绝对PPT态的光谱优化：纯度、熵与体积衰减
Spectral Optimization for Absolutely PPT States: Purity, Entropy, and Volume Decay
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中文总结 AI 辅助
本文通过光谱多胞形优化，给出了APPT态纯度的改进上界和熵下界，并精确刻画了qubit-qudit光谱体积的指数衰减率。
中文摘要 AI 辅助
我们研究绝对正部分转置（APPT）态的最大纯度和最小冯·诺依曼熵，以及其光谱集的相对体积。Hildebrand准则的一个推论给出了一个显式外部光谱多胞形，我们对其顶点进行了分类。在该多胞形上优化纯度，结合Song-Chen对$2\otimes3$系统的结果，对每个$mn\ge6$，给出了APPT态纯度的显式上界，该上界渐近于$4/(3mn)$。这改进了绝对可分态之前的$2/(mn)$上界。新界适用于每个APPT态（包含所有绝对可分态的类别），并且对每个$n\ge3$的$2\otimes n$系统是紧的。然而，对于每个$n\ge3$的$3\otimes n$系统，唯一的外部多胞形最大化子不是APPT，因此该界是严格的，并反驳了Dũng-Khôi qutrit-qudit猜想。该多胞形还给出了任意二分维度下的显式熵下界，并结合Song-Chen的极端点分类，给出了每个$2\otimes n$系统的精确最小熵。最后，内多胞形和外部光谱多胞形的相对体积的精确公式给出了qubit-qudit相对光谱体积$a_n$的显式双侧界，其比率小于4且趋于3。因此，$a_n=\Theta\\!\left(\sqrt n(4/27)^n\right)$，且qubit-qudit APPT光谱集（等价地，绝对可分光谱集）的相对体积具有精确的指数衰减率$\ln(27/4)$。
英文摘要
We study maximum purity and minimum von Neumann entropy of absolutely positive partial transpose (APPT) states, together with the relative volume of their spectral sets. A consequence of Hildebrand's criterion gives an explicit outer spectral polytope, whose vertices we classify. Optimizing purity over this polytope, together with the Song--Chen result for the $2\otimes3$ system, yields, for every $mn\ge6$, an explicit upper bound on the purity of APPT states that is asymptotic to $4/(3mn)$. This improves the previous $2/(mn)$ upper bound for absolutely separable states. The new bound applies to every APPT state, a class containing all absolutely separable states, and is sharp for every $2\otimes n$ system with $n\ge3$. For every $3\otimes n$ system with $n\ge3$, however, the unique outer-polytope maximizer is not APPT, so the bound is strict and disproves the Dũng--Khôi qutrit--qudit conjecture. The polytope also gives an explicit entropy lower bound in arbitrary bipartite dimensions and, together with the Song--Chen extreme-point classification, the exact minimum entropy for every $2\otimes n$ system. Finally, exact formulas for the relative volumes of an inner polytope and the outer spectral polytope give explicit two-sided bounds on the qubit--qudit relative spectral volume $a_n$ whose ratio is less than $4$ and tends to $3$. Consequently, $a_n=Θ\!\left(\sqrt n(4/27)^n\right)$, and the relative volume of the qubit--qudit APPT spectral set (equivalently, the absolutely separable spectral set) has exact exponential decay rate $\ln(27/4)$.
发表机构
- The University of Texas at Dallas（德克萨斯大学达拉斯分校）
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