Self-testing refers to a device-independent way to uniquely identify an unknown quantum device based only on the observed statistics. Earlier results on self-testing of multipartite state were restricted either to Dicke states or Graph states. In this paper, we propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of $W$ state and $GHZ$ state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical bases, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semidefinite programming (SDP) in high precision.

中文翻译：

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对称三量子位态的自测

自检是指一种独立于设备的方式，仅根据观察到的统计数据来唯一识别未知量子设备。多方状态自测的早期结果仅限于 Dicke 状态或 Graph 状态。在本文中，我们为一大类对称三量子位态提出了自测方案，即 $W$ 状态和 $GHZ$ 状态。通过同时设计部分纠缠态和最大纠缠态的子系统自测，我们首先提出并分析证明了具有等规范基系数的特殊对称态的自测准则。然后我们在一般情况下证明，可以通过结合半定规划（SDP）的交换方法对状态进行高精度的数值自测。