Time operators for quantum walks

Abstract

We study time operators for discrete-time quantum systems. Quantum walks are typical examples. We construct time operators for one-dimensional homogeneous quantum walks and determine their deficiency indices and spectra. Our time operators always have self-adjoint extensions. This is in contrast to the fact that time operators for continuous-time quantum systems generally have no self-adjoint extensions. The uniqueness of the extensions relates to the winding numbers corresponding to the system. If it is unique, its spectrum becomes a discrete set of real numbers, i.e., the time operator is quantized.

A Fundamental properties of time operators of a unitary operator

The following are fundamental properties of time operators of a unitary operator.

Proposition A.1

Let T be a strong time operator of a unitary operator U. Then the following hold.

• (1) Its closure ${\bar{T}}$ is a strong time operator of U as well.

• (2) $[T,U]=U$ holds on a dense subspace D(T). In particular, T is a time operator of U. The converse is not true, i.e., not every time operator of U is a strong time operator of U.

• (3) $\sigma (T)=\sigma (T+1)$ holds. In particular, T is unbounded.

• (4) If T is essentially self-adjoint, then $\sigma (U)={\mathbb {T}}$

Proof

(1) follows from a simple limiting argument. The first half of (2) is straightforward. For the second half of (2), see Example 4.6. (3) is obvious. We show (4). It follows that

$$\begin{aligned} U^*e^{it{\bar{T}}}U = e^{it({\bar{T}}+1)}= e^{it}e^{it{\bar{T}}} \end{aligned}$$

for any $t\in {\mathbb {R}}$. Thus, we obtain $e^{it{\bar{T}}}Ue^{-it{\bar{T}}}=e^{it}U$. This means that $\sigma (U)=\sigma (e^{it}U)$ for all $t\in {\mathbb {R}}$. Hence, $\sigma (U)={\mathbb {T}}$ holds. $\square$

Theorem A.2

Let U be a unitary operator admitting a strong time operator T. Then U has no eigenvalues.

Proof

The proof is same as the proof of [9, Corollary 4.3], and thus we omit it. $\square$

A strong time operator governs the decay rate of the transition probability as follows:

Theorem A.3

Let T be a strong time operator of a unitary operator U. Then for any $n\in {\mathbb {N}}$, $\psi \in D(T^n)$ and $\phi \in D\bigl ((T^*)^n\bigr )$, there exists a constant $C_n(\phi ,\psi )>0$ such that

$$\begin{aligned} |\langle \phi ,U^t\psi \rangle | \le \frac{C_n(\phi ,\psi )}{|t|^n},\ \ \ \ \ t\in {\mathbb {Z}}\setminus \{0\} \end{aligned}$$

holds.

Proof

The proof is same as the proof of [3, Theorem 8.5], and thus we omit it. $\square$