In the quantum context, Uhlmann proved that if [rho], [sigma] [member of] S(H) and dim H < [infinity], then [mathematical expression not reproducible], where [phi] is some mixed unitary quantum operation, i.e., [phi]([rho]) = [[summation].sup.k.sub.i=1][[lambda].sub.i][U.sub.i][rho][U.sup.*.sub.i] ([for all][rho] [member of] S(H)), where k < [infinity], [[lambda].sub.i] > 0, [[summation].sup.k.sub.i=1] [[lambda].sub.i] = 1 and all [U.sub.i] are

unitary operators acting on H.

First, we recall some known results on numerical ranges of operators and a class of very important

unitary operators on the Fock space.

This action is known to be ergodic (see for example [FTP82] and [FTP83]), but since the measure is not preserved, no theorem on the convergence of means of the corresponding

unitary operators had been proved.

Let [pi] : G [right arrow] B(H) be a projective, unitary representation of the group G into the

unitary operators on the Hilbert space H.

Motivated by the works of Mustari and Taylor [15,17], we have recently begun to study the mean ergodic theorem under the framework of RN modules [18] to obtain the mean ergodic theorem in the sense of convergence in probability, where we proved the mean ergodic theorem for a strongly continuous semigroup of random

unitary operators defined on complete random inner product modules (briefly, complete RIP modules).

Therefore, it is mandatory that the

unitary operators (which are used for modelling quantum computations) are proved to have implementations that are based only on fault tolerant quantum gates.

He explains vector spaces and bases, linear transformations and operators, eigenvalues, circles and ellipses, inner products, adjoints, Hermitian operators,

unitary operators, the wave equation, continuous spectre and the Dirac delta function, Fourier transforms, Green's and functions, and includes an appendix on matrix operations (new to this edition) and a full chapter on crucial applications.

So let M and T be

unitary operators in [L.sup.2]([??]) defined by

In order to construct

unitary operators that disperse the position of a particle, it is necessary to introduce an extra degree of freedom, known as chirality.

Finally, we recall the Weyl

unitary operators [BC] defined on [L.sup.2](d[micro]) or [H.sup.2](d[micro]) by ([W.sub.a]f)(z)=[k.sub.a](z)f(z-a).

One can verify that

unitary operators [U.sub.n], n [greater than or equal to] 0, commute mutually; namely, [U.sub.m] [U.sub.n] = [U.sub.n] [U.sub.m] for all m, n [greater than or equal to] 0.