Asymptotic velocity of a position-dependent quantum walk

作者：Suzuki Akito^*

来源：Quantum Information Processing, 2016, 15(1): 103-119.

DOI：10.1007/s11128-015-1183-x

• Quantum walk
• Asymptotic velocity
• Weak limit

We consider a position-dependent coined quantum walk on Z and assume that the coin operator C(x) satisfies
parallel to C(x) - C-0 parallel to <= c(1) vertical bar x vertical bar(-1-epsilon), x is an element of Z \ {0}
with positive c(1) and epsilon and C-0 is an element of U(2). We show that the Heisenberg operator (x) over cap (t) ofthe position operator converges to the asymptotic velocity operator (v) over cap+ so that
s- lim(t ->infinity) exp(i xi (x) over capt/t) = Pi(p) (U) + exp(i xi (v) over cap (+)) Pi(ac) (U)
provided that U has no singular continuous spectrum. Here Pi(p)(U) ((resp., Pi(ac)(U)) is the orthogonal projection onto the direct sum of all eigenspaces (resp., the subspace of absolute continuity) of U. We also prove that for the random variable X-1 denoting the position of a quantum walker at time t is an element of N, X-t/t converges in law to a random variable V with the probability distribution
mu(V) = parallel to Pi(p) (U)Psi(0)parallel to(2) delta(0) + parallel to E(v) over cap+ (.) Pi(ac) (U)Psi(0)parallel to(2) ,
where Psi(0) is the initial state, delta(0) the Dirac measure at zero, and E(v) over cap+ the spectral measure of (v) over cap (+).

• 出版日期2016-1