Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, and $\mathcal{ F}$ be a field. A vector $u\in\mathcal{ F}^{ d_1}\otimes\cdots\otimes\mathcal{ F}^{ d_n}$ is called a product vector if $u=u^{ [1]}\otimes\cdots\otimes u^{ [n]}$ for some $u^{ [1]}\in\mathcal{ F}^{ d_1},\ldots,u^{ [n]}\in\mathcal{ F}^{ d_n}$. A basis composed of product vectors is called a product basis. In this paper, we show that the maximum dimension of subspaces of $\mathcal{ F}^{ d_1}\otimes\cdots\otimes\mathcal{ F}^{ d_n}$ with no product basis is equal to $d_1d_2\cdots d_n-2$ if either (i) $n=2$ or (ii) $n\ge3$ and $\#\mathcal{ F}>\max\{ d_i : i\not=n_1,n_2\}$ for some $n_1$ and $n_2$. When $\mathcal{ F}=\mathbb{ C}$, this result is related to the maximum number of simultaneously distinguishable states in general probabilistic theories (GPTs).
Categories: math.CO quant-ph