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arXiv · 1609.01327

Almost all multipartite qubit quantum states have trivial stabilizer

Gilad Gour·Barbara Kraus·Nolan R. Wallach
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Abstract

The stabilizer group of an n-qubit state \psi is the set of all matrices of the form g=g_1\otimes\cdots\otimes g_n, with g_1,...,g_n being any 2x2 invertible complex matrices, that satisfy g\psi=\psi. We show that for 5 or more qubits, except for a set of states of zero measure, the stabilizer group of multipartite entangled states is trivial; that is, containing only the identity element. We use this result to show that for 5 or more qubits, the action of deterministic local operations and classical communication (LOCC) can almost always be simulated simply by local unitary (LU) operations. This proves that almost all n-qubit states with n>4 are isolated, that is they can neither be reached nor converted into any other (n-partite entangled), LU-inequivalent state via deterministic LOCC. We also find a simple and elegant expression for the maximal probability to convert one multi-qubit entangled state to another for this generic set of states.

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Gilad Gour, Barbara Kraus, Nolan R. Wallach. 2016-09-05. Almost all multipartite qubit quantum states have trivial stabilizer. https://doi.org/10.1063/1.5003015

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