论文标题
Grothendieck在单个量子系统中绑定
Grothendieck bound in a single quantum system
论文作者
论文摘要
Grothendieck的界限用于单个量子系统的上下文，与以前的工作相比，该工作将其用于多部分纠缠系统和违反类似钟形的不平等现象。粗略地说Grothendieck Theorem认为使用单位光盘中的复数数字$ {\ cal c} $，并将值小于$ 1 $。然后证明，如果复数被希尔伯特空间的单位球中的向量替换，则“量子”二次形式$ {\ cal q} $可能会使值大于$ 1 $，直至复杂的grothendieck常数$ k_g $。 Grothendieck定理在此根据任意矩阵（将其乘以适当的归一化预成分倍数）进行重新校正，以便它直接适用于量子量。本文中的重点是在“ grothendieck区域” $（1，k_g）$中，这是一个经典的禁区，从某种意义上说，$ {\ cal c} $无法在其中占据价值。给出了$ {\ cal Q} $在Grothendieck区域中取值的必要条件（但不够）。两个示例涉及$ 6 $和$ 12 $二维的希尔伯特空间的系统中的物理数量，显示出在Grothendieck区域$（1，k_g）$的$ {\ cal Q} $中。它们涉及解决身份并具有离散各向同性的新型广义状态的重叠的投影仪。
Grothendieck's bound is used in the context of a single quantum system, in contrast to previous work which used it for multipartite entangled systems and the violation of Bell-like inequalities. Roughly speaking the Grothendieck theorem considers a `classical' quadratic form ${\cal C}$ that uses complex numbers in the unit disc, and takes values less than $1$. It then proves that if the complex numbers are replaced with vectors in the unit ball of the Hilbert space, then the `quantum' quadratic form ${\cal Q}$ might take values greater than $1$, up to the complex Grothendieck constant $k_G$. The Grothendieck theorem is reformulated here in terms of arbitrary matrices (which are multiplied with appropriate normalisation prefactors), so that it is directly applicable to quantum quantities. The emphasis in the paper is in the `Grothendieck region' $(1,k_G)$, which is a classically forbidden region in the sense that ${\cal C}$ cannot take values in it. Necessary (but not sufficient) conditions for ${\cal Q}$ taking values in the Grothendieck region are given. Two examples that involve physical quantities in systems with $6$ and $12$-dimensional Hilbert space, are shown to lead to ${\cal Q}$ in the Grothendieck region $(1,k_G)$. They involve projectors of the overlaps of novel generalised coherent states that resolve the identity and have a discrete isotropy.
