Gaussian-to-Gaussian超算子的规范形分解
Normal form decomposition for Gaussian-to-Gaussian superoperators
摘要：
在本文中，我们探讨了一组线性映射发送到自己的量子高斯态的集合。这些映射一般不是正的，这是一个可以用来检验给定的量子态是否属于高斯态凸包的特征（如果所考虑的映射之一将其发送到非正算子中，则上述状态被证明不属于集合）。推广的结果已知是有效的假设下，完全积极的，我们提供了一个表征这些高斯到高斯（不一定是积极的）超算子的特征函数的输入。对于单模映射的特殊情况下，我们还表明，任何高斯到高斯超算子可以表示为相空间膨胀的串联，其次是一个完全正的高斯通道的行动，可能与换位组成。虽然一个类似的分解失败的多模式的情况下，我们证明，它仍然持有至少在协方差矩阵上的均匀行动的进一步假设。
英文摘要：
In this paper we explore the set of linear maps sending the set of quantum Gaussian states into itself. These maps are in general not positive, a feature which can be exploited as a test to check whether a given quantum state belongs to the convex hull of Gaussian states (if one of the considered maps sends it into a non positive operator, the above state is certified not to belong to the set). Generalizing a result known to be valid under the assumption of complete positivity, we provide a characterization of these Gaussian-to-Gaussian (not necessarily positive) superoperators in terms of their action on the characteristic function of the inputs. For the special case of one-mode mappings we also show that any Gaussian-to-Gaussian superoperator can be expressed as a concatenation of a phase-space dilatation, followed by the action of a completely positive Gaussian channel, possibly composed with a transposition. While a similar decomposition is shown to fail in the multi-mode scenario, we prove that it still holds at least under the further hypothesis of homogeneous action on the covariance matrix.
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