基于两步测量方法及其最少观测次数的任意量子纯态估计
Estimation of arbitrary quantum pure states based on the two-step measurement method and the minimum observations

量子状态层析所需要的完备观测次数d2(d = 2n)随着状态的量子位数n的增加呈指数增长, 这使得对高维 量子态的层析变得十分困难. 本文提出一种基于两步测量的量子态估计方法, 可以对任意量子纯态的估计提供最少 的观测次数. 本文证明: 当选择泡利观测算符, 采用本文所提出的量子态估计方法对d = 2n维希尔伯特空间中的任 意n量子位纯态进行重构时, 如果为本征态, 那么所需最少观测次数me min仅为me min = n; 对于包含l(2 6 l 6 d)个 非零本征值的叠加态, 重构所需最少观测次数ms min满足ms min = d + 2l ?? 3, 此数目远小于压缩传感理论给出的量 子态重构所需测量配置数目O(rd log d), 以及目前已发表论文给出的纯态唯一确定所需最少观测次数4d ?? 5. 同时 给出最少观测次数对应的最优观测算符集的构建方案, 并通过仿真实验对本文所提出的量子态估计方法进行验证, 实验中重构保真度均达到97%以上.
The number of complete observables required in quantum state tomography is d2(d = 2n), which increases exponentially with the qubit number n of the quantum system, makes the reconstruction of the high dimensional quantum state become very difficult. In this paper, we propose a quantum two-step measurement method of the estimation of arbitrary quantum pure states with the minimum number of observables. We prove when choosing the observables of Pauli operators and the two steps measurement method proposed in this paper, the minimum number of observables required for the estimation of an n-qubit eigenstate is me min = n, and the minimum number of a superposition state consisting of l(2 6 l 6 d) nonzero eigenvalues satisfies ms min = d + 2l ?? 3. Either the number of eigenstate or super-position state is far less than the number of measurement configurations required by compressive sensing O(rd log d), and the minimum number of observables for pure states uniquely determination 4d??5 in published papers up to now. We also give the method of selecting the corresponding observable sets, called the optimal observable set in this paper. Mathematical simulation experiments are carried out to validate the method of pure state reconstruction based on adaptive measurements. The fidelities in our experiments are all over 97%.