Decomposability of Linear Maps under Tensor Products
Decomposability of Linear Maps under Tensor Products
复制标题
张量积下线性映射的可分解性
DOI：
10.1063/1.5045559
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
Alexander Muller
中科院分区：
文献类型：
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作者：
Alexander Muller
完全正映射和完全余正映射在张量幂下都是可分解的，即在线性映射自身张量的情况下是可分解的。但是，有没有其他具有这种属性的地图的例子？我们证明了情况并非如此：任何既不是完全正的也不是完全余正的可分解映射，在取足够的张量次方后，最终将失去可分解性。此外，我们建立了明确的界限来量化这种情况发生的时间。为了证明这些结果，我们使用了来自纠缠蒸馏理论的对称化技术，并分析了某些对称映射在取张量次方后变得不可分解的情况。最后，我们应用我们的结果构造了不可分解正映射的新例子，并建立了与PPT平方猜想的联系。
Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually after taking enough tensor powers. Moreover, we establish explicit bounds to quantify when this happens. To prove these results we use a symmetrization technique from the theory of entanglement distillation, and analyze when certain symmetric maps become non-decomposable after taking tensor powers. Finally, we apply our results to construct new examples of non-decomposable positive maps, and establish a connection to the PPT squared conjecture.