Genuinely multipartite entangled states in higher dimensions: a generalization of balancedness

TL;DR

This paper extends the concept of balancedness to higher-dimensional qudits, characterizing maximally entangled states as balanced and establishing their properties and classifications within local unitary transformations.

Contribution

It generalizes the notion of balancedness to arbitrary qudit dimensions and links maximally entangled states to balanced states and local $SL(d)$-equivalence.

Findings

Maximally entangled states are balanced and stochastic.

All irreducibly balanced states are genuinely multi-qudit entangled.

Genuinely multi-qudit entangled states form $SU(d)$-orbits.

Abstract

I generalize the concept of balancedness to qudits with arbitrary dimension $d$. It is an extension of the concept of balancedness in New J. Phys. {\bf 12}, 075025 (2010) [1]. At first, I define maximally entangled states as being the stochastic states (with local reduced density matrices $\id/d$ for a $d$-dimensional local Hilbert space) that are not product states and show that every so-defined maximal genuinely multi-qudit entangled state is balanced. Furthermore, all irreducibly balanced states are genuinely multi-qudit entangled and are locally equivalent with respect to $SL(d)$ transformations (i.e. the local filtering transformations (LFO)) to a maximally entangled state. In particular the concept given here gives the maximal genuinely multi-qudit entangled states for general local Hilbert space dimension $d$. All genuinely multi-qudit entangled states are an element of the partly balanced $SU(d)$-orbits.