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 $-\tfrac{\gamma}{2}[L,[L,\rho]]$ 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.