Entanglement distillation is a well-studied problem in quantum information, where one typically starts with $n$ noisy Bell pairs and distills $k$ Bell pairs of higher fidelity. While distilling Bell pairs is the canonical setting, it is important to study the distillation of multipartite entangled states because these can be useful for realizing distributed algorithms on quantum networks. In this paper, we study the distillation of GHZ states using quantum error correcting codes (QECCs). Using the stabilizer formalism, we begin by explaining the QECC-based Bell pair distillation protocol in arXiv:0708.3699, which relies particularly on the transpose symmetry between Alice's and Bob's qubits in Bell states. Extending this idea, we show that, given $n$ GHZ states, performing a matrix on Alice's qubits is equivalent to performing a "stretched" version of the transpose of the matrix on the qubits of Bob and Charlie. We call this mapping to the stretched version of the matrix the GHZ-map, and show that it is an algebra homomorphism. Using this property, we show that Alice projecting her qubits onto an $[[n,k]]$ stabilizer code implies the simultaneous projection of Bob's and Charlie's qubits onto an induced $[[2n,k]]$ stabilizer code. Guided by this insight, we develop a GHZ distillation protocol based on local operations and classical communication that uses any stabilizer code. Inspired by stabilizer measurements on GHZ states, we also develop a new algorithm to generate logical Pauli operators of any stabilizer code and use it in the protocol. Since quantum codes with finite rate and almost linear minimum distance have recently been discovered, this paper paves the way for high-rate high-output-fidelity GHZ distillation. We provide simulation results on the $5$-qubit perfect code to emphasize the importance of the placement of a certain local Clifford operation in the protocol.

中文翻译：

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使用稳定器代码提取 GHZ 状态

纠缠蒸馏是量子信息中一个经过充分研究的问题，其中通常从 $n$ 个嘈杂的 Bell 对开始，然后蒸馏 $k$ 个更高保真度的 Bell 对。虽然提炼贝尔对是规范设置，但研究多方纠缠态的提炼很重要，因为这对于在量子网络上实现分布式算法非常有用。在本文中，我们使用量子纠错码 (QECC) 研究了 GHZ 状态的蒸馏。使用稳定器形式，我们首先解释 arXiv:0708.3699 中基于 QECC 的贝尔对蒸馏协议，该协议特别依赖于贝尔状态下 Alice 和 Bob 的量子位之间的转置对称性。扩展这个想法，我们表明，给定 $n$ GHZ 状态，在 Alice 的量子位上执行矩阵等效于执行“ 本文为高速高输出保真GHZ蒸馏铺平了道路。我们提供了 $5$-qubit 完美代码的模拟结果，以强调在协议中放置某个本地 Clifford 操作的重要性。