Multipartite-entanglement monotones and polynomial invariants

TL;DR

This paper characterizes the conditions under which polynomial invariants serve as valid entanglement monotones in multipartite quantum systems, revealing limitations on the degree of such invariants and unifying various known invariants.

Contribution

It establishes that only homogeneous functions of degree four or less define valid entanglement monotones and provides a unified formalism for degree-four invariants across multipartite systems.

Findings

Power greater than one of N-tangle is not an entanglement monotone.

Degree-four invariants unify N-tangle and other polynomial invariants.

The results extend to multi-partite systems with even numbers of qudits.

Abstract

We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an N-qubit entanglement monotone if and only if the homogeneous degree is not larger than four. In particular this implies that any power larger than one of the well-known N-tangle (N > 2) is not an entanglement monotone anymore. We then describe a common basis and formalism for the N-tangle and other known invariant polynomials of degree four. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration. Finally we prove that an analogous statement holds for any multi-partite system with even number of qudits.