Linear Open Quantum Walks

• Linear open quantum walks are discrete-time quantum stochastic processes driven by CPTP dynamics and structured via Kraus operators.
• They exhibit rich recurrence, mixing, and fluctuation phenomena influenced by spectral properties and environmental dissipation.
• Applications span quantum state engineering, dissipative quantum computation, and decoherence control through tailored coin designs.

A linear open quantum walk (OQW) is a discrete-time quantum stochastic process in which a quantum walker with internal degrees of freedom propagates on a graph under the action of a completely positive trace-preserving (CPTP) map, with evolution entirely governed by environmental dissipation rather than unitary dynamics. The linear OQW framework is mathematically equivalent to a quantum Markov chain on the state space $\mathcal{H}\otimes \mathcal{K}$, where $\mathcal{H}$ is the coin (internal) Hilbert space and $\mathcal{K}$ the position Hilbert space associated to vertex set $\mathcal{V}$ (or, for the integer line, $\mathbb{Z}$). The key ingredients are Kraus operators constructed by factoring site-to-site transitions into coin transformations and position shifts. This structure underlies rich recurrence, mixing, and fluctuation phenomena distinct from those seen in unitary quantum walks, with behavior strongly influenced by the spectral properties of underlying superoperators and environmental coupling.

1. CPTP Evolution and Kraus Formalism

Let $\mathcal{H}$ be a finite-dimensional internal Hilbert space and $\mathcal{K}$ a position space with basis $\{|i\rangle\}_{i\in\mathcal{V}}$. The state at discrete time $n$ is a density matrix $$\rho^{[n]}\in\mathcal{B}(\mathcal{H}\otimes\mathcal{K})$$, frequently of block-diagonal form:

$$\rho^{[n]} = \sum_{i\in\mathcal{V}} \rho_i^{[n]} \otimes |i\rangle\langle i|.$$

Transitions correspond to the application of a CPTP map:

$$\rho \mapsto \mathcal{M}(\rho) = \sum_{i,j\in\mathcal{V}} M_{ij} \rho M_{ij}^\dagger,$$

with Kraus operators

$M_{ij} = B^i_j \otimes |i\rangle\langle j|,$

where each $B^i_j\in\mathcal{B}(\mathcal{H})$ is the coin-jump operator associated with move $j\rightarrow i$. The normalization condition

$$\sum_{i} (B^i_j)^\dagger B^i_j = I_\mathcal{H}\quad \forall j\in\mathcal{V}$$

ensures trace preservation, and the map is linear and completely positive by construction (Sinayskiy et al., 2014).

2. Site Recurrence Notions: Monitored and Pólya (SJK)

Site recurrence in OQWs is analyzed via two notions:

• Monitored recurrence: One tracks the first-return to a site by monitoring at each step. Define $\pi_r(i,i)$ as the set of operator strings corresponding to first hitting $i$ at time $r$, and for initial coin state $\rho_i$:

$$R_\text{mon}^{|i\rangle}(\rho_i) = \sum_{r=1}^\infty \sum_{C\in\pi_r(i,i)} \mathrm{Tr}(C\,\rho_i\,C^*).$$

The site $|i\rangle$ is monitored-recurrent if $R_\text{mon}^{|i\rangle}(\rho_i)=1$ for all internal densities.

• Pólya (SJK) recurrence: Let $p_i(n;\rho)$ be the probability of occupying site $i$ at step $n$ starting from density $\rho$ at $i$, and define the Pólya number:

$P(\rho,i) = \prod_{n=1}^\infty (1-p_i(n;\rho)).$

Site $i$ is SJK-recurrent for $(\rho,i)$ if $P(\rho,i)=0$, which is equivalent to $\sum_{n=0}^\infty p_i(n;\rho)=\infty$ (Carvalho et al., 2016). This mirrors classical return criteria extended to CPTP dynamics.

3. Relation to Unitary Quantum Walks: Interference Effects and Monitored Recurrence

For walks induced by the same pair of matrices $L,R$, one can analyze the correspondence between OQW and unitary coined quantum walks (UQWs). If $U=L+R$ is unitary, the UQW's monitored-return probability for pure initial state $|\psi\rangle$ at site $0$ is:

$$R_u(\psi) = \sum_{n=1}^\infty \|a_n(|\psi\rangle\otimes|0\rangle)\|^2,$$

where $a_n$ are appropriate return operators involving projections onto the monitored site. It is shown that

$R_u(\psi) = R_o(\psi) + \alpha(\psi),$

where $R_o(\psi)$ is the OQW monitored-return probability, and $\alpha(\psi)$ is an additive "interference term" resulting from coherences between distinct return paths. If $\alpha(\psi)\le 0$, monitored recurrence in the unitary walk implies monitored recurrence in the open walk, and vice versa, highlighting the role of environmental decoherence in modifying recurrence thresholds (Carvalho et al., 2016).

4. Open-Quantum Kac Lemma: Expected Return Times

An "open quantum" version of Kac's lemma relates expected first-return times to stationary probabilities. For an irreducible OQW with a unique stationary state $\pi=\sum_i \pi(i)\otimes |i\rangle\langle i|$, and internal density $\rho_x$ at site $x$,

$E_R(\rho_x) = \frac{1}{\mathrm{Tr}\,\pi(x)},$

where $E_R(\rho_x)$ is the expected return time at $x$ for a monitored recurrent state. This generalizes classical Kac's lemma, adapting it to CPTP dynamics and block-diagonal stationary states (Carvalho et al., 2016).

5. Examples of Site Recurrence Criteria

Several explicit constructions illustrate recurrence and transience in linear OQWs:

• Hadamard OQW on $\mathbb{Z}$ (coin dimension $d=2$): Operators

$R=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\0&0\end{pmatrix},\quad L=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0\1&-1\end{pmatrix}$

yield monitored-recurrence at the origin for all coin states, but the associated unitary coined walk is not monitored-recurrent due to destructive interference.

• Bit-flip OQW: With

$$L=\sqrt{p}\,I_2,\quad R=\sqrt{1-p}\,\sigma_x,\, p\in [0,1],$$

the site is recurrent iff $p=\frac12$, corresponding to balanced coin parameters.

• Classical coin $d=1$ example (finite three-site chain): Explicit stationary distribution and Kac lemma formulae determine expected return times, with $E_R(0)=2/p$ for site $0$ if transition probability $p$ from $1$ to $0$ (Carvalho et al., 2016).

6. Recurrence Criteria, Spectral Analysis, and Drift

From a spectral perspective, recurrence/transience in linear OQWs hinges on the drift induced by the underlying auxiliary coin map $\mathcal{L}(\sigma) = L \sigma L^* + R \sigma R^*$ and the structure of invariant states. For a homogeneous OQW on $\mathbb{Z}$:

• If the drift $$m = \mathrm{Tr}(L\,\sigma_\infty L^*) - \mathrm{Tr}(R\,\sigma_\infty R^*)$$ vanishes for the unique steady-state coin $\sigma_\infty$, the walk is recurrent.
• Nonzero drift yields transience, with ballistic escape and finite total return probabilities (Loebens, 2 Jan 2025).

The recurrence classification can be generalized via block decompositions in reducible coins, with site recurrence determined locally in each invariant sector.

7. Physical Implications and Generalizations

Site recurrence properties are closely tied to transport and localization in dissipative quantum systems, impacting topics from quantum state engineering to dissipative quantum computation (Sinayskiy et al., 2014, Attal et al., 2014). OQWs interpolate between classical random walks and unitary quantum walks, acting as an archetype for quantum stochastic processes with environmental noise. These recurrence phenomena can be tuned via coin design, topology, and dissipation rates, with applications to mixing speed, decoherence control, and quantum algorithm efficiency. The formalism connects deeply to renewal theory, operator-valued generating functions, and effective "thermodynamic" behavior under CPTP dynamics (Grünbaum et al., 2017).

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References:

• (Sinayskiy et al., 2014) Open Quantum Walks: a short introduction
• (Carvalho et al., 2016) Site recurrence of open and unitary quantum walks on the line
• (Loebens, 2 Jan 2025) Recurrence Criteria for Reducible Homogeneous Open Quantum Walks on the Line
• (Grünbaum et al., 2017) A generalization of Schur functions: applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks