Abstract
Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.
Highlights
Due to the relevance of multipartite entanglement in many areas of physics, entanglement theory has developed into an important research topic over the last decades [1, 2]
It is apparent that any state |ψ, which can be deterministically converted to another state |φ via Local quantum Operations assisted by Classical Communication (LOCC), must be at least as entangled as |φ according to any reasonable measure of entanglement
In Subsection 4.4, we perform a complete characterization of the orbit types of stochastic LOCC (SLOCC) classes in 2 × m × n systems, which we summarize in form of a flowchart in Subsection 4.5
Summary
Due to the relevance of multipartite entanglement in many areas of physics, entanglement theory has developed into an important research topic over the last decades [1, 2]. In [39, 40], it has been shown that a numerical algorithm, which follows the gradient of the sum of the linear entropies of the eigenvalues of ρi, allows to determine the critical state within a polystable SLOCC class, or the critical state within the closure of a strictly semistable SLOCC class. The latter method allows to distinguish certain SLOCC classes within the null-cone [38, 40,41,42, 45, 46].
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