In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements in phase space. However, quadrature operators $x$ and $p$ are noncommutative conjugate observables, whose simultaneous measurement is prohibited by the uncertainty principle. Gottesman-Kitaev-Preskill (GKP) error correction deals with this problem using complex non-Gaussian states called GKP states. On the other hand, simultaneous estimation of displacement using experimentally feasible non-Gaussian states has not been well studied. In this paper, we consider a multiparameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing postselection of measurement outcomes. We derive a lower bound for the estimation error when only Gaussian operations are used and show that even simple non-Gaussian states such as single-photon states can beat this bound. Based on Ghosh's bound, we also obtain a lower bound for the estimation error when the maximum photon number of the input state is given. Our results reveal the role of non-Gaussianity in the estimation of displacements and pave the way toward the error correction of Gaussian errors using experimentally feasible non-Gaussian states.

中文翻译：

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使用非高斯状态估计高斯随机位移

在连续变量量子信息处理中，高斯误差的量子误差校正需要同时估计相空间中位移的两个正交分量。然而，正交算子$X$ 和 $p$是不可交换的共轭可观测量，不确定性原理禁止其同时测量。Gottesman-Kitaev-Preskill (GKP) 纠错使用称为 GKP 状态的复杂非高斯状态处理此问题。另一方面，使用实验可行的非高斯状态同时估计位移还没有得到很好的研究。在本文中，我们考虑位移的多参数估计问题，假设各向同性高斯先验分布并允许测量结果的后选择。当仅使用高斯运算时，我们推导出估计误差的下限，并表明即使是简单的非高斯状态（例如单光子状态）也能超过此界限。基于 Ghosh 的界限，当给定输入状态的最大光子数时，我们还获得了估计误差的下限。我们的结果揭示了非高斯性在位移估计中的作用，并为使用实验可行的非高斯状态对高斯误差进行误差校正铺平了道路。