Browsing by Subject "invariant theory"
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Item Open Access Hyperdeterminants, entangled states, and invariant theory(Bilkent University, 2013) Şen, EmreIn [1] and [2], A. Klyachko connects quantum entanglement and invariant theory so that entangled state of a quantum system can be explained by invariants of the system. After representing states in multidimensional matrices, this relation turns into finding multidimensional matrix invariants so called hyperdeterminants. Here we provide a necessary and sufficient condition for existence of a hyperdeterminant of a multidimensional matrix of an arbitrary format. The answer is given in terms of the so called castling transform that relates hyperdeterminants of different formats. Among castling equivalent formats there is a unique castling reduced one, that has minimal number of entries. We prove the following theorem: “Multidimensional matrices of a given format admit a non-constant hyperdeterminant if and only if logarithm of dimensions of the castling reduced format satisfy polygonal inequalities.”