论文标题
没有产品基础的子空间的最大尺寸
Maximum Dimension of Subspaces with No Product Basis
论文作者
论文摘要
令$ n \ ge2 $和$ d_1，\ ldots，d_n \ ge2 $为整数，而$ \ mathcal {f} $是一个字段。 vector $ u \ in \ Mathcal {f}^{d_1} \ otimes \ cdots \ cdots \ otimes \ Mathcal \ Mathcal {f}^{d_n} $，如果$ u = u = u^{[1]} $ u^{[1]} \ in \ mathcal {f}^{d_1}，\ ldots，u^{[n]} \ in \ Mathcal {f}^{f}^{d_n} $。由产品向量组成的基础称为产品基础。在本文中，我们表明$ \ Mathcal {f}^{d_1} \ otimes \ cdots \ cdots \ otimes \ otimimes \ Mathcal {f}^{d_n} $的最大维$ \＃\ MATHCAL {F}> \ MAX \ {d_i：i \ not = n_1，n_2 \} $对于某些$ n_1 $和$ n_2 $。当$ \ mathcal {f} = \ mathbb {c} $时，此结果与一般概率理论（GPTS）中同时可区分状态的最大数量有关。
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).
