Title: 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.