Maximum Dimension of Subspaces with No Product Basis

Abstract

Let $n\ge 2$ and $d_1,\dots ,d_n\ge 2$ 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 $u^{[1]}\in {\mathcal{F}}^{d_1},\dots ,u^{[n]}\in {\mathcal{F}}^{d_n}$. 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 $d_1{d}_2\cdots d_n-2$ if either (i) $n=2$ or (ii) $n\ge 3$ and $\#\mathcal{F}>\max \{d_i:i\ne n_1,n_2\}$ 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).

14 pages

References in corpus (2)

• A derivation of quantum theory from physical requirements
• Entangled states close to the maximally mixed state