Let ( , F, P) be a probability space and Z = (Z K ) k∈N a Bernoulli noise on ( , F, P) which has the chaotic representation property. In this paper, we investigate a special family of functionals of Z , which we call the coherent states. First, with the help of Z , we construct a mapping φ from l 2 (N) to L 2 ( , F, P) which is called the coherent mapping. We prove that φ has the continuity property and other properties of operation. We then define functionals of the form φ( f ) with f ∈ l 2 (N) as the coherent states and prove that all the coherent states are total in L 2 ( , F, P). We also show that φ can be used to factorize L 2 ( , F, P). Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.2010 Mathematics subject classification: primary 60H40; secondary 46E30.

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Let M = (M) n∈N be a discrete-time normal martingale satisfying some mild requirements. In this paper we show that through the full Wiener integral introduced by Wang et al. ('An alternative approach to Privault's discrete-time chaotic calculus', J. Math. Anal. Appl. 373 (2011), 643-654), one can define a multiplication-type operation on square integrable functionals of M, which we call the convolution. We examine algebraic and analytical properties of the convolution and, in particular, we prove that the convolution can be used to represent a certain family of conditional expectation operators associated with M. We also present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.2010 Mathematics subject classification: primary 60H40; secondary 46E30.

In this paper, on the basis of constructing a new shift operator [Formula: see text] and choosing the Grover coin [Formula: see text] as coin operator [Formula: see text], we get the standard evolution operator [Formula: see text] on cycles. Using [Formula: see text], we not only got the analytical expression of wavefunction [Formula: see text], but also obtained the conclusion that the limit distribution [Formula: see text] of [Formula: see text], which is not uniform distribution, regardless of [Formula: see text] is odd or even.

In this paper, we successfully obtain an explicit expression of the limit distribution [Formula: see text] of three-state quantum walks on cycles, the total variation distance between [Formula: see text] and the average probability [Formula: see text], and lower bound on the difference between two eigenvalues, among others. Based on the above conclusions, we finally get the mixing time [Formula: see text] of the quantum walk of Grover coin on the N-cycle. [Formula: see text] is the time required to characterize [Formula: see text] approaching [Formula: see text]. Our results show that the average probability of a three-state quantum walk on a cycle can approach its limit distribution faster than that of a two-state quantum walk, which might be of significance to quantum computation.