通过SWAP测试实现未知混合量子比特态的最优线性速率转换
Optimal Linear-Rate Conversion of Unknown Mixed Qubit States via SWAP Tests
- Duke University（杜克大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究确定了通过SWAP测试实现未知混合量子比特态间最优线性转换速率，该速率由复RLD Fisher信息矩阵特征值决定，并揭示了其虚部的几何意义。
AI中文摘要：
通过消耗未知量子比特态的多个副本，可以在保持其布洛赫矢量方向的同时修改其纯度。我们确定了不同纯度的量子比特态可以相互转换的最大线性速率，允许非零误差（例如以迹距离量化），前提是该误差在无穷多副本的极限下消失。有趣的是，最优转换速率由与量子比特态的$\mathrm{SU}(2)$旋转相关联的复右对数导数（RLD）Fisher信息矩阵的两个特征值决定。当输出量子比特具有更高纯度（对应于浓缩）时，最优速率由输入和输出RLD矩阵的最大特征值之比给出。相反，当输出量子比特具有较低纯度（对应于稀释）时，最优速率由它们的最小特征值之比给出。值得注意的是，浓缩和稀释都可以仅使用SWAP测试作为唯一的非平凡双量子比特测量原语，并辅以初始制备在最大混合态中的辅助量子比特来实现，而无需任何额外的双量子比特门。因此，我们的工作为完整的复RLD Fisher信息矩阵提供了一种新颖的操作性解释。至关重要的是，其反对称的纯虚部编码了超出密度算子之间统计距离的几何信息，并在确定最优态转换速率中起着至关重要的作用。
英文摘要：
By consuming multiple copies of an unknown qubit state, one can modify its purity while preserving the direction of its Bloch vector. We determine the maximum linear rate at which qubit states of different purities can be interconverted, allowing a nonzero error, quantified, for instance, by the trace distance, provided that it vanishes in the limit of infinitely many copies. Interestingly, the optimal conversion rate is determined by the two eigenvalues of the complex right-logarithmic-derivative (RLD) Fisher information matrix associated with $\mathrm{SU}(2)$ rotations of the qubit state. When the output qubits have higher purity, corresponding to concentration, the optimal rate is given by the ratio of the maximum eigenvalues of the input and output RLD matrices. In contrast, when the output qubits have lower purity, corresponding to dilution, the optimal rate is given by the ratio of their minimum eigenvalues. Remarkably, both concentration and dilution can be implemented using SWAP tests as the only nontrivial two-qubit measurement primitive, together with ancillary qubits initially prepared in maximally mixed states, without requiring any additional two-qubit gates. Our work thus provides a novel operational interpretation of the full complex RLD Fisher information matrix. Crucially, its antisymmetric, purely imaginary part encodes geometric information beyond the statistical distance between density operators and plays an essential role in determining the optimal state-conversion rates.
