paper

Existence of locally maximally entangled quantum states via geometric invariant theory

arXiv:1708.01645 · doi:10.1007/s00023-018-0682-6

Abstract

We study a question which has natural interpretations in both quantum mechanics and in geometry. Let $V_{1}, ..., V_{n}$ be complex vector spaces of dimension $d_{1}, ..., d_{n}$ and let $G = S L_{d_{1}} \times \dots \times S L_{d_{n}}$ . Geometrically, we ask given $(d_{1}, ..., d_{n})$ , when is the geometric invariant theory quotient $P (V_{1} \otimes \dots \otimes V_{n}) // G$ non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space $V_{1} \otimes \dots \otimes V_{n}$ 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 $R (d_{1}, ..., d_{n}) ⩾ 0$ where \[ R(d_1,...,d_n) = \prod_i d_i +\sum_{k=1}^n (-1)^k \sum_{1\leq i_1<\dotsb <i_k\leq n} (\gcd(d_{i_1},\dotsc ,d_{i_k}) )^{2}. \] 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.

Improved the exposition and streamlined some proofs using results of Littelmann and Sato-Kimura