Abstract
We study a question which has natural interpretations both in quantum mechanics and in geometry. Let \(V_{1},\cdots , V_{n}\) be complex vector spaces of dimension \(d_{1},\ldots ,d_{n}\) and let \(G= {\text {SL}}_{d_{1}} \times \cdots \times {\text {SL}}_{d_{n}}\). Geometrically, we ask: Given \((d_{1},\ldots ,d_{n})\), when is the geometric invariant theory quotient \(\mathbb {P}(V_{1}\otimes \cdots \otimes V_{n})/\!/G\) non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space \(V_{1}\otimes \cdots \otimes V_{n}\) has a locally maximally entangled state, i.e., a state such that the density matrix for each elementary subsystem is a multiple of the identity. We show that the answer to this question is yes if and only if \(R(d_{1},\cdots ,d_{n})\geqslant 0\) where
We also provide a simple recursive algorithm which determines the answer to the question, and we compute the dimension of the resulting quotient in the non-empty cases.
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Communicated by David Pérez-García.
Bryan and Reichstein were partially supported by Grants from the National Science and Engineering Council of Canada (NSERC). Van Raamsdonk was partially supported by Grants from NSERC and the Simons Foundation.
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Bryan, J., Reichstein, Z. & Van Raamsdonk, M. Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory. Ann. Henri Poincaré 19, 2491–2511 (2018). https://doi.org/10.1007/s00023-018-0682-6
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DOI: https://doi.org/10.1007/s00023-018-0682-6