Abstract
We give an explicit formula of the generator of an abstract Szegedy evolution operator in terms of the discriminant operator of the evolution. We also characterize the asymptotic behavior of a quantum walker through the spectral property of the discriminant operator by using the discrete analog of the RAGE theorem.
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Acknowledgments
The authors thank H. Ohno and Y. Matsuzawa for their useful comments. ES and AS also acknowledge financial supports of the Grant-in-Aid for Young Scientists (B) of Japan Society for the Promotion of Science (Grants No. 25800088 and No. 26800054, respectively). ES is also supported by the Japan-Korea Basic Scientific Cooperation Program “Non-commutative Stochastic Analysis: New Prospects of Quantum White Noise and Quantum Walk” (2015–2016).
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Appendix
Appendix
1.1 Proof of Proposition 2.3
We present a proof of Proposition 2.3. Let H be the generator of an evolution \((U, \{\mathcal {H}_v \}_{v \in V}) \in \mathscr {F}_\mathrm{QW}\). Throughout this subsection, we assume that \(\mathrm{dim}\mathcal {H}_v < \infty \) (\(v \in V\)). Let \(\mathcal {H}_1\) be the set of vectors \(\Psi _0 \in \mathcal {H}\) satisfying
for any finite subset R of V, and \(\mathcal {H}_2\) the set of vectors \(\Psi _0 \in \mathcal {H}\) satisfying
for any sequence \(\{R_m\}\) of finite subsets of V such that \(R_m \subset R_{m+1}\) and \(V=\cup _m R_m\). Because \(\nu _n^{\alpha \Psi _0 + \beta \Phi _0}(R) \le 2 \left( |\alpha |^2 \nu _n^{\Psi _0}(R) + |\beta |^2 \nu _n^{\Phi _0}(R) \right) \), we know that \(\mathcal {H}_1\) and \(\mathcal {H}_2\) are subspaces of \(\mathcal {H}\). Let \(P_R = \sum _{x \in R} P_x\) (\(R \subset V\)). Then,
Lemma 6.1
\(\mathcal {H}_1 \perp \mathcal {H}_2\).
Proof
Let \(\Psi _0 \in \mathcal {H}_1\) and \(\Phi _0 \in \mathcal {H}_2\). Then, for all \(R \subset V\),
We first estimate the first term. By the Cauchy–Schwarz inequality,
The second term is estimated as follows:
Combining these inequalities yields the result that
Let \(\epsilon >0\) and \(\{R_m\}_{m\ge 1}\) be a family of finite subsets of V such that \(R_m \subset R_{m+1}\) and \(V = \cup _{m \ge 1} R_m\). Because \(\Phi _0 \in \mathcal {H}_2\), there exists an \(m_0 \in \mathbb {N}\) such that \(\nu _n^{\Phi _0}(R_m^\mathrm{c}) < \epsilon ^2/\Vert \Psi _0\Vert ^2\) (\(m \ge m_0\)). Because \(\Psi _0 \in \mathcal {H}_1\), it follows from (7.1) that
which completes the proof. \(\square \)
Lemma 6.2
-
(i)
\(\mathcal {H}_\mathrm{c}(H) \subset \mathcal {H}_1\);
-
(ii)
\(\mathcal {H}_\mathrm{p}(H) \subset \mathcal {H}_2\).
Proof
Let \(\Psi _0 \in \mathcal {H}_\mathrm{c}(H)\). For any finite set R,
where \(\{\phi _{x,j}\}\) is a complete orthonormal system of \(\mathcal {H}_x\) and \(\bar{\nu }_N (\phi )\) \(:= \frac{1}{N} \sum _{n=0}^{N-1}\) \(|\langle \phi , \mathrm{e}^{inH}\Psi _0 \rangle |^2\). Because, by assumption, the sum in (6.2) runs over a finite set, it suffices to show that \(\lim _{N \rightarrow \infty } \bar{\nu }_N (\phi ) = 0\). Let \(\omega (x) = \mathrm{e}^{in x}\) and \(g_N(\omega ) = \frac{1}{N}\sum _{n=0}^{N-1} \omega ^n\). Then, \(g_N(\omega ) = \frac{1-\omega ^N}{N(1-\omega )}\) if \(\omega \not =1\) and \(g_N(1) = 1\). By the Fubini theorem,
where \(P_\mathrm{c}(H)\) is the projection onto \(\mathcal {H}_\mathrm{c}(H)\). By the polarization identity, there exists \(\{\psi _j\}_{j=1,2,3,4} \subset \mathcal {H}_\mathrm{c}(H)\) such that
Because \(F_j := \Vert E_H(\cdot ) \psi _j\Vert ^2\) is continuous,
as \(\epsilon \rightarrow 0\). Because \(\sup _{|\omega |=1} |g_N(\omega )| \le 1\) and \(\lim _{N \rightarrow \infty } g_N(\omega (\lambda -\mu )) = 0\) (\(\lambda \not =\mu \)), we obtain \(\lim _{N\rightarrow 0} \bar{\nu }_N(\phi ) =0\) by the dominated convergence theorem. This completes the proof of (i).
Let \(\Psi _0 \in \mathcal {H}_\mathrm{p}(H)\). For any \(\epsilon > 0\), there exist eigenvectors \(\{ \phi _j\}_{j=1}^M\) (\(M \in \mathbb {N}\)) of H such that \(\Vert \Psi _0 - \sum _{j=1}^M \langle \phi _j, \Psi _0 \rangle \phi _j \Vert < \epsilon \). Let \(\{R_m\}\) be a sequence of finite subsets of V such that \(R_m \subset R_{m+1}\) and \(\cup _m R_m = V\). It follows that
which proves \(\lim _{m \rightarrow \infty } \sup _n \nu _n^{\Psi _0}(R_m^\mathrm{c}) = 0\). Hence, we have (ii). \(\square \)
Proof of Proposition 2.3
Combining Lemmas 6.1 and 6.2 yields the result that
which proves the proposition. \(\square \)
1.2 Proof of Equation (2.4)
In this subsection, we prove the following:
Lemma 6.3
Let \((U, \{\mathcal {H}_v\}_{v \in V}) \in \mathscr {F}_{QW}\) and \(\Psi _0 \in \mathcal {H}\) satisfy
for an increasing sequence \(\{R_m \}\) of finite subsets of V. Then, (2.4) holds. In particular, (2.4) holds for all \(\Psi _0 \in \mathcal {H}_\mathrm{p}(H)\).
Proof
By assumption, we know that for any \(\epsilon > 0\), there exists \(m_0 \in \mathbb {N}\) such that \(\sup _n \nu _n^{\Psi _0}(R_{m_0}^\mathrm{c}) < \epsilon \). Hence,
If \(\limsup _n \nu _n^{\Psi _0}(x) = 0\) for any \(x \in R_{m_0}\), then
which contradicts (6.3). Therefore, (2.4) holds for some \(x \in R_{m_0}\). \(\square \)
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Segawa, E., Suzuki, A. Generator of an abstract quantum walk. Quantum Stud.: Math. Found. 3, 11–30 (2016). https://doi.org/10.1007/s40509-016-0070-1
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DOI: https://doi.org/10.1007/s40509-016-0070-1