Quantum Evolution of Hopf Algebra Hamiltonians

TL;DR

This paper investigates whether deformed Hopf algebra symmetries can lead to physically consistent Lindblad-type decoherence in quantum systems, concluding that such a framework cannot be established for the considered models.

Contribution

It provides a detailed analysis showing that deformed Hopf algebra symmetries do not yield physically viable Lindblad evolutions for qubits.

Findings

Generalized adjoint actions ensure von Neumann dynamics

Deformed spacetime symmetries do not produce viable Lindblad evolution

The framework for decoherence via Hopf algebra deformation is not physically consistent

Abstract

In recent years, growing attention has been devoted to the possibility that theories with deformed symmetries, associated with certain models of non-commutative spacetime, may encode a fundamental form of decoherence. This effect should be described by a Lindblad-like evolution governed by the non-trivial Hopf algebra structure of the time-evolution generators. In this work we provide a detailed analysis of such possibility for similar Hopf algebra deformations of the Hamiltonian of a qubit. Starting from a critical examination of the very definition of time evolution through the generalized adjoint action, we explore whether a coherent and physically viable framework can be established. In particular, our analysis shows that a more general combination of adjoint actions always guarantees a von Neumann dynamics and, also in the case of deformed spacetime symmetries considered in the literature, a physically viable Lindblad evolution cannot be established.