Stellar Representation of Multipartite Antisymmetric States

Symmetric Multiqudit States: Stars, Entanglement, Rotosensors

“…A second duality relation is also mentioned in [48], namely , ( ) = ,2 +1− ( ), where , ( ) is the generating function for the multiplicities in the fully antisymmetric case. This latter relation is made obvious if one realizes that totally antisymmetricpartite ∧-factorizable spin-states (i.e., -fold wedge products of spin-states) are in 1-to-1 correspondence with -planes in the spin-Hilbert space, and the fact that a -plane and its orthogonal complement, a (2 + 1 − )-plane, carry the same geometrical information (for more details see, e.g., [33]). Since a general antisymmetric state is a linear combination of ∧-factorizable ones, the above isomorphism, which we denote by , extends to the entire  ∧ ,…”

“…The problem of the decomposition of the -fold tensor product of the spin-irrep of (2) has been considered before, both by physicists and mathematicians. Related material, from a physical perspective, can be found in [50,51], the analogous problem for fully antisymmetric states is examined in [33], while a more mathematical approach is undertaken in [52]. Both the fully symmetric and antisymmetric cases considered here and in [33] are particular cases of the general plethysm problem, which is still open (see, e.g., [53,54]).…”

“…(2) transformations of the state result in (corresponding) rigid rotations of the constellation -more details can be consulted in [11,26,33,39].…”

“…For an alternative approach to the problem of encoding the complex weights see [32]. By an argument identical to the one given in [33] for the case of permutation antisymmetric states, it can be shown that a particular choice of the reference kets | ⟩, which we will call the canonical section, results in being invariant under rotations (except for a subset of states in  ∨ of measure zero), while rotate as Majorana constellations do. Since we will follow that recipe in the examples that follow, we summarize the procedure here, referring the reader to [33] for further details and proofs (keep in mind though that the notation and the exposition here is slightly different, aiming at improved clarity).…”

“…The stellar representation of Majorana, originally designed for pure states of a monopartite system [1], can be also generalized for mixed quantum states [32]. In the case of multipartite systems, the stellar representation is directly applicable to symmetric states of a system composed of several qubits, but a generalization applicable to antisymmetric states has also been established [33].…”