Maximum dimension of subspaces with no product basis
Maximum dimension of subspaces with no product basis
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DOI：
10.1016/j.laa.2021.03.001
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发表时间：
2020-10
影响因子：
1.1
通讯作者：
Yuuya Yoshida
中科院分区：
文献类型：
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作者：
Yuuya Yoshida
设n≥ 2，d1，…，dn ≥ 2为整数，F为域.一个向量u∈ F d 1 F d n称为积向量，如果u= u [1] u [n]，其中u [1]∈ F d 1，.，u [n]∈ F d n.由乘积向量组成的基称为乘积基。本文证明了无乘积基的Fd 1 <$$> Fdn的子空间的最大维数等于d1 d2 <$dn − 2，如果（i）n= 2或（ii）n≥ 3且对某个n1和n2，# F> max <${di：i <$n1，n2}.当F= C时，这一结果与一般概率理论（GPTs）中同时可区分状态的最大数目有关。
Let n≥ 2 and d 1,…, d n≥ 2 be integers, and F be a field. A vector u∈ F d 1⊗⋯⊗ F d n is called a product vector if u= u [1]⊗⋯⊗ u [n] for some u [1]∈ F d 1,…, u [n]∈ 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⊗⋯⊗ F d n with no product basis is equal to d 1 d 2⋯ d n− 2 if either (i) n= 2 or (ii) n≥ 3 and# F> max{d i: i≠ 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).