置换不变贝尔算子的stoquasticity锥的精确半定刻画

中文摘要 AI 辅助

贝尔相关性已在从数百到数十万粒子的原子系综中被检测到，这由置换不变（PI）贝尔算子所见证。对于迄今为止实现的所有算子，最大违反态可以在合适的基下转化为非负振幅形式。在该基下，贝尔算子是stoquastic的。参数化这些算子且同时导致stoquasticity的系数构成一个凸锥，即所谓的stoquasticity锥。其刻画在二体算子之外一直悬而未决，因为对于多体算子，不等式的数量随系统规模$n$增长。在这项工作中，我们表明在热力学极限下，stoquasticity锥允许一个精确的半定刻画，该刻画在任何算子阶数下都有效。关键观察是，沿每个非对角元，矩阵元素是单个单变量多项式的值。由于是单变量的，该条件等价于一个线性矩阵不等式，无需松弛层次结构。因此，判定stoquasticity变为一个单一的半定可行性问题。作为其最低阶实例，该描述恢复了对二体算子的已知三超平面刻画，并为三体算子产生了一个弯曲边界。

英文摘要

Bell correlations have been detected in atomic ensembles ranging from hundreds to hundreds of thousands of particles, witnessed by the permutationally invariant (PI) Bell operators. For all the operators realized so far, the maximally violating state can be brought, in a suitable basis, into a nonnegative-amplitude form. In that basis, the Bell operator is stoquastic. The coefficients parametrising these operators that at the same time lead to stoquasticity form a convex cone, the so-called stoquasticity cone. Its characterization had remained open beyond two-body operators, as for many-body operators the number of inequalities grows with the system size $n$. In this work we show that, in the thermodynamic limit, the stoquasticity cone admits an exact semidefinite characterization, valid at any operator order. The key observation is that, along each off-diagonal, the matrix elements are the values of a single univariate polynomial. Being univariate, this condition is equivalent to a linear matrix inequality, with no relaxation hierarchy. Therefore, deciding stoquasticity becomes a single semidefinite feasibility problem. As its lowest-degree instances, this description recovers the known three-hyperplane characterization for two-body operators and yields, for three-body operators, a curved boundary.

发表机构

• Universiteit Leiden（莱顿大学）
• Instituut-Lorentz, Universiteit Leiden（莱顿大学洛伦兹研究所）
• LIACS, Universiteit Leiden（莱顿大学计算科学研究所）

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