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.
 M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, Cambridge, 2009).
 D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Info. Comput. 7, 401 (2007).
 B. Kraus, Phys. Rev. Lett. 104, 020504 (2010a).
 E. Chitambar, Phys. Rev. Lett. 107, 190502 (2011).
 T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano, Phys. Rev. Lett. 103, 070503 (2009).
 C. Eltschka and J. Siewert, Journal of Physics A: Mathematical and Theoretical 47, 424005 (2014).
 G. Gour and N. R. Wallach, Phys. Rev. Lett. 111, 060502 (2013).
 G. Kempf and L. Ness, in Algebraic Geometry, edited by K. Lønsted (Springer Berlin Heidelberg, Berlin, Heidelberg, 1979) pp. 233–243.
 A. Osterloh and J. Siewert, New Journal of Physics 12, 075025 (2010).
 G. Gour and N. R. Wallach, New Journal of Physics 13, 073013 (2011).
 T. Maciążek and A. Sawicki, Journal of Physics A: Mathematical and Theoretical 48, 045305 (2015).
 L. Kronecker, Sitzungsberichte d. Preußischen Adad. d. Wissenschaften , 763 (1890).
 M. Brion, Les cours du CIRM 1, 1 (2010).
 P. Tauvel and R. Yu, Introduction to actions of algebraic groups (Springer, 2015).
 F. Gantmacher, The Theory of Matrices, Vol. 1 and 2 (Chelsea Publishing Company, 1959).
 D. Sauerwein, A. Molnar, J. I. Cirac, and B. Kraus, Phys. Rev. Lett. 123, 170504 (2019).
 V. Hoskins, Lecture notes, Zurich (2012).
 Antoine Neven, David Gunn, Martin Hebenstreit, and Barbara Kraus, "Local Transformations of Multiple Multipartite States", arXiv:2007.06256.
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