Abstract
We study a question which has natural interpretations in both quantum mechanics and in geometry. Let V₁,..., V_n be complex vector spaces of dimension d₁,...,d_n and let G= SL_d₁ × … × SL_d_n. Geometrically, we ask given (d₁,...,d_n), when is the geometric invariant theory quotient P(V₁ ⊗ … ⊗ V_n)// G non-empty? This is equivalent to the quantum mechanical question of whether the multipart quantum system with Hilbert space V₁⊗ … ⊗ 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(d₁,...,d_n) 0 where R(d₁,...,d_n) = Π_i d_i +Σ_k=1ⁿ (-1)^k Σ_1≤ i₁< <i_k≤ n (gcd(d_i₁, ,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.