Permutationally invariant processes in open multiqudit systems

TL;DR

This paper develops a comprehensive theoretical framework for describing the dynamics of permutationally invariant states in open multiqudit systems, leveraging Schur-Weyl duality to simplify analysis and avoid complex transformations.

Contribution

It introduces a novel approach that operates directly within the PI operator subspace, enabling efficient analysis of large multiqudit systems without computing the Schur transform.

Findings

Framework applies to both Markovian and non-Markovian dynamics.

Operates within a polynomially scaled subspace, improving computational efficiency.

Introduces the concept of $3 u$-symbol matrix for analysis.

Abstract

We establish the comprehensive theoretical framework for an exact description of the open system dynamics of permutationally invariant (PI) states in arbitrary $N$-qudit systems when this dynamics preserves the PI symmetry over time. Thanks to the powerful Schur-Weyl duality formalism, we unveil the internal links between the canonical time-local Lindblad-like master equation and the Markovian or non-Markovian dynamics of each permutationally-invariant degree of freedom (Schur subspaces). Our approach does not require one to compute the Schur transform as it operates directly within the restricted PI operator subspace of the Liouville space, whose dimension only scales polynomially with the number of qudits. We introduce the concept of $3\nu$-symbol matrix, where $\nu$ here denotes an integer partition, that proves to be very useful in this context.