The Entanglement Hierarchy of 2 x m x n Systems

We consider three partite pure states in the Hilbert space of dimensions 2,m,n and investigate to which states a given state can be locally transformed with a non-vanishing probability. Whenever the initial and final state are elements of the same Hilbert space, the problem is solved via the characterization of the SLOCC classes. However, when considering transformations from higher to lower dimensional Hilbert spaces, a hierarchy among the states can be found. We build on results presented in [1], where a connection to linear matrix pencils has been drawn in order to study SLOCC classes in 2,m,n systems. We first show that a generic set of states of dimensions 2,m,n, where n=m is the union of infinitely many SLOCC classes. However, for n≠m, there exists a single SLOCC class which is generic. Using this result, we derive a hierarchy of SLOCC classes for generic states. We also investigate common resource states, which are those states which can be transformed to any state (not excluding any zero-measure set) in the smaller dimensional Hilbert space.

[1] E. Chitambar, C.A. Miller, and Y. Shi, J. Math. Phys. 51, 072205 (2010)