Mathematical Research of Quantum Information and Quantum Computing

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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).