Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory

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

$$\begin{aligned} R(d_{1},\cdots ,d_{n}) = \prod _{i}d_{i} +\sum _{k=1}^{n} (-1)^{k}\sum _{1\leqslant i_{1}<\cdots <i_{k}\leqslant n} \left( \gcd (d_{i_{1}},\cdots ,d_{i_{k}}) \right) ^{2}. \end{aligned}$$

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.