Maximum Dimension of Subspaces with No Product Basis

Yuuya Yoshida

doi:10.1016/j.laa.2021.03.001

paper

Maximum Dimension of Subspaces with No Product Basis

arXiv:2010.16293 · doi:10.1016/j.laa.2021.03.001

Abstract

Let $n \geq 2$ and $d_{1}, \dots, d_{n} \geq 2$ be integers, and $F$ be a field. A vector $u \in F^{d_{1}} \otimes \dots \otimes F^{d_{n}}$ is called a product vector if $u = u^{[1]} \otimes \dots \otimes u^{[n]}$ for some $u^{[1]} \in F^{d_{1}}, \dots, u^{[n]} \in F^{d_{n}}$ . A basis composed of product vectors is called a product basis. In this paper, we show that the maximum dimension of subspaces of $F^{d_{1}} \otimes \dots \otimes F^{d_{n}}$ with no product basis is equal to $d_{1} d_{2} \dots d_{n} - 2$ if either (i) $n = 2$ or (ii) $n \geq 3$ and $# F > max {d_{i} : i \neq = n_{1}, n_{2}}$ for some $n_{1}$ and $n_{2}$ . When $F = C$ , this result is related to the maximum number of simultaneously distinguishable states in general probabilistic theories (GPTs).

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