Passage Times, Exit Times and Dirichlet Problems for Open Quantum Walks

Abstract

We consider an open quantum walk on a graph, and the random variables defined as the passage time and number of visits at a given point of the graph. We study in particular the probability that the passage time is finite, the expectation of that passage time, the expectation of the number of visits, and discuss the notion of recurrence for open quantum walks. We also study exit times and exit probabilities from a finite domain, and use them to solve Dirichlet problems and to determine harmonic measures. We consider in particular the case of irreducible open quantum walks. The results we obtain extend those for classical Markov chains.

Appendices

Appendix 1: Proofs for Section 2

Proof of Lemma 2.5:

Conditionally on $(x_n,\rho _n)$, one has for all i in V

$$\begin{aligned} m_{n+1}=\mathrm {Tr}(\rho _{n+1} A_{x_{n+1}}) = \mathrm {Tr}\Big (\frac{L_{i,x_n}\rho _{n}L_{i,x_n}^*}{\mathrm {Tr}(L_{i,x_n}\rho _{n}L_{i,x_n}^*)} A_{i}\Big ) \end{aligned}$$

with probability $\mathrm {Tr}(L_{i,x_n}\rho _{n}L_{i,x_n}^*)$, so that

$$\begin{aligned} {\mathbb {E}}\left( \mathrm {Tr}(\rho _{n+1} A_{x_{n+1}})| x_n,\rho _n\right)&= \sum _{i\in V} \mathrm {Tr}\left( {L_{i,x_n}\rho _{n}L_{i,x_n}^*} A_{i}\right) \\&= \mathrm {Tr}\big (\rho _{n} \sum _{i\in V} L_{i,x_n}^* A_{i} L_{i,x_n}\big )\\&= \mathrm {Tr}(\rho _n A_{x_n})=m_n. \end{aligned}$$

$\square$

Appendix 2: Proofs for Section 3

We start by computing simple expressions for the quantities ${\mathbb {P}}_{i,\rho }(t_j<\infty )$ and ${\mathbb {E}}_{i,\rho }(n_j)$:

Lemma 8.7

We have the identities

$$\begin{aligned} {\mathbb {P}}_{i,\rho }(t_j<\infty ) = \sum _{\pi \in {\mathcal P}^{V\setminus \{j\}}(i,j)} \mathrm {Tr}(L_\pi \rho L_\pi ^*),\qquad {\mathbb {E}}_{i,\rho }(n_j) = \sum _{\pi \in {\mathcal P}(i,j)} \mathrm {Tr}(L_\pi \rho L_\pi ^*), \end{aligned}$$

where the second expression is possibly $\infty$.

Proof

We have ${\mathbb {P}}_{i,\rho }(x_1=i_1,\ldots ,x_\ell =i_\ell )=\mathrm {Tr}(L_\pi \rho L_\pi ^*)$ where $\pi =(i,i_1,\ldots ,i_\ell )$. In addition,

$$\begin{aligned} {\mathbb {P}}_{i,\rho }(t_j<\infty )= \sum _{k\ge 0}\ \sum _{i_1,\ldots ,i_k\in V\setminus \{j\}} {\mathbb {P}}_{i,\rho }(x_1=i_1,\ldots ,x_k=i_k,x_{k+1}=j) \end{aligned}$$

which leads to the first formula. We also have immediately ${\mathbb {P}}_{i,\rho }(x_k=j)=\sum _{\pi \in {\mathcal P}_k(i,j)} \mathrm {Tr}(L_\pi \rho L_\pi ^*)$, and the second formula follows from

$$\begin{aligned} {\mathbb {E}}_{i,\rho }(n_j)=\sum _{k= 1}^\infty {\mathbb {P}}_{i,\rho }(x_k=j). \end{aligned}$$

$\square$

Proof of Proposition 3.3

We begin with the definition of ${\mathfrak P}_{i,j}$. For any $\rho$ in $\mathcal I_1({\mathfrak h}_i)\setminus \{0\},$ the triangle inequality for the trace norm implies that

$$\begin{aligned} \mathrm {Tr}\big (\big |\sum _{\pi \in \mathcal P^{V\setminus \{j\}}(i,j)} L_\pi \rho L_\pi ^*\, \mathbbm {1}_{\ell (\pi )\le n}\big |\big )&\le \sum _{\pi \in \mathcal P^{V\setminus \{j\}}(i,j)} \mathrm {Tr}\, \big |L_\pi \rho L_\pi ^*\big |\, \mathbbm {1}_{\ell (\pi )\le n}\\&= \sum _{\pi \in \mathcal P^{V\setminus \{j\}}(i,j)} \mathrm {Tr}\, (L_\pi |\rho | L_\pi ^*)\, \mathbbm {1}_{\ell (\pi )\le n}\\&= \mathrm {Tr}|\rho | \times {\mathbb {P}}_{i,\frac{|\rho |}{\mathrm {Tr}(|\rho |)}}(t_j\le n) \\&\le \mathrm {Tr}|\rho |, \end{aligned}$$

so that

$$\begin{aligned} \sup _n \mathrm {Tr}\Big (\big |\sum _{\pi \in \mathcal P^{V\setminus \{j\}}(i,j)} L_\pi \rho L_\pi ^* \mathbbm {1}_{\ell (\pi )\le n}\big |\Big ) <\infty . \end{aligned}$$

Consequently, by the Banach–Steinhaus Theorem, the operator on $\mathcal I_1({\mathfrak h}_i)$ defined by

$$\begin{aligned} {\mathfrak P}_{j,i}(\rho )= \lim _{n\rightarrow \infty }\quad \sum _{\pi \in \mathcal P^{V\setminus \{j\}}(i,j)} L_\pi \rho L_\pi ^* \mathbbm {1}_{\ell (\pi )\le n} \end{aligned}$$

is everywhere defined and bounded.

This proves the first identity in Proposition 3.3. To prove the second we need a series of technical results. Our strategy is the same as in the classical case: we introduce a weight on the length of paths, in order to tame the possible divergence of the series giving ${\mathbb {E}}_{i,\rho }(n_j)$ in Lemma 8.7. First note that, for any $i,j\in V$ and any $\alpha \in (0,1)$, there exists a bounded, completely positive map ${\mathfrak N}_{j,i}^{(\alpha )}$ from $\mathcal I_1({\mathfrak h}_i)$ to $\mathcal I_1({\mathfrak h}_j)$ such that

$$\begin{aligned} \sum _{\pi \in \mathcal P(i,j)}\alpha ^{\ell (\pi )} \mathrm {Tr}\,L_\pi \rho L_\pi ^* = \mathrm {Tr}\big ({\mathfrak N}_{j,i}^{(\alpha )}(\rho )\big ). \end{aligned}$$

In particular, the following limit holds in $[0,\infty ]$:

$$\begin{aligned} {\mathbb {E}}_{i,\rho }(n_j)=\lim _{\alpha \rightarrow 1}\mathrm {Tr}\big ( {\mathfrak N}_{j,i}^{(\alpha )}(\rho )\big ). \end{aligned}$$

This operator ${\mathfrak N}_{j,i}^{(\alpha )}$ is defined by

$$\begin{aligned} {\mathfrak N}_{j,i}^{(\alpha )}(\rho )=\lim _{n\rightarrow \infty } \sum _{\pi \in {\mathcal P}(i,j)} \alpha ^{\ell (\pi )} L_\pi \rho L_\pi ^* \mathbbm {1}_{\ell (\pi )\le n}, \end{aligned}$$

using the Banach–Steinhaus Theorem and the simple bound

$$\begin{aligned} \mathrm {Tr}\big (\big |\sum _{\pi \in {\mathcal P}(i,j)} \alpha ^{\ell (\pi )}\, L_\pi \rho L_\pi ^* \mathbbm {1}_{\ell (\pi )\le n} \big |\big )&\le \sum _{\pi \in {\mathcal P}(i,j)} \alpha ^{\ell (\pi )}\, \mathrm {Tr}\,L_\pi |\rho | L_\pi ^* \, \mathbbm {1}_{\ell (\pi )\le n}\\&= \sum _{k= 0}^n \alpha ^k \,{\mathbb {P}}_{i,\rho } (x_k=j)\\&\le (1-\alpha )^{-1}. \end{aligned}$$

We also define

$$\begin{aligned} {\mathfrak P}_{j,i}^{(\alpha )}(\rho )= \lim _{n\rightarrow \infty }\, \sum _{\pi \in \mathcal P^{V\setminus \{j\}}(i,j)} \alpha ^{\ell (\pi )}\,L_\pi \rho L_\pi ^* \,\mathbbm {1}_{\ell (\pi )\le n}. \end{aligned}$$

Since any $\pi \in {\mathcal P}(i,j)$ is a concatenation of $\pi _0 \in {\mathcal P}^{V\setminus \{j\}}(i,j)$ and $\pi _1,\ldots ,\pi _k$ in ${\mathcal P}^{V\setminus \{j\}}(j,j)$, and

$$\begin{aligned} L_\pi = L_{\pi _k}\circ \ldots \circ L_{\pi _1} \circ L_{\pi _0},\qquad \ell (\pi )= \ell (\pi _k)+\cdots + \ell (\pi _1)+\ell (\pi _0), \end{aligned}$$

we have

$$\begin{aligned}&\sum _{\pi \in {\mathcal P}(i,j) }\alpha ^{\ell (\pi )} L_\pi \rho L_\pi ^* \mathbbm {1}_{\ell (\pi )\le n}\\&\quad =\sum _{k\ge 0}\quad \sum _{\begin{array}{c} \pi _0\in {\mathcal P}^{V\setminus \{j\}}(i,j),\\ \pi _1,\ldots ,\pi _k\in {\mathcal P}^{V\setminus \{j\}}(j,j) \end{array}} \alpha ^{\sum _{r=0}^k \ell (\pi _r)}\, L_{\pi _k} \ldots L_{\pi _1} L_{\pi _0} \rho L_{\pi _0}^* L_{\pi _1}^* \ldots L_{\pi _k}^*\, \mathbbm {1}_{\sum _{r=0}^k \ell (\pi _r)\le n}. \end{aligned}$$

Because both sides define bounded operators as $n\rightarrow \infty$, we have

$$\begin{aligned} {\mathfrak N}_{j,i}^{(\alpha )}(\rho )=\sum _{k\ge 0} {\mathfrak P}_{j,j}^{(\alpha )\,k} \circ {\mathfrak P}_{j,i}(\rho )= \left( \mathrm {Id}-{\mathfrak P}_{j,j}^{(\alpha )}\right) ^{-1} \circ {\mathfrak P}_{j,i}^{(\alpha )}(\rho ). \end{aligned}$$

Since $\alpha \mapsto {\mathfrak P}_{j,j}^{(\alpha )}(\rho )$ is monotone increasing for $\rho \ge 0$, the right-hand side is monotone increasing as well, and the second identity follows. $\square$

Proof of Equation (16) By definition, we have

$$\begin{aligned} {\mathbb {E}}_{i,\rho }(\rho _{t_j}\,|\, t_j<\infty )&= \frac{{\mathbb {E}}_{i,\rho }(\rho _{t_j} \mathbbm {1}_{t_j<\infty })}{{\mathbb {P}}_{i,\rho }(t_j<\infty )}\\&= \frac{1}{{\mathbb {P}}_{i,\rho }(t_j<\infty )} \, \sum _{\pi \in {\mathcal P}^{V\setminus \{j\}}(i,j)} \frac{L_\pi \rho L_\pi ^*}{\mathrm {Tr}\,L_\pi \rho L_\pi ^*} \, \mathrm {Tr}\,L_\pi \rho L_\pi ^*\\&= \frac{{\mathfrak P}_{j,i}(\rho )}{\mathrm {Tr}\big ({\mathfrak P}_{j,i}(\rho )\big )}. \end{aligned}$$

Proof of Corollary 3.5

• 1. Let $i,j\in V$ and $\rho \in \mathcal {S}({\mathfrak h}_i)$. By Proposition 3.3, we have ${\mathbb {P}}_{i,\rho }(t_j<\infty )= \mathrm {Tr}\, \rho \,{\mathfrak P}^*_{j,i}(\mathrm {Id}_{{\mathfrak h}_{j}})$ and, since $\mathrm {Tr}\,\rho =1$, we have ${\mathbb {P}}_{i,\rho }(t_j<\infty )~=~1$ if and only if $P_\rho {\mathfrak P}_{j,i}^*(\mathrm {Id}_{{\mathfrak h}_j}) P_\rho = P_\rho$, where $P_\rho$ is the orthogonal projection on the support of $\rho$. Write ${\mathfrak P}_{j,i}^*(\mathrm {Id}_{{\mathfrak h}_j})$ as ${\mathfrak P}_{j,i}^*(\mathrm {Id}_{{\mathfrak h}_j})=\left( \begin{array}{ll} \mathrm {Id}_{\mathrm {Ran}\,\rho } &{} A \\ A^* &{} B \end{array}\right)$ in the decomposition ${\mathfrak h}_i=\mathrm {Ran}\,\rho \oplus ({\mathrm {Ran}}\,\rho )^\perp$. Then the property ${\mathfrak P}_{j,i}^*(\mathrm {Id}_{{\mathfrak h}_j})\le \mathrm {Id}_{{\mathfrak h}_i}$ implies that $\left( \begin{array}{ll} 0 &{} -A \\ -A^* &{} \mathrm {Id}_{{\mathrm {Ker}}\,\rho } - B \end{array}\right) \ge 0$, so that necessarily $A=0$. In particular, if $\rho$ is faithful, then ${\mathbb {P}}_{i,\rho }(t_j<\infty )=1$ if and only if ${\mathfrak P}_{j,i}^*(\mathrm {Id}_{{\mathfrak h}_j})=\mathrm {Id}_{{\mathfrak h}_i}$. In that case, ${\mathbb {P}}_{i,\rho '}(t_j<\infty )=1$ for any $\rho '$ in $\mathcal {S}({\mathfrak h}_i)$.

• 2. Consequently, if this is the case for $j=i$, then for any $\rho '$ in $\mathcal {S}({\mathfrak h}_i)$ one has ${\mathbb {E}}_{i,\rho '}(n_i)=\infty$, since by Proposition 3.3 we have

  $$\begin{aligned} {\mathbb {E}}_{i,\rho '}(n_i)=\sum _{k\ge 1} \mathrm {Tr}\big (\rho '\, {\mathfrak P}_{i,i}^{*\, k}(\mathrm {Id}_{{\mathfrak h}_i})\big ). \end{aligned}$$

• 3. If ${\mathbb {E}}_{i,\rho }(n_j)<\infty$ with $\rho$ faithful and $\mathrm {dim}\,{\mathfrak h}_i<\infty$, then for any $\alpha \in (0,1)$,

  $$\begin{aligned} \mathrm {Tr}\big ({\mathfrak N}_{j,i}(\rho )\big )\ge \mathrm {Tr}\big (\rho \,{\mathfrak N}_{j,i}^{(\alpha )\, *}(\mathrm {Id}_{{\mathfrak h}_j})\big ) \ge \inf \, (\rho ) \times \Vert {\mathfrak N}_{j,i}^{(\alpha )\, *}(\mathrm {Id}_{{\mathfrak h}_i})\Vert , \end{aligned}$$

  so that ${\mathfrak N}_{j,i}^{(\alpha )\, *}(\mathrm {Id}_{{\mathfrak h}_j})$ is uniformly (in $\alpha$) bounded in norm. The monotone increasing function $\alpha \mapsto {\mathfrak N}_{j,i}^{(\alpha )\, *}(\mathrm {Id}_{{\mathfrak h}_j})$ therefore has a limit and, by Proposition 3.3, ${\mathbb {E}}_{i,\rho '}(n_j)<\infty$ for any $\rho '$.

• 4. The construction of ${\mathfrak N}_{j,i}$ when ${\mathbb {E}}_{i,\rho }(n_j)<\infty$ for any $\rho$ is obtained by a Banach–Steinhaus argument.

$\square$

Proof of Proposition 3.7

Recall that ${\mathbb {P}}_{i,\rho }(t_i<\infty )=\Vert {\mathfrak P}_{i,i}(\rho )\Vert$. By Proposition 3.3, the map ${\mathfrak P}_{i,i}$ is bounded, and since $\mathcal S({\mathfrak h}_i)$ is compact, the supremum $p=\sup _{\rho \in \mathcal S({\mathfrak h}_i)} \mathrm {Tr}\big ({\mathfrak P}_{i,i}(\rho )\big )$ satisfies $p<1$. A standard application of the strong Markov property for the chain $(x_n,\rho _n)_n$ shows that ${\mathbb {P}}_{i,\rho }(n_i=k)\le p^k$ and by a direct computation ${\mathbb {E}}_{i,\rho }(n_i) \le p(1-p)^{-2}$, which gives the result. $\square$

Proof of Proposition 3.9 and Corollary 3.10

We start with two simple lemmata: $\square$

Lemma 8.8

Assume that $\mathfrak M$ is an irreducible open quantum walk and let i, j in V be such that $\mathrm {dim}\,{\mathfrak h}_i<\infty$. Then

$$\begin{aligned} \inf _{\rho \in \mathcal S({\mathfrak h}_i)}{\mathbb {P}}_{i,\rho }(t_j<\infty )>0. \end{aligned}$$

Proof of Lemma 8.8

For any $\rho$ in $\mathcal S({\mathfrak h}_i)$, there exists a unit vector $\varphi$ in ${\mathfrak h}_i$ and $\lambda > 0$ such that $\rho \ge \lambda |\varphi \rangle \langle \varphi |$. By irreducibility, there exists a path $\pi$ in $\mathcal P(i,j)$ such that $\Vert L_\pi \varphi \Vert ^2>0$, so that ${\mathbb {P}}_{i,\rho }(t_{j}<\infty )>0$. By continuity of ${\mathfrak P}_{j,i}$ and compactness of $\mathcal S({\mathfrak h}_j)$, one has the result. $\square$

Lemma 8.9

Assume that $\mathfrak M$ is an irreducible open quantum walk and let i, j be in V. If $\mathrm {dim}\,{\mathfrak h}_j<\infty$ and $\rho \in \mathcal S({\mathfrak h}_i)$ is such that ${\mathbb {E}}_{i,\rho }(n_j)=\infty$, then for any $j' \in V$ one has ${\mathbb {E}}_{i,\rho }(n_{j'})=\infty$.

Proof of Lemma 8.9

By Lemma 8.8, one has $\inf _{\rho '\in \mathcal S({\mathfrak h}_j)}{\mathbb {P}}_{j,\rho '}(t_{j'}<\infty )>0$ for any $j'\in V$. Now, a standard markovianity argument shows that ${\mathbb {E}}_{i,\rho }(n_j)=\infty$ implies ${\mathbb {E}}_{i,\rho }(n_{j'})=\infty$. $\square$

Remark 8.10

Here we used only a weaker version of irreducibility, namely the fact that for any k, l in V, any $\varphi$ in ${\mathfrak h}_k$, there exists a path $\pi$ in ${\mathcal P}(k,l)$ such that $L_\pi \varphi \ne 0$.

Let us go back to the proof of Proposition 3.9 and Corollary 3.10. Define for j in V

$$\begin{aligned} D^n(j)=\big \{\varphi =\sum _{i\in V}\varphi _i\otimes |i\rangle \ \text{ s.t. } \sum _{i\in V}\sum _{\pi \in {\mathcal P}(i,j)} \Vert L_\pi \varphi _i\Vert ^2 < \infty \big \}. \end{aligned}$$

(41)

It is immediate that $D^n(j)$ is a vector space, and that $(L_{k,l}\otimes |k\rangle \langle l|) D^n(j)\subset D^n(j)$ for any k, l in V. In the language of [10], this means that $\overline{D^n (j)}$ is an enclosure for $\mathfrak M$. Moreover, the only possible enclosures for an irreducible $\mathfrak M$ are $\{0\}$ and ${\mathcal H}$. Therefore, either ${D^n(j)}=\{0\}$ or $\overline{D^n(j)}={\mathcal H}$. Define for i in V ${\mathfrak d}^n _{j,i}=D^n(j)\cap {\mathfrak h}_i$ (with a slight abuse of notation). Then either for every i the subspace ${\mathfrak d}^n_{j,i}$ is dense in ${\mathfrak h}_i$ or for every i it is $\{0\}$. Remark that by Lemma 8.7, $\sum _{\pi \in {\mathcal P}(i,j)} \Vert L_\pi \varphi _i\Vert ^2 = {\mathbb {E}}_{i,|\varphi _i\rangle \langle \varphi _i|}(n_j)$. By linearity of ${\mathbb {E}}_{i,\rho }(n_j)$ in $\rho$, if ${\mathfrak d}^n_{j,i}=\{0\}$ then ${\mathbb {E}}_{i,\rho }(n_j)=\infty$ for any $\rho$ in $\mathcal S({\mathfrak h}_i)$, and if ${\mathfrak d}^n_{j,i}$ is dense then ${\mathbb {E}}_{i,\rho }(n_j)<\infty$ for any $\rho$ with finite range in ${\mathfrak d}^n_{j,i}$. This concludes the proof of Proposition 3.9.

Now, if $\mathrm {dim}\,{\mathfrak h}_i<\infty$, then in situation 2. of Proposition 3.9 one has ${\mathbb {E}}_{i,\rho }(n_j)=\infty$ for any i in V and $\rho$ in $\mathcal S({\mathfrak h}_i)$. Now, Lemma 8.9 forbids the situation where for $j\ne j'$ one has ${\mathbb {E}}_{i,\rho }(n_j)=\infty$ and ${\mathbb {E}}_{i,\rho }(n_{j'})<\infty$ for every $\rho$ in $\mathcal S({\mathfrak h}_i)$, and this proves Corollary 3.10.

Remark 8.11

This proof is essentially due to [20].

Proof of Proposition 3.12

By Definition 2.1 of irreducibility, there is no nontrivial invariant subspace of ${\mathfrak h}_j$ left invariant by all $L_\pi$, $\pi \in {\mathcal P}(j,j)$. Since any $\pi \in {\mathcal P}(j,j)$ is a concatenation of paths in ${\mathcal P}^{V\setminus \{j\}}(j,j)$, there is also no nontrivial invariant subspace of ${\mathfrak h}_j$ left invariant by all $L_\pi$, $\pi \in {\mathcal P}^{V\setminus \{j\}}(j,j)$, and this means that ${\mathfrak P}_{j,j}$ is a completely positive irreducible map on $\mathcal I_1({\mathfrak h}_j)$. In addition, we know from the Russo–Dye Theorem that $\Vert {\mathfrak P}_{j,j}\Vert =\Vert {\mathfrak P}^*_{j,j}(\mathrm {Id})\Vert \le 1$, so that the eigenvalue $\lambda$ of ${\mathfrak P}_{j,j}$ of largest modulus satisfies $|\lambda |\le 1$. By the Perron–Frobenius Theorem for completely positive maps acting on the set of trace-class operators of a finite-dimensional space (see Theorem 3.1 and Remark 3.1 in [35], which are essentially proven in [19]), there exists a faithful state $\rho _{\mathrm {f}}$ on ${\mathfrak h}_j$ such that ${\mathfrak P}_{j,j}(\rho _{\mathrm {f}})=|\lambda |\rho _{\mathrm {f}}$. If $|\lambda |<1$, then by Proposition 3.3 one has ${\mathbb {E}}_{j,\rho _{\mathrm {f}}}(n_j)<\infty$. However, by Proposition 3.9, the assumption ${\mathbb {E}}_{i,\rho }(n_j)=\infty$ implies ${\mathbb {E}}_{j,\rho _{\mathrm {f}}}(n_j)=\infty$, a contradiction. Therefore $|\lambda |=1$, $\rho _{\mathrm {f}}$ is a faithful invariant state and $\mathrm {Tr}\, {\mathfrak P}_{j,j}(\rho _{\mathrm {f}})=\mathrm {Tr}\,\rho _{\mathrm {f}}=1$. By Corollary 3.5, we have that ${\mathbb {P}}_{j,\rho }(t_j<\infty )=1$ for any $\rho$ in $\mathcal S({\mathfrak h}_i)$. $\square$

Appendix 3: Proofs for Section 4

Proof of Proposition 4.1

The expansion of ${\mathbb {E}}_{i,\rho }(t_j)$ and the construction of ${\mathfrak T}_{j,i}$ are obtained by now standard Banach–Steinhaus arguments. $\square$

Proof of Proposition 4.4

Proposition 4.4 is proved like Proposition 3.9, by introducing

$$\begin{aligned} D^t(j)=\big \{\varphi =\sum _{i\in V}\varphi _i\otimes |i\rangle \ \text{ s.t. } \sum _{i\in V}\sum _{\pi \in {\mathcal P}^{V\setminus \{j\}}(i,j)} \ell (\pi ) \,\Vert L_\pi \varphi _i\Vert ^2 < \infty \big \} \end{aligned}$$

(42)

and remarking that $D^t(j)$ is an enclosure. $\square$

Proof of Theorem 4.3

Define $\mathfrak d^t_{j,i}=D^t(j)\cap {\mathfrak h}_i$. Remark that in the case of a semifinite OQW, by Proposition 4.4, for every j in V either $\mathfrak d^t_{j,i}=\{0\}$ for every i; or $\mathfrak d^t_{j,i}={\mathfrak h}_i$ for every i. If for some j one has $\mathfrak d^t_{j,i}={\mathfrak h}_i$ for every i, then we have in particular ${\mathbb {E}}_{j,\rho }(t_{j})<\infty$ for any $\rho$ in $\mathcal S({\mathfrak h}_j)$; for any $j'$, applying Lemma 8.8 again one has $\inf _{\rho \in \mathcal S({\mathfrak h}_j)}{\mathbb {P}}_{j,\rho }(t_{j'}<\infty )>0$. By a markovianity argument, one obtains that ${\mathbb {E}}_{j,\rho '}(t_{j'})<\infty$ for any $j'$ in V and $\rho '\in \mathcal S({\mathfrak h}_j)$. $\square$

Proof of Theorem 4.5

Let ${\tau ^{\mathrm {inv}}}=\sum _{i\in V} {\tau ^{\mathrm {inv}}}(i)\otimes |i\rangle \langle i|$ be an invariant state for $\mathfrak M$. Then by the infinite-dimensional extension of the Kümmerer–Maassen ergodic Theorem (see [31]), one has, for any $i\in V$ and $\rho \in \mathcal S({\mathfrak h}_i)$, the ${\mathbb {P}}_{i,\rho }$-almost-sure convergence

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^{n} \rho _k \otimes |x_k\rangle \langle x_k|\underset{n\rightarrow \infty }{\longrightarrow } \sum _{j\in V}{\tau ^{\mathrm {inv}}}(j)\otimes |j\rangle \langle j|, \end{aligned}$$

(43)

where convergence is in the weak-* sense. This implies in particular that

$$\begin{aligned} n_j^{(k)} = \mathrm {card}\{n\le k\,|\, x_n=j\} \end{aligned}$$

satisfies, for any $j\in V$, ${n_j^{(k)}}/k\underset{k\rightarrow \infty }{\rightarrow }\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)$, ${\mathbb {P}}_{i,\rho }$-almost-surely. Therefore, $t_j^{(k)}<\infty$ but $t_j^{(k)}\underset{k\rightarrow \infty }{\rightarrow }\infty$. Considering $m=t_j^{(k)}$, we have ${n_j^{(m)}}/{m}={k}/{t_j^{(k)}}$ and therefore, ${\mathbb {P}}_{i,\rho }$-almost-surely, $t_j^{(k)}/k \rightarrow \big (\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)\big )^{-1}$ .

Observe now that, as shown in Example 5.3, our assumptions imply in particular that ${\mathbb {P}}_{j,\rho }(t_j<\infty )=1$ for any $\rho$ in $\mathcal {S}({\mathfrak h}_j)$, so that ${\mathfrak P}_{j,j}$ is a completely positive, trace-preserving map, with Kraus decomposition

$$\begin{aligned} {\mathfrak P}_{j,j}(\rho )=\sum _{\pi \in \mathcal P^{V\setminus \{j\}}} L_\pi \rho L_\pi ^*. \end{aligned}$$

In addition, we have ${\mathbb {P}}_{j,\rho }$-almost-surely from (43)

$$\begin{aligned} \frac{1}{n} \sum _k\rho _{t_{j}^{(k)}} \mathbbm {1}_{t_j^{(k)}\le n} \underset{n\rightarrow \infty }{\longrightarrow } {\tau ^{\mathrm {inv}}}(j) \end{aligned}$$

(the convergence needs not be specified, as ${\mathfrak h}_j$ is finite-dimensional), but the Kümmerer–Maassen ergodic Theorem applied to ${\mathfrak P}_{j,j}$ shows that $\frac{1}{n_j^{(m)}} \sum _{k=1}^{n_j^{(m)}} \rho _{t_{j}^{(k)}}$ converges almost-surely to an invariant of ${\mathfrak P}_{j,j}$. Therefore, $\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)}$ is an invariant state for ${\mathfrak P}_{j,j}$ and ${\mathbb {P}}_{j,\rho }$-almost-surely,

$$\begin{aligned} \frac{1}{n_j^{(m)}} \sum _{k=1}^{n_j^{(m)}} \rho _{t_{i}^{(k)}} \underset{m\rightarrow \infty }{\longrightarrow }\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)} \end{aligned}$$

(44)

In addition, since ${{\tau ^{\mathrm {inv}}}(j)}$ is faithful on ${\mathfrak h}_j$, one has by necessity that ${\mathfrak P}_{j,j}$ is irreducible: if there existed an invariant subspace for all $L_\pi$, $\pi \in \mathcal P^{V\setminus \{j\}}$, then there would exist an invariant state $\rho '_j$ for ${\mathfrak P}_{j,j}$ with support on this invariant subspace, and considering initial data $(j,\rho _j')$ in (44) above would show that ${\tau ^{\mathrm {inv}}}(j)$ has support no larger than the support of $\rho '_j$, a contradiction.

We now define a new probability space by $\Omega ^{(j)}= \big (\mathcal P^{V\setminus \{j\}}(j,j)\big )^{\otimes \mathbb {N}}$, and let

$$\begin{aligned} {\mathbb {P}}^{(j)}(\pi _1,\ldots ,\pi _m)= \mathrm {Tr}\,\, \big (L_{\pi _m}\ldots L_{\pi _1} \frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)} L_{\pi _1}^* \ldots L_{\pi _m}^*\big ). \end{aligned}$$

The trace-preserving property of ${\mathfrak P}_{j,j}$ shows that this defines a consistent family and by the Daniell–Kolmogorov extension Theorem this defines a probability ${\mathbb {P}}^{(j)}$ on $\Omega ^{(j)}$. In addition, the invariance of $\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)}$ by ${\mathfrak P}_{j,j}$ implies

$$\begin{aligned} \sum _{\pi _1 \in \mathcal P^{V\setminus \{j\}}(j,j)} {\mathbb {P}}^{(j)}(\pi _1,\ldots ,\pi _m)= {\mathbb {P}}^{(j)}(\pi _2,\ldots ,\pi _m), \end{aligned}$$

which shows that ${\mathbb {P}}^{(j)}$ is invariant by the left shift

$$\begin{aligned} \begin{array}{cccc} \Theta : &{} \Omega ^{(j)} &{} \rightarrow &{} \Omega ^{(j)}\\ &{} (\pi _1,\pi _2,\ldots )&{}\mapsto &{} (\pi _2,\pi _3,\ldots ) \end{array} \end{aligned}$$

Now, the Perron–Frobenius Theorem implies that 1 is a simple eigenvalue for ${\mathfrak P}_{j,j}$. This immediately shows that for any two cylinder sets E and F,

$$\begin{aligned} \frac{1}{m} \sum _{k=1}^m {\mathbb {P}}^{(j)}\big (E \cap \Theta ^{-k}(F)\big )\underset{m\rightarrow \infty }{\longrightarrow }{\mathbb {P}}^{(j)}(E) \,{\mathbb {P}}^{(j)}(F), \end{aligned}$$

so that $(\Omega ^{(j)},{\mathbb {P}}^{(j)})$ is ergodic for $\Theta$. Now, if we consider the map $\ell ^{(k)}$ defined by

$$\begin{aligned} \ell ^{(k)}(\pi _1,\pi _2,\ldots )=\ell (\pi _1)+\ldots +\ell (\pi _k), \end{aligned}$$

then this map is an additive functional, i.e. satisfies $\ell ^{(k+k')}=\ell ^{(k)}+\ell ^{(k')}\circ \Theta ^k$. By Birkhoff’s ergodic Theorem one has ${\mathbb {P}}^{(j)}$-almost-sure convergence of ${\ell ^{(k)}}/k$ to the expectation of $\ell ^{(1)}$ for ${\mathbb {P}}^{(j)}$. It is immediate, however, that the distribution of $\ell ^{(k)}$ under ${\mathbb {P}}^{(j)}$ is the same as the distribution of $t_j^{(k)}$ under ${\mathbb {P}}_{j,\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)}}$. We therefore have

$$\begin{aligned} {t_j^{(k)}}/{k}\underset{k\rightarrow \infty }{\longrightarrow }{\mathbb {E}}_{j,\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)}}(t_j^{(1)}), \end{aligned}$$

where convergence is almost-sure and in the $\mathrm L^1$ sense, with respect to ${\mathbb {P}}_{j,\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)}}$. The first part of the proof shows that

$$\begin{aligned} {\mathbb {E}}_{j,\frac{{\tau ^{\mathrm {inv}}}(j)}{\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)}}(t_j^{(1)})=\big (\mathrm {Tr}\,{\tau ^{\mathrm {inv}}}(j)\big )^{-1}, \end{aligned}$$

and this concludes the proof. $\square$

Appendix 4: Proofs for Section 6

Proof of Proposition 6.1

Consider $A=\sum _{i\in V} A_i\otimes |i\rangle \langle i|$ such that for any i in V, $\sum _{j\in V}\Vert {\mathfrak N}_{j,i}^*(A_j)\Vert <\infty$. Then (25) defines an operator Z. Proving that Z satisfies (24) is then a straightforward computation. By linearity it is enough to assume that $A=A_k\otimes |k\rangle \langle k|$. We then have

$$\begin{aligned} \mathfrak M^*(Z)&= \sum _{i\in V} \Big (\sum _{j\in V} L_{j,i}^* \Big (\mathbbm {1}_{j=k}\,A_k+\sum _{\pi \in {\mathcal P}(j,k)}L_\pi ^* A_k L_\pi \Big ) L_{j,i}\Big )\otimes |j\rangle \langle j|. \end{aligned}$$

Since the set of paths obtained by concatenating one step from a given i to a variable j, then some $\pi$ from j to k, is exactly the set of paths from i to k of length $\ge 2$, and (i, k) is the only path from i to k of length 1, we obtain $\mathfrak M^*(Z)=Z-A_k\otimes |k\rangle \langle k|$, so that $(\mathrm {Id}-\mathfrak M^*)(Z)=A$. If $Z'$ is another solution of (24), then $Y=Z'-Z$ satisfies $\mathfrak M^*(Y)=Y$ and by the Perron–Frobenius Theorem of [23] applied to the irreducible map $\mathfrak M^*$, we have $Y\in \mathbb {C}\mathrm {Id}_{\mathcal H}$. $\square$

Proof of Proposition 6.3

It is now a routine argument to construct ${\mathfrak P}_{i,j}^D$ using the Banach–Steinhaus Theorem, as

$$\begin{aligned} {\mathfrak P}_{j,i}^D(\rho )=\sum _{\pi \in {\mathcal P}^{D\backslash \{j\}}(i,j)} L_\pi \rho L_\pi ^*. \end{aligned}$$

One then has by definition ${\mathbb {P}}_{i,\rho }(t_j\le t_{\partial \! D}<\infty )=\mathrm {Tr}\, {\mathfrak P}^D_{j,i}(\rho )$, and the second identity follows from ${\mathbb {P}}_{i,\rho }(t_{\partial \! D}<\infty )=\sum _{j\in {\partial \! D}}{\mathbb {P}}_{i,\rho }(t_j\le t_{\partial \! D}<\infty )$. Relations (28) and (29) are obtained as Equation (16). $\square$

Proof of Proposition 6.5

We define

$$\begin{aligned} p= \inf _{i\in D}\inf _{\rho \in \mathcal S(\mathfrak h_i)} {\mathbb {P}}_{i,\rho }( t_{{\partial \! D}} < +\infty ). \end{aligned}$$

We will show independently that $p>0$ and that $p\in \{0,1\}$, therefore proving Proposition 6.5.

To prove that $p>0$, we use a simple adaptation of Lemma 8.8. Fix some $\rho$ in $\mathcal S (\mathfrak h_i)$; there exist a unit vector $\varphi$ in $\mathfrak h_i$ and $\lambda >0$ such that $\rho \ge \lambda |\varphi \rangle \langle \varphi |$. By irreducibility, for any j in ${\partial \! D}$ there exists a path $\pi$ in ${\mathcal P}(i,j)$ such that $L_\pi \,\varphi \ne 0$. There exists $j'$ in ${\partial \! D}$ (the first point of ${\partial \! D}$ visited by the trajectory $\pi$) and a subpath $\pi '$ of $\pi$ belonging to ${\mathcal P}^{D}(i,j')$, with necessarily $L_{\pi '} \,\varphi \ne 0$. We have shown $\mathrm {Tr}\,{\mathfrak P}^D_{i}(\rho )>0$ and, ${\mathfrak P}^D_{i}$ being continuous, we have by a compactness argument that $\inf _{\rho \in \mathcal S({\mathfrak h}_i)}\mathrm {Tr}\,{\mathfrak P}^D_{i}(\rho )>0$, and therefore $p>0$ as D is finite.

We next prove that $p \in \{0,1\}$. By the strong Markov property, for any n one has

$$\begin{aligned} 1-p&=\sup _{i\in D}\sup _{\rho \in \mathcal S(\mathfrak h_i)} {\mathbb {P}}_{i,\rho }(t_{{\partial \! D}}=+\infty )\\&= \sup _{i,\rho } {\mathbb {E}}_{i,\rho } \big ( \mathbbm {1}_{x_1,\ldots , x_n \in D} \, {\mathbb {P}}_{x_n,\rho _n} (t_{{\partial \! D}}=+\infty ) \big )\\&\le (1-p) \, {\mathbb {P}}_{i,\rho } ( x_1,\ldots , x_n \in D), \end{aligned}$$

and taking $n\rightarrow \infty$ leads to $(1-p)\le (1-p)^2$, so that $p\in \{0,1\}$. This concludes our proof. $\square$

Proof of Lemma 6.6

Let j in V with $\mathrm {dim}\,{\mathfrak h}_j<\infty$. By irreducibility, there exists a path $\pi$ in $\mathcal P^D(j,k)$ for some $k\in {\partial \! D}$ such that $\mathrm {Tr}L_\pi \rho L_\pi ^*\ne 0$. There exists $k'$ in ${\partial \! D}$ and a subpath $\pi '$ of $\pi$ which belongs to $\mathcal P^{D\setminus \{j\}}(j,k')$ such that $\mathrm {Tr}L_{\pi '} \rho L_{\pi '}^*\ne 0$, which implies that ${\mathbb {P}}_{j,\rho }(t_j\le t_{\partial \! D})~<~1$. In particular, $\mathrm {Tr}\,{\mathfrak P}^D_{j,j}(\rho )<1$ for any $\rho$ in $\mathcal S({\mathfrak h}_j)$, so that $\Vert {\mathfrak P}^D_{j,j}\Vert <1$. The same discussion that allowed us to construct ${\mathfrak N}_{j,i}$ shows that ${\mathfrak N}^D_{j,i}$ is well-defined by ${\mathfrak N}^D_{j,i}=(\mathrm {Id}- {\mathfrak P}^D_{j,j})^{-1}\circ {\mathfrak P}^D_{j,i}$ and satisfies relations (30) and (31). $\square$

Proof of Proposition 6.8

By Lemma 6.6, all operators ${\mathfrak N}^D_{j,i}$ and therefore the operator Z, are well-defined. Obviously $Z_j=B_j$ for $j\in {\partial \! D}$; the proof that $\big ((\mathrm {Id}-\mathfrak M^*)(Y)\big )_i=A_i$ for $i\in D$ is similar to that for Proposition 6.1. Now consider two solutions Z and $Z'$; then $Y=Z-Z'$ satisfies $Y_j=0$ for $j\in {\partial \! D}$ and $\big ((\mathrm {Id}-\mathfrak M^*)(Y)\big )_i=0$ for $i\in D$. As in Lemma 2.5 we can prove that, if $m_n=\big (\mathrm {Tr}(\rho _n Y_{x_n})\big )_n$, then $m^D_n=m_{\inf (n,t_{\partial \! D})}$ is a ${\mathbb {P}}_{i,\rho }$-martingale for any i in D and $\rho$ in $\mathcal S({\mathfrak h}_i)$. The optional sampling Theorem applied to the bounded martingale $\mathrm {Tr}(\rho _n Y_{x_n})$ and the stopping time $t_{\partial \! D}$ implies that

$$\begin{aligned} \mathrm {Tr}\big (\rho \,Y_i\big )={\mathbb {E}}_{i,\rho }\big (\mathrm {Tr}(\rho _{t_{\partial \! D}} Y_{x_{t_{\partial \! D}}})\big )=0. \end{aligned}$$

Since this is true for any $\rho$ in $\mathcal S({\mathfrak h}_i)$, we obtain that $Y_i=0$, for any $i\in D$. $\square$

Appendix 5: Proof for Section 8

Proof of Lemma 8.3

Since $\Vert \mathfrak M^*\Vert =1$, the quantum detailed balance condition implies that the spectrum of $\mathfrak M^*$ is contained in $[-1,+1]$, so that $I-\mathfrak M^*$ is a positive operator and $\mathcal {E}(X)\ge 0$ for all $X\in \mathcal {B}(\mathcal {H})$. In addition, $\mathcal {E}(X)=0$ if and only if $\mathfrak M^*(X)=X$. If $\mathfrak M$ is irreducible, then by the Perron–Frobenius Theorem for operators on a C*-algebra (see [23]) applied to the irreducible map $\mathfrak M^*$, the identity $\mathfrak M^*(X)=X$ is equivalent with $X\in \mathbb {C}\mathrm {Id}_{\mathcal H}$. $\square$

Proof of Theorem 8.5

Let us write $Z=B+X+X'$ with $X\in \mathcal B(\mathcal H_D)$ and $X'\in \mathcal B(\mathcal H_{V\setminus (D\cup {\partial \! D})})$. By definition of ${\partial \! D}$, one has $(\mathrm {Id}-\mathfrak M^*)(X')\in \mathcal B(\mathcal H_{V\setminus D})$. Denoting $C=(\mathrm {Id}-\mathfrak M^*)(B)$ we have that Z is a solution of (26) if and only if $\big ((\mathrm {Id}-\mathfrak M^*)(X)\big )_k=(A-C)_k$ for $k\in D$, or equivalently if

$$\begin{aligned} \mathcal E(T,X)=\langle T, A-C\rangle _\diamond \quad \text{ for } \text{ any } T\in \mathcal B(\mathcal H_D). \end{aligned}$$

(45)

By Lemma 8.3, ${\mathcal E}(X,X)$ is non-negative and vanishes only if $X\in \mathbb {C}\mathrm {Id}_{\mathcal H}$. However, since ${\partial \! D}\ne \emptyset$, $\mathrm {Id}_{\mathcal H}\not \in \mathcal B(\mathcal H_D)$ and one has $\mathcal E(X,X)>0$ for any $X\in \mathcal B(\mathcal H_D)$. Consequently, by a compactness argument, there exists $\lambda >0$ such that $\mathcal E(X,X)\ge \lambda \Vert X\Vert _\diamond ^2$ for $X\in \mathcal B(\mathcal H_D)$. One can then apply the Lax–Milgram Theorem (see [8]): there exists a unique $X_0$ satisfying (45), which in addition is the minimizer of

$$\begin{aligned} \mathcal B(\mathcal H_D) \ni X\mapsto \frac{1}{2}\,\mathcal E(X,X) - \langle X,A-C\rangle _\diamond = \frac{1}{2}\,\mathcal E(X,X) + \mathcal E(X,B)- \langle X,A\rangle _\diamond . \end{aligned}$$

The solutions of Equation (26) are therefore the operators of the form

$$\begin{aligned} Z=B+X_0+X' \end{aligned}$$

for $X'\in \mathcal B(\mathcal H_{V\setminus (D\cup {\partial \! D})})$. $\square$

Proof of Proposition 8.6

The proof is simply a matter of computation. For doubly stochastic OQW, $L_{ij}=L^*_{ji}$, the invariant state $\tau _\diamond$ is the identity and the Dirichlet form reads

$$\begin{aligned} \mathcal {E}(X)= \mathrm {Tr}\big (X^*(\mathrm {Id}-\mathfrak {M})X\big ) =\sum _{i,j\in V}\mathrm {Tr}\big (X^*_i \delta _{ij} X_j - X_i^*L_{ij}X_jL_{ji}\big ). \end{aligned}$$

On the other hand we have

$$\begin{aligned} \frac{1}{2} \Vert (\nabla X)\Vert _V^2= & {} \frac{1}{2} \sum _{i,j\in V} \mathrm {Tr}\big (\big (X_iL_{ij}-L_{ij}X_j\big )\big (L_{ji}X_i^*-X_j^*L_{ji}\big )\big )\\= & {} \frac{1}{2}\sum _{i,j\in V} \mathrm {Tr}\left( X_iL_{ij}L_{ji}X_i^* +X^*_iL_{ij}L_{ji}X_i - 2 L_{ij}X_jL_{ji}X_i^*\right) . \end{aligned}$$

The two formulas coincide since $\sum _{j\in V} L_{ij}L_{ji}=\mathrm {Id}$ for doubly stochastic OQW. $\square$