We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad-type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn–Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. Furthermore, we have devised another method of detection NM of various qubit channels, by using projective measurements in the Bell basis. This can be done without the knowledge of what specific kind of channel we are dealing with. This gives us a huge operational advantage for NM detection. We further investigate the geometric structure of the Markovian Choi states to find that they do not form a polytope. This presents a platform to consider nonlinear improvement of non-Markovianity witnesses.

我们开发了一种用于检测非马尔可夫性的线性见证理论，基于所有具有 Lindblad 型生成器的马尔可夫进化的 Choi 状态集的几何结构。我们表明所有这些马尔可夫崔状态的集合在小时间间隔近似下形成一个凸的和紧的集合。调用几何 Hahn-Banach 定理，我们构造线性见证以将给定的非马尔可夫 Choi 状态与马尔可夫 Choi 状态集合分开。在量子比特的情况下，我们展示了用于去相位通道和泡利通道的此类见证的示例。此外，我们设计了另一种检测各种量子比特通道 NM 的方法，通过在 Bell 基础中使用投影测量。这可以在不知道我们正在处理的特定渠道类型的情况下完成。这为我们提供了 NM 检测的巨大操作优势。我们进一步研究 Markovian Choi 状态的几何结构，发现它们不形成多面体。这提供了一个平台来考虑非马尔可夫证人的非线性改进。