Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

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 $G= SL_{d_1} \times \dots \times SL_{d_n}$. Geometrically, we ask given $(d_1,...,d_n)$, when is the geometric invariant theory quotient $\mathbb{P}(V_1 \otimes \dots \otimes V_n)// G$ 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 $R(d_1,...,d_n)\geqslant 0$ 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.