Postselection induced localization and coherence in quantum walks on heterogeneous networks

Abstract

Postselection of quantum trajectories is known effectively introduce nonlinearity into dynamics of open quantum systems. We study the effect of such non-linearity in continuous-time quantum walks (CTQWs) on networks with homogeneous and heterogeneous degree distributions. Using the recently proposed nonlinear Lindblad master equation (NLME), we investigate the dynamics under two decoherence mechanisms: Haken-Strobl and quantum stochastic walk (QSW). Our analysis reveals a striking dichotomy: under Haken-Strobl decoherence the nonlinear contributions precisely cancel, yielding a uniform steady state independent of postselection details. In stark contrast, QSW decoherence permits postselection to break dynamical balance on heterogeneous networks, inducing robust localization preferentially at low-degree (peripheral) nodes. Remarkably, this localized state maintains finite quantum coherence. Extending our results to many-body spin systems, we demonstrate that degree heterogeneity similarly stabilizes localization of spin-up excitations in spin-down backgrounds, enhancing entanglement preservation. These findings establish degree heterogeneity and postselection as joint control parameters for engineering quantum transport and localization in dissipative dynamics.

Figures (10)

• Figure 1: Schematic illustration of the postselected of CTQW on a line graph (a) QSW decoherence arises from sysetm environment coupling (strength $p$) through jump operators that cause on-site dephasing ($P_{kk}$) as well as hopping between nodes ($P_{kj}$), both of which are monitored. The graph is color-coded to highlight the postselected steady state: low-degree edge nodes are shown in red (indicating localization), while bulk nodes are blue. (b) Haken-Strobl decoherence (strength $\gamma$) which cause only onsite dephasing ($P_{k}$), which is registered by the detectors. In both cases, the decoherence-induced jumps are depicted by yellow wavy lines, representing the transitions monitored with efficiency $\eta$.
• Figure 2: Schematic representation of the three network topologies studied. (a) The cylinder topology, having two distinct boundaries composed of degree-3 edge nodes (red). (b) The Möbius strip, which possesses a single continuous boundary of degree-3 edge nodes (red). (c) The torus, a regular graph with no boundaries, where all nodes are part of the bulk and have a uniform degree of 4 (blue). Majority of the nodes shown in blue colors has the same degree 4.
• Figure 3: Steady-state probability distributions and $\ell$-norm of coherence for CTQW under QSW decoherence. (a) On the Cylinder and (b) the Möbius strip, postselection $(\eta>0)$ induces strong localization on the low-degree edge nodes (degree 3). The elevated probability on the first layer of bulk nodes is due to their connection to these highly populated edge nodes. (c) On the edgeless Torus, where all nodes are in the bulk (degree 4), the distribution remains uniform, showing that degree heterogeneity is essential for the localization. In all cases, the initial state was localized at a bulk node and decoherence strength, $p=0.5$.
• Figure 4: Numerical verification of the derived steady-state constraint (Eq. \ref{['eq:25']}). The steady-state probabilities $\rho^{ss}_{kk}$ obtained from time evolution (markers) are compared against the RHS of analytically derived Eq. \ref{['eq:25']} (dashed lines) for the cylinder graph. The plot show that the simulated steady state strictly obeys the derived analytical condition for all tested values of $\eta$ (p=0.5).
• Figure 5: Steady-state probability distributions ($\rho^{ss}_{kk}$) of a CTQW across network nodes under varying decoherence and postselection. The plots illustrate the distribution of steady-state probabilities for a cylinder topology with initial localization is towards one edge. (a)-(c) Steady-state probability distributions for varying postselection efficiency ($\eta$) at fixed strong decoherence ($p = 0.9$) (d)-(f) Distributions for varying decoherence strength ($p$) at a fixed strong postselection ($\eta=0.8$). Under sufficiently strong decoherence and postselection, the system breaks spatial symmetry and preferentially populates the edge closest to the initial excitation as indicated by peak probabilities.