Abstract
In continuous-variable quantum information processing, quantum error correction of Gaussian errors requires simultaneous estimation of both quadrature components of displacements on phase space. However, quadrature operators $x$ and $p$ are non-commutative 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 multi-parameter estimation problem of displacements assuming an isotropic Gaussian prior distribution and allowing post-selection 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.
Highlights
Continuous-variable optical quantum information processing has attracted much attention due to its extent scalability achieved by large-scale cluster states [1,2] which enable universal Gaussian operations [3,4]
The Cramér-Rao bound [16,17,18,38,39] is often used to obtain a lower bound on the estimation error
We have shown that there is a lower bound on the estimation error v when only Gaussian states and Gaussian operations are used
Summary
Continuous-variable optical quantum information processing has attracted much attention due to its extent scalability achieved by large-scale cluster states [1,2] which enable universal Gaussian operations [3,4]. Gaussian error is the most common type of error in optical systems It includes photon losses and the Gaussian quantum channel [6], which is defined as phase-space displacements following an isotropic Gaussian distribution. It has been proven, that Gaussian errors imposed on Gaussian states cannot be corrected using only Gaussian operations [7,8]. We show that this bound can be beaten for some range of the prior variance even with only simple non-Gaussian states such as single-photon states This result reveals the role of non-Gaussianity in the estimation of displacements and opens up the possibility of correcting Gaussian errors using experimentally feasible non-Gaussian states.
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