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

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.