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 by Osterloh and Siewert (2010 New J. Phys. 12 075025). At first, I define maximally entangled states as being the stochastic states (with local reduced density matrices. 1/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) to a maximally entangled state. In particular the concept given here gives a condition that all maximal genuinely multi-qudit entangled states for general local Hilbert space dimension d have to satisfy. A general genuinely multi-qudit entangled state is an element of the partly balanced SU(d)-orbits.