论文标题
分解的Pauli对角线图和量子图的张量正方形
Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps
论文作者
论文摘要
由于E.størmer，每个正值图都可以分解为完全正面映射和完全共同的映射的总和，这是一个众所周知的结果。在这里，我们将此结果推广到量子图的张量正方形。具体而言，我们表明，带有自身的量子图的任何阳性张量产品都是可分解的。这解决了S. Fillipov和K. Magadov最近的猜想。我们将此结果与不可分解的正图作为两个不同的量子图的张量产物或可分解地图的张量平方的示例进行对比。为了显示我们的主要结果，我们将问题减少到Pauli对角线图。然后，我们通过确定所有252张极端射线的木马对角度图，表征可分解的木quart Pauli对角线图的锥体，这些图均完全正面且完全共同。这些极端射线在天然对称组下分为三个不连接轨道，其中两个轨道仅包含纠缠折断图。最后，我们开发了一种通用组合方法，以确定保利对角图的极端射线，该射线使用其Choi矩阵的有序光谱在多量器系统之间完全正面且完全共同呈阳性。将这些极端射线分类以外的木quart是一个空旷的问题。
It is a well-known result due to E. Størmer that every positive qubit map is decomposable into a sum of a completely positive map and a completely copositive map. Here, we generalize this result to tensor squares of qubit maps. Specifically, we show that any positive tensor product of a qubit map with itself is decomposable. This solves a recent conjecture by S. Fillipov and K. Magadov. We contrast this result with examples of non-decomposable positive maps arising as the tensor product of two distinct qubit maps or as the tensor square of a decomposable map from a qubit to a ququart. To show our main result, we reduce the problem to Pauli diagonal maps. We then characterize the cone of decomposable ququart Pauli diagonal maps by determining all 252 extremal rays of ququart Pauli diagonal maps that are both completely positive and completely copositive. These extremal rays split into three disjoint orbits under a natural symmetry group, and two of these orbits contain only entanglement breaking maps. Finally, we develop a general combinatorial method to determine the extremal rays of Pauli diagonal maps that are both completely positive and completely copositive between multi-qubit systems using the ordered spectra of their Choi matrices. Classifying these extremal rays beyond ququarts is left as an open problem.
