Maximum Dimension of Subspaces with No Product Basis

TL;DR

This paper determines the maximum dimension of subspaces in tensor product spaces over a field that contain no product basis, revealing a precise upper bound under certain conditions.

Contribution

It establishes an exact maximum dimension for subspaces without product bases in tensor spaces, extending previous understanding in quantum and probabilistic theories.

Findings

Maximum dimension is $d_1d_2\cdots d_n - 2$ for certain cases.

The result applies to complex fields and relates to distinguishable states in GPTs.

Provides bounds for subspaces lacking product bases in tensor product spaces.

Abstract

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).