Correlation and Entanglement partners in Gaussian systems

TL;DR

This paper develops a framework to identify and characterize the distribution of correlations and entanglement across subsystems in Gaussian quantum systems, revealing unique partner modes for pure states and a split into classical and quantum correlation partners for mixed states.

Contribution

It introduces a novel method to identify correlation and entanglement partners in Gaussian systems, including explicit constructions for pure and mixed states, extending to multi-mode subsystems.

Findings

Pure Gaussian states have unique single-mode correlation partners.

Mixed Gaussian states have separate classical and entanglement partners.

The framework applies to complex many-body Gaussian systems.

Abstract

We introduce a framework to identify where the total correlations and entanglement with a chosen degree of freedom reside within the rest of a system, in the context of bosonic many-body Gaussian quantum systems. Our results are organized into two main propositions. First, for pure Gaussian states, we show that every correlated mode possesses a unique single-degree-of-freedom partner that fully captures its correlations (consisting of entanglement), and we provide an explicit construction of this partner from the complex structure of the system's state. Second, for mixed Gaussian states, we constructively demonstrate that the notion of a partner subsystem splits into two: a correlation partner, which contains all classical and quantum correlations and need not correspond to a single degree of freedom, and an entanglement partner, which is always at most single-mode. Finally, we extend the construction of partners to multi-mode subsystems. Together, these results provide conceptual practical tools to study how bipartite correlations and entanglement are structured and where they can be found in complex Gaussian many-body systems.