Partitioned Density Matrices and Entanglement Correlators

TL;DR

This paper introduces a framework for analyzing the dynamics of partitioned density matrices and entanglement correlators in multi-subsystem quantum systems, providing a hierarchy of equations of motion and detailed examples for spin-1/2 systems.

Contribution

It develops a hierarchy of equations of motion for partitioned density matrices and entanglement correlators, extending methods analogous to quantum field theory hierarchies.

Findings

Derived a hierarchy of equations linking subsystem dynamics

Reformulated equations in terms of entanglement correlators

Applied framework to coupled spin-1/2 systems

Abstract

The density matrix of a non-relativistic quantum system, divided into $N$ sub-systems, is rewritten in terms of the set of all partitioned density matrices for the system. For the case where the different sub-systems are distinguishable, we derive a hierarchy of equations of motion linking the dynamics of all the partitioned density matrices, analogous to the "Schwinger-Dyson" hierarchy in quantum field theory. The special case of a set of $N$ coupled spin-$1/2$ "qubits" is worked out in detail. The equations are then rewritten in terms of a set of "entanglement correlators", which comprise all the possible correlation functions for the system - this case is worked out for coupled spin systems. The equations of motion for these correlators can be written in terms of a first-order differential equation for an entanglement correlator supervector.