Mathematical Research of Quantum Information and Quantum Computing

1.重新考虑了Heisenberg的测不准原理，它给出了测量噪声和干扰之间的折衷。阐明了其有效性的局限性，建立了普遍有效的推广（泛测不准原理）。如果任何测量仪器测量一个具有噪声ε（A）和干扰η（B）的可观测量A，我们有关系ε（A）η（B）+ ε（A）σ（B）+ σ（A）η（B）<$（1/2）|<[A，B]>| 2.证明了如果噪声ε（A）和扰动η（B）与被测物体的初始状态统计无关，则有原始的海森堡关系ε（A）η（B）（1/2）|<[A，B]>| 3.从普适测不准原理出发，将1952年Wigner-Araki-Yanase（WAY）在守恒律下提出的测量极限，定量地推广为测量均方噪声与守恒量方差之间的倒数关系。已经表明，角动量守恒定律对量子门操作的精度设置了限制，由与n量子位辅助系统的相互作用控制的CNOT门的精度被限制为具有至少1/（4n^2）的错误概率，并且平均光子数N的电磁场控制的CNOT被限制为误差概率至少为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).