Entanglement criteria for the bosonic and fermionic induced ensembles

TL;DR

This paper introduces bosonic and fermionic ensembles of density matrices, analyzing their entanglement properties and providing criteria for detecting entanglement in these quantum states.

Contribution

It develops new entanglement criteria for bosonic and fermionic ensembles and analyzes their properties, including the typical entanglement behavior and PPT criterion failure conditions.

Findings

Fermionic ensembles are typically entangled due to non-positive partial transposition.

In the bosonic case, entanglement detection is complicated by a large positive eigenvalue.

The asymptotic ratio for the environment size where bosonic states are entangled is computed.

Abstract

We introduce the bosonic and fermionic ensembles of density matrices and study their entanglement. In the fermionic case, we show that random bipartite fermionic density matrices have non-positive partial transposition, hence they are typically entangled. The similar analysis in the bosonic case is more delicate, due to a large positive outlier eigenvalue. We compute the asymptotic ratio between the size of the environment and the size of the system Hilbert space for which random bipartite bosonic density matrices fail the PPT criterion, being thus entangled. We also relate moment computations for tensor-symmetric random matrices to evaluations of the circuit-counting and interlace graph polynomials for directed graphs.