Quantum Channels on Graphs: a Resonant Tunneling Perspective

TL;DR

The paper tackles how coherent information propagates on quantum graphs by recasting graph scattering as a quantum channel between input and output ports and by constructing a global scattering matrix S_G from local nodes via the Redheffer star product.A key concept, resonant concatenation (RC), arises from internal back-reflections and leads to a nonlinear composition of channels that can suppress noise and even produce super-activation of quantum capacity when individual components have zero capacity.The authors develop a state-dependent erasure-channel framework to bound the quantum capacity of graph channels and demonstrate RC with a concrete resonant tunneling model involving two barriers and a lossy element, showing robust SA effects as system parameters and energy are varied.The approach provides a general methodology for analyzing coherent information flow in structured quantum environments, with potential applications in quantum communication, control, and simulation on networks.

Abstract

Quantum transport on structured networks is strongly influenced by interference effects, which can dramatically modify how information propagates through a system. We develop a quantum-information-theoretic framework for scattering on graphs in which a full network of connected scattering sites is treated as a quantum channel linking designated input and output ports. Using the Redheffer star product to construct global scattering matrices from local ones, we identify resonant concatenation, a nonlinear composition rule generated by internal back-reflections. In contrast to ordinary channel concatenation, resonant concatenation can suppress noise and even produce super-activation of the quantum capacity, yielding positive capacity in configurations where each constituent channel individually has zero capacity. We illustrate these effects through models exhibiting resonant-tunneling-enhanced transport. Our approach provides a general methodology for analyzing coherent information flow in quantum graphs, with relevance for quantum communication, control, and simulation in structured environments.

Figures (10)

• Figure 1: Panel a): Schematic representation of a quantum channel $\Phi_{\mathbb S}$ associated with the scattering of a quantum particle $P$ (green element in the figure), prepared by Alice in the internal state $\hat{\rho}$ and received by Bob. The arrows pointing toward and away from the scattering center $S$ indicate the possible spatial directions the particle can take. The symbol $\O$ on an incoming line denotes that no particle enters from that particular direction. Panel b): Direct and resonant concatenation scenarios for two scattering events. In the absence of back-scattering, the resulting quantum channel corresponds to the direct composition $\Phi_{{\mathbb S}_2 \circ {\mathbb S}_1}$ of the individual maps $\Phi_{{\mathbb S}_1}$ and $\Phi_{{\mathbb S}_2}$. When back-scattering from ${\mathbb S}_2$ to ${\mathbb S}_1$ is allowed (red dotted line), the resulting map is described by the RC channel $\Phi_{{\mathbb S}_2 \star {\mathbb S}_1}$. Panel c): Super-activation induced by RC. Here, ${\mathbb S}_1$ and ${\mathbb S}_2$ represent potential barriers that can reflect $P$, while ${\mathbb S}_{\eta}$ serves as a scatterer that deflects $P$ from the horizontal paths into the vertical ones, thereby modeling losses.
• Figure 2: Top: Quantum capacities $Q(\Phi_{{\mathbb S}_2\star({\mathbb S}_{\eta}\circ {\mathbb S}_1)})$ (orange) and $Q(\Phi_{{\mathbb S}_{\eta}\circ {\mathbb S}_1})$ (blue) for spin-independent potentials ($\epsilon=0$) as a function of the input energy $E$, for fixed loss. Bottom: lower bound $Q_{\rm up}(\Phi_{{\mathbb S}_2\star({\mathbb S}_{\eta}\circ {\mathbb S}_1)})$ of Eq. \ref{['eq:Qerasureineq']} for the quantum capacity of $\Phi_{{\mathbb S}_2\star({\mathbb S}_{\eta}\circ {\mathbb S}_1)}$ (orange), and upper bound $Q_{\rm low}(\Phi_{{\mathbb S}_{\eta}\circ {\mathbb S}_1})$ for the quantum capacity of $\Phi_{{\mathbb S}_{\eta}\circ {\mathbb S}_1}$ (blue) in the spin-dependent case with $\epsilon=0.1$. In all plots $d=2$, $\eta=0.1$, $a= 0.06 \sqrt{20}$ and $w= 10\sqrt{20}$ where $a$ and $w$ are intended in units of $k_0^{-1} = \frac{\hbar}{\sqrt{2mV_0}}$. Red arrows indicate the energy intervals where the model exhibit SA (i.e. $Q(\Phi_{{\mathbb S}_2\star({\mathbb S}_{\eta}\circ {\mathbb S}_1)})>0$ while $Q(\Phi_{{\mathbb S}_{\eta}\circ {\mathbb S}_1})=0$).
• Figure 3: Panel a): Scattering center with $2k$ input edges ($\{ e^{({\rm in})}_{L_1}, e^{({\rm in})}_{L_2}, \cdots e^{({\rm in})}_{L_k} \}$ on the left, and $\{ e^{({\rm in})}_{R_1}, e^{({\rm in})}_{R_2}, \cdots e^{({\rm in})}_{R_k} \}$ on the right) and $2k$ output edges ($\{ e^{({\rm out})}_{L_1}, e^{({\rm out})}_{L_2}, \cdots e^{({\rm out})}_{L_k} \}$ on the left, and $\{ e^{({\rm out})}_{R_1}, e^{({\rm out})}_{R_2}, \cdots e^{({\rm out})}_{R_k} \}$ on the right). Panel b): Example of star product composition of two scattering matrices ${\mathbb S}_1$ and ${\mathbb S}_2$ with the same number of dangling edges ($2k$ incoming and $2k$ outgoing) and $2k$ intra-node connections, $k$ from ${\mathbb S}_1$ to ${\mathbb S}_2$, and $k$ from ${\mathbb S}_2$ to ${\mathbb S}_1$.
• Figure 4: Example of a two-vertex quantum graph with non-homogeneous edge distributions: here the number of edges (black arrows) from ${\mathbb S}_1$ and ${\mathbb S}_2$ is larger than the number of edges from ${\mathbb S}_2$ to ${\mathbb S}_1$. The graph can be transformed into a homogeneous graph similar to the one in panel a) of Fig. \ref{['figureSM2']} by adding extra fictitious edges (red arrows). In the present example, the value of $\bar{k}$ is given by $k_{1,R}^{(\rm out)}=k_{2,L}^{(\rm in)}$ so no fictitious edges that leaves ${\mathbb S}_1$ and enters into ${\mathbb S}_2$ are added. Furthermore according to \ref{['SMdefkbar']} we have $\bar{k}_{i,L/R}^{({\rm in/out})}=\bar{k}$ for all $i=1,2$.
• Figure 5: Example of a two-vertex quantum graph with full asymmetric concatenation: in this case there are no loops between the two vertices and the star product reduces to direct concatenation, see Eq. \ref{['SMEQiv']}. Notice also that here $\bar{k}=k$. Red arrows represent fictitious edges, black arrows the physical ones.