Reduced density matrix of nonlocal identical particles

TL;DR

This paper explores the theoretical links between three approaches to analyzing entanglement in identical particles, demonstrating how reduced density matrices can be consistently computed across these frameworks.

Contribution

It establishes the equivalence of symmetrized partial trace, interior product in SEA, and subalgebra restriction in GNS algebra, unifying different perspectives on entanglement analysis.

Findings

Symmetrized partial trace corresponds to interior product in SEA.

Interior product in SEA aligns with subalgebra restriction in GNS.

The work bridges different theoretical approaches to quantum entanglement of identical particles.

Abstract

We probe the theoretical connection among three different approaches to analyze the entanglement of identical particles, i.e., the first quantization language (1QL), elementary-symmetric/exterior products (which has the mathematical equivalence to no-labeling approaches), and the algebraic approach based on the GNS construction. Among several methods to quantify the entanglement of identical particles, we focus on the computation of reduced density matrices, which can be achieved by the concept of \emph{symmetrized partial trace} defined in 1QL. We show that the symmetrized partial trace corresponds to the interior product in symmetric and exterior algebra (SEA), which also corresponds to the subalgebra restriction in the algebraic approach based on GNS representation. Our research bridges different viewpoints for understanding the quantum correlation of identical particles in a consistent manner.