Entanglement classes of permutation-symmetric qudit states: symmetric operations suffice

TL;DR

This paper classifies permutation-symmetric qudit states into entanglement classes using symmetric local operations, introducing new state representatives and invariants that generalize well-known qubit states.

Contribution

It demonstrates that entanglement classification can be restricted to symmetric operations and introduces excitation states as generalizations of W states for higher-dimensional systems.

Findings

Symmetric operations suffice for entanglement classification.

Jordan form of one-particle operators is a SLOCC invariant.

Introduces excitation states as generalizations of W states.

Abstract

We analyse entanglement classes for permutation-symmetric states for n qudits (i.e. d-level systems), with respect to local unitary operations (LU-equivalence) and stochastic local operations and classical communication (SLOCC equivalence). In both cases, we show that the search can be restricted to operations where the same local operation acts on all qudits, and we provide an explicit construction for it. Stabilizers of states in the form of one-particle operations preserving permutation symmetry are shown to provide a coarse-grained classification of entanglement classes. We prove that the Jordan form of such one-particle operator is a SLOCC invariant. We find, as representatives of those classes, a discrete set of entangled states that generalize the GHZ and W state for the many-particle qudit case. In the later case, we introduce "excitation states" as a natural generalization of the W state for d>2.