环形核证书用于PPT平方对角正交协变信道
Toroidal-Core Certificates for PPT-Squared Diagonal Orthogonal Covariant Channels
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中文总结 AI 辅助
针对PPT平方猜想，在DOC类信道中推导了任意有限维的充分条件，通过构造TCP核和逐项不等式保证复合为EB，并在四维及连续族中验证，提供秩敏感改进。
中文摘要 AI 辅助
正部分转置平方（PPT-squared）猜想询问两个正部分转置（PPT）量子信道的复合是否必须是纠缠破坏（EB）的。该问题在一般情况下仍然开放，并且在对角正交协变（DOC）类中，第一个未解决的确定性维度是四。针对任意有限维度的DOC映射，推导了PPT平方复合的充分条件。证明将量子相干数据隔离为两个相关矩阵，将每个矩阵与恒等矩阵混合，直到其位于秩一相关矩阵的凸包中，并将所得对象转换为显式的三重组完全正（TCP）核。剩余贡献纯粹是经典的，并且每当满足简单的逐项混合不等式时即为TCP。对于在$M_d$上的PPT DOC映射$\Phi_{A,B,C}$和$\Phi_{D,E,F}$，对所有$i,j$满足逐项界$(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}$保证了$\Phi_{A,B,C}\circ\Phi_{D,E,F}$是EB的。在维度四中，通用系数变为四。然后构造了一个连续的双随机PPT信道族；族中每个具有$a\ne1$的成员不是EB的，而族内每对成员的复合被证明是EB的。秩敏感的细化明确显示了低相干秩如何改善通用系数。所得证书是充分而非必要的，并且与因子宽度和半定层次方法互补。
英文摘要
The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps $Φ_{A,B,C}$ and $Φ_{D,E,F}$ on $M_d$, the entrywise bound $(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}$ for all $i,j$ guarantees that $Φ_{A,B,C}\circΦ_{D,E,F}$ is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with $a\ne1$ is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.
发表机构
- Rutgers University（罗格斯大学）
机构由 AI 辅助整理，请以论文原文为准。
