In this short note, we show two completely opposite methods of constructing bipartite entangled states. Given a bipartite state $$\gamma \in M_k\otimes M_k$$ γ ∈ M k ⊗ M k , define $$\gamma _S=(Id+F)\gamma (Id+F)$$ γ S = ( I d + F ) γ ( I d + F ) , $$\gamma _A=(Id-F)\gamma (Id-F)$$ γ A = ( I d - F ) γ ( I d - F ) , where $$F\in M_k\otimes M_k$$ F ∈ M k ⊗ M k is the flip operator. In the first method, entanglement is a consequence of the inequality $$ {\text {rank}}(\gamma _S)<\sqrt{{\text {rank}}(\gamma _A)}$$ rank ( γ S ) < rank ( γ A ) . In the second method, there is no correlation between $$\gamma _S$$ γ S and $$\gamma _A$$ γ A . These two methods show how diverse is quantum entanglement. We show that any bipartite state $$\gamma \in M_k\otimes M_k$$ γ ∈ M k ⊗ M k satisfies $$\begin{aligned} \displaystyle \mathrm{SN}(\gamma )\ge \max \left\{ \frac{ {\text {rank}}(\gamma _L)}{ {\text {rank}}(\gamma )}, \frac{ {\text {rank}}(\gamma _R)}{ {\text {rank}}(\gamma )}, \frac{\mathrm{SN}(\gamma _S)}{2}, \frac{\mathrm{SN}(\gamma _A)}{2} \right\} , \end{aligned}$$ SN ( γ ) ≥ max rank ( γ L ) rank ( γ ) , rank ( γ R ) rank ( γ ) , SN ( γ S ) 2 , SN ( γ A ) 2 , where $$\mathrm{SN}(\gamma )$$ SN ( γ ) stands for the Schmidt number of $$\gamma $$ γ and $$\gamma _L$$ γ L and $$\gamma _R$$ γ R are the marginal states of $$\gamma $$ γ . These inequalities are useful to compute the Schmidt number of many bipartite states. We prove that $$\mathrm{SN}(\gamma )=\min \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}$$ SN ( γ ) = min { rank ( γ L ) , rank ( γ R ) } , if $$\displaystyle {\text {rank}}(\gamma )= \frac{\max \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}}{\min \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}}$$ rank ( γ ) = max { rank ( γ L ) , rank ( γ R ) } min { rank ( γ L ) , rank ( γ R ) } . We also present a family of PPT states in $$M_k\otimes M_k$$ M k ⊗ M k , whose members have Schmidt number equal to n , for any given $$\displaystyle 1\le n\le \left\lfloor \frac{k}{2}\right\rfloor $$ 1 ≤ n ≤ k 2 . This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.

中文翻译：

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二分状态施密特数的不等式

在这个简短的说明中，我们展示了构建二分纠缠态的两种完全相反的方法。给定一个二分状态 $$\gamma\in M_k\otimes M_k$$ γ ∈ M k ⊗ M k ，定义 $$\gamma _S=(Id+F)\gamma (Id+F)$$ γ S = ( I d + F ) γ ( I d + F ) , $$\gamma _A=(Id-F)\gamma (Id-F)$$ γ A = ( I d - F ) γ ( I d - F ) ，其中$$F\in M_k\otimes M_k$$ F ∈ M k ⊗ M k 是翻转算子。在第一种方法中，纠缠是不等式 $$ {\text {rank}}(\gamma _S)<\sqrt{{\text {rank}}(\gamma _A)}$$ rank ( γ S ) 的结果<等级（γA）。在第二种方法中，$$\gamma _S$$ γ S 和$$\gamma _A$$ γ A 之间没有相关性。这两种方法显示了量子纠缠的多样性。我们证明任何二分状态 $$\gamma \in M_k\otimes M_k$$ γ ∈ M k ⊗ M k 满足 $$\begin{aligned} \displaystyle \mathrm{SN}(\gamma )\ge \max \left \{ \frac{ {\text {rank}}(\gamma _L)}{ {\text {rank}}(\gamma )}, \frac{ {\text {rank}}(\gamma _R)}{ { \text {rank}}(\gamma )}, \frac{\mathrm{SN}(\gamma _S)}{2}, \frac{\mathrm{SN}(\gamma _A)}{2} \right\ } , \end{aligned}$$ SN ( γ ) ≥ max rank ( γ L ) rank ( γ ) , rank ( γ R ) rank ( γ ) , SN ( γ S ) 2 , SN ( γ A ) 2 , 其中$$\mathrm{SN}(\gamma )$$ SN ( γ ) 表示 $$\gamma $$ γ 和 $$\gamma _L$$ γ L 和 $$\gamma _R$$ γ R 的施密特数是 $$\gamma $$ γ 的边缘状态。这些不等式可用于计算许多二分状态的施密特数。我们证明 $$\mathrm{SN}(\gamma )=\min \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}$$ SN ( γ ) = min { rank ( γ L ) , rank ( γ R ) } ，如果 $$\displaystyle {\text {rank}}(\ gamma )= \frac{\max \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}}{\min \{ {\text {rank}} (\gamma _L), {\text {rank}}(\gamma _R)\}}$$ rank ( γ ) = max { rank ( γ L ) , rank ( γ R ) } min { rank ( γ L ) ,秩(γR)}。我们还在 $$M_k\otimes M_k$$ M k ⊗ M k 中展示了一系列 PPT 状态，其成员的施密特数等于 n ，对于任何给定的 $$\displaystyle 1\le n\le \left\lfloor \ frac{k}{2}\right\rfloor $$ 1 ≤ n ≤ k 2 。这是对寻找 PPT 状态的最佳施密特数这一开放问题的新贡献。{\text {rank}}(\gamma _R)\}}$$ rank ( γ ) = max { rank ( γ L ) , rank ( γ R ) } min { rank ( γ L ) , rank ( γ R ) } . 我们还在 $$M_k\otimes M_k$$ M k ⊗ M k 中展示了一系列 PPT 状态，其成员的施密特数等于 n ，对于任何给定的 $$\displaystyle 1\le n\le \left\lfloor \ frac{k}{2}\right\rfloor $$ 1 ≤ n ≤ k 2 。这是对寻找 PPT 状态的最佳施密特数这一开放问题的新贡献。{\text {rank}}(\gamma _R)\}}$$ rank ( γ ) = max { rank ( γ L ) , rank ( γ R ) } min { rank ( γ L ) , rank ( γ R ) } . 我们还在 $$M_k\otimes M_k$$ M k ⊗ M k 中展示了一系列 PPT 状态，其成员的施密特数等于 n ，对于任何给定的 $$\displaystyle 1\le n\le \left\lfloor \ frac{k}{2}\right\rfloor $$ 1 ≤ n ≤ k 2 。这是对寻找 PPT 状态的最佳施密特数这一开放问题的新贡献。