Mathematical Research of Quantum Information and Quantum Computing

1.重新考虑了海森堡测不准原理，该原理给出了测量噪声和干扰之间的权衡。澄清了其有效性的局限性，建立了普遍有效的概括（普遍不确定性原理）。如果任何测量仪器测量到一个具有噪声ε(A)和扰动η(B)的可观测A，我们有ε(A)η(B)+ε(A)σ(B)+σ(A)η(B)≧(1/2)|<[A,B]>|的关系式，其中σ表示测量前状态的标准差。结果表明，如果噪声ε(A)和扰动η(B)在统计上与被测物体的初始状态无关，则得到原始海森堡关系ε(A)η(B)≧(1/2)|<[A,B]>|.3。1952年发现的Wigner-Araki-Yanase （WAY）从普遍的测不准原理出发，设定了守恒定律下测量的限制，并将其定量地推广为测量的均方噪声与守恒量的方差之间的反比关系。由普遍不确定性原理可知，角动量守恒定律对量子门操作精度有限制，与n个量子比特辅助系统相互作用控制的CNOT门精度有限制，误差概率至少为1/(4n^2)，平均光子n的电磁场控制的CNOT门精度有限制，误差概率至少为1/(16N)。

1.Heisenberg's uncertainty principle, which gives the trade-off between measurement noise and disturbance, has been reconsidered. Its limitation of the validity has been clarified and the universally valid generalization (universal uncertainty principle) has been established. If any measuring apparatus measures an observable A with noise ε(A) and disturbance η(B) on B, we have the relation ε(A)η(B)+ε(A)σ(B)+σ(A)η(B)≧(1/2)|<[A,B]>|, where σ stands for the standard deviation in the state just before the measurement.2.It has been shown that, if the noise ε(A) and the disturbance η(B) are statistically independent from the initial state of the measured object, we have the original Heisenberg relation ε(A)η(B)≧(1/2)|<[A,B]>|.3.From the universal uncertainty principle, Wigner-Araki-Yanase (WAY), found in 1952 to set a limitation of measurements under conservation law, has been quantitatively generalized to a reciprocal relation between the mean-square-noise of the measurement and the variance of the conserved quantity.4.As a consequence from the universal uncertainty principle, it has been shown that the angular momentum conservation law sets a limitation to the accuracy of quantum gate operations, the accuracy of the CNOT gate controlled by an interaction with n qubit ancilla system is limited with error probability at least 1/(4n^2), and the CNOT controlled by the electro-magnetic field with average photon N is limited with error probability at least 1/(16N).