Quantum Markov Chains Associated with Open Quantum Random Walks

Abstract

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties of the underlying dynamics such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc. So, the quantum Markov chain machinery opens many new features of the dynamics. On the other hand, as will be shown in this paper, the open quantum random walks serves as a very interesting nontrivial model for which one can construct the associated quantum Markov chains. Here, after constructing the quantum Markov chain associated with the open quantum random walks, we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We see also that the classical Markov chains can be reconstructed as quantum Markov chains.

Equivalence of Concepts of Reducibility/Irreducibility of OQRWs Defined in [4] and in This Paper

First of all we recall the definition of reducibility/irreducibility used in [4]. Let $\Phi$ be a positive map on the ideal $\mathcal {I}_1(\mathfrak h)$ of trace class operators on a Hilbert space $\mathfrak h$. When we come to our model, $\mathfrak h$ is ${\mathcal {H}}\otimes {\mathcal {K}}$ and $\Phi$ is ${\mathcal {M}}$. $\Phi$ is said to be irreducible (see [4, Definition 3.1]) if the only orthogonal projections p reducing $\Phi$, i.e. such that $\Phi (p\mathcal {I}_1(\mathfrak h)p)\subset p\mathcal {I}_1(\mathfrak h)p$, are $p=0$ and I. Applying to OQRWs, Carbone and Pautrat have shown (terminology in our language):

Proposition A.1

([4, Proposition 3.8]) The completely positive and trace preserving map ${\mathcal {M}}$ is irreducible if and only if for any $i,j\in \Lambda$ and any $\psi ,\xi \in {\mathcal {H}}\setminus \{0\}$, there is a path $\pi \in {\mathcal {P}}(i,j)$ such that $\langle \xi ,B_\pi \psi \rangle \ne 0$.

Now we show the definitions of reducibility/irreducibility of OQRWs given in [4] and in the present paper are equivalent. First we remark that as given by [4, Proposition 6.1, item 3], once an OQRW is reducible (in the sense of [4]) one can always find a reducing projection p of the block-diagonal form: $p=\sum _j p(j)\otimes |j\rangle \langle j|$. Conversely speaking, if there is no nontrivial block-diagonal reducing projection the OQRW is irreducible. Suppose the OQRW is reducible in the sense of [4] with a reducing projection $p=\sum _j p(j)\otimes |j\rangle \langle j|$. By [4, Proposition 6.2], it holds that for any $i,j\in \Lambda$,

$$\begin{aligned} B_j^ip(j)=p(i)B_j^ip(j). \end{aligned}$$

(A.1)

Take an initial state $\rho ^{(0)}=\sum _j\rho ^{(0)}_j\otimes |j\rangle \langle j|$ such that $p(j)\rho ^{(0)}_jp(j)=\rho ^{(0)}_j$ for all $j\in \Lambda$. We can show by induction that for all $n\ge 0$ and $j\in \Lambda$,

$$\begin{aligned} p(j)\rho ^{(n)}_jp(j)=\rho ^{(n)}_j. \end{aligned}$$

(A.2)

In fact, suppose (A.2) holds for $n=0,\cdots ,k$. Then, by the assumption hypothesis and (A.1)

$$\begin{aligned} p(j)\rho ^{(k+1)}_jp(j)= & {} \sum _ip(j)B_i^j\rho ^{(k)}_i{B_i^j}^*p(j)\\= & {} \sum _ip(j)B_i^jp(i)\rho ^{(k)}_ip(i){B_i^j}^*p(j)\\= & {} \sum _iB_i^jp(i)\rho ^{(k)}_ip(i){B_i^j}^*\\= & {} \sum _iB_i^j\rho ^{(k)}_i{B_i^j}^*=\rho ^{(k+1)}_j. \end{aligned}$$

Now (A.2) holds and by Theorem 4.12 the OQRW is reducible in the sense of this paper (recall (A.2) is equivalent to $\rho ^{(n)}_jp(j)=\rho ^{(n)}_j$).

Conversely, suppose that the OQRW is reducible in the sense of present paper. By Theorem 4.12, there is a nontrivial projection $p=\sum _j p(j)\otimes |j\rangle \langle j|$ such that (A.2) holds for $n\ge n_0$ for some $n_0$. Find a $j\in \Lambda$ such that $p(j)\ne I_{\mathcal {H}}$. By the assumption we have for any $k\ge 0$,

$$\begin{aligned} {\mathrm {Tr}}(\rho ^{(n_0+k)}_jp(j)^\perp )={\mathrm {Tr}}(\rho ^{(n_0+k)}_jp(j)p(j)^\perp )=0. \end{aligned}$$

Take an $i\in \Lambda$ such that $\rho ^{(n_0)}_i\ne 0$. From the above relation we have

$$\begin{aligned} 0= & {} {\mathrm {Tr}}(\rho ^{(n_0+k)}_jp(j)^\perp )\\= & {} \sum _{i_0,\ldots ,i_{k-1}}{\mathrm {Tr}}\left( B_{\pi (i_0,\ldots ,i_{k-1},j)}\rho ^{(n_0)}_{i_0}B^*_{\pi (i_0,\ldots ,i_{k-1},j)}p(j)^\perp \right) \\\ge & {} {\mathrm {Tr}}\left( B_{\pi }\rho ^{(n_0)}_{i}B^*_{\pi }p(j)^\perp \right) ={\mathrm {Tr}}\left( \rho ^{(n_0)}_{i}B^*_{\pi }p(j)^\perp B_{\pi }\right) \ge 0, \end{aligned}$$

for any path $\pi \in {\mathcal {P}}(i,j)$ of length k. Thus for any $0\ne \psi \in {\mathcal {H}}$ lying in the spectral projection of $\rho ^{(n_0)}_i$ away from zero, e.g., any eigenvector of $\rho ^{(n_0)}_i$ corresponding to nonzero eigenvalue,

$$\begin{aligned} \langle \psi ,B^*_{\pi }p(j)^\perp B_{\pi }\psi \rangle =0. \end{aligned}$$

Therefore, for any such a vector $0\ne \psi$ and $0\ne \xi \in p(j)^\perp$, and for any path $\pi \in {\mathcal {P}}(i,j)$,

$$\begin{aligned} |\langle \xi ,B_\pi \psi \rangle |= & {} |\langle \xi ,p(j)^\perp B_\pi \psi \rangle |\\\le & {} \Vert \xi \Vert \langle p(j)^\perp B_\pi \psi ,p(j)^\perp B_\pi \psi \rangle ^{1/2}=0. \end{aligned}$$

By Proposition A.1, it says that the OQRW is reducible in the sense of [4]. This completes the proof of equivalence.