Detection of many-body entanglement partitions in a quantum computer

TL;DR

This paper introduces a symmetry-based method for detecting and characterizing entanglement partitions in multipartite quantum systems, applicable to systems of three to four parties and scalable to larger systems, with implications for quantum computing and mathematics.

Contribution

It develops a novel symmetry-based framework for identifying all entanglement partitions and witnesses in multipartite states, including bound entangled states, and extends mathematical inequalities related to matrix immanants.

Findings

Characterized all entanglement partitions for three- and four-partite states.

Provided a family of witnesses for detecting many-body entanglement in large systems.

Established new inequalities between matrix immanants.

Abstract

We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, $m$-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.