Energy-space quantum walks: Thermalization without state convergence

Abstract: We introduce energy-space quantum walks as a minimal framework to investigate equilibration, thermalization, and irreversibility from an effective-dynamics perspective. By mapping the configuration space of a walk onto a ladder of energy eigenlevels, we reinterpret thermalization as transport in energy space, independently of microscopic system--bath details. At the classical level, the resulting birth--death--lazy dynamics leads to equilibration of the energy distribution and, under suitable conditions, to a Gibbs stationary state. We then embed this dynamics into a unitary, collision-assisted model in which coherence is controlled by a single parameter. A central result is a structural decoupling between population dynamics and coherence generation: while the populations evolve according to the classical process and relax to the Gibbs distribution, the full quantum state exhibits a persistent coherence-induced deviation from the thermal manifold. This establishes a minimal scenario of thermalization without state convergence, where equilibration occurs at the level of populations but not at the level of the full density operator. We quantify this effect using the thermal distance to the Gibbs state and derive perturbative bounds that relate the long-time deviation to classical transport properties. Our results show that coherence acts as a controllable and quantitatively bounded source of nonthermal behavior, providing a clear separation between classical equilibration and genuinely quantum corrections.

• The paper demonstrates that quantum walks in energy space can decouple population thermalization from full state convergence, establishing a framework for nonthermal behavior.
• It employs classical birth-death-lazy stochastic dynamics integrated with collision-assisted unitary maps to achieve Gibbsian equilibrium alongside persistent coherence.
• Perturbative analysis and numerical simulations quantify coherence effects, offering insights for quantum simulation and thermodynamic protocols.

Energy-Space Quantum Walks: Structural Decoupling of Thermalization and State Convergence

Introduction and Motivation

The paper "Energy-space quantum walks: Thermalization without state convergence" (2605.15339) establishes energy-space quantum walks as an effective minimal framework for probing how equilibration, thermalization, and irreversibility emerge independent of microscopic system-bath models. The central thesis is that thermalization can be reframed as a transport phenomenon in an energy-space ladder, with a structural decoupling between population dynamics (energy distribution) and quantum coherence (off-diagonal elements in the density matrix). The work systematically demonstrates that classical birth-death-lazy stochastic dynamics on this ladder leads to Gibbs equilibrium distributions, while embedding this process in a collision-assisted quantum walk enables persistent coherence-induced deviation from the thermal manifold, even after population equilibration. This separation permits thermalization of populations without convergence of the full quantum state, quantifying the role of coherence as a controllable contribution to nonthermal stationary behavior.

Classical Energy-Space Dynamics and Gibbs Equilibration

The classical regime is formulated as stochastic energy-space transport, with transition probabilities for absorption ($p_+$), emission ($p_-$), and lazy steps ($p_0$). The effective dynamics preserve diagonality and is encapsulated in a CPTP map $\Phi$, leading to birth-death-lazy population evolution. Nonunitality ($p_+\neq p_-$) induces bias and irreversibility, with the stationary solution determined by current conservation:

$$\wp_\infty(n+1) = \left(\frac{p_+}{p_-}\right) \wp_\infty(n)$$

which yields a geometric stationary distribution. For equally spaced spectra ($E_n = nE$), this is re-expressed as a canonical Gibbs distribution:

$$\wp_\infty(n) \propto e^{-\beta E_n}, \quad \frac{p_+}{p_-} = e^{-\beta E}$$

with the effective temperature set by the ladder statistics. Stationary populations encode irreversibility and the emergence of Gibbs equilibrium, independent of explicit thermal bath modeling.

Figure 1: Equilibration and thermalization in the incoherent energy-space dynamics for equally spaced energies, illustrating exponential relaxation of populations toward the stationary (Gibbs) distribution and the effect of bias on relaxation times.

Temporal averaging yields robust equilibration diagnostics, while trace distance and thermal distance metrics quantify convergence rates and distinguish genuine thermalization from mere stationary population equilibration. With transition rates satisfying local detailed balance ($p_+(n)/p_-(n+1) = e^{-\beta(E_{n+1}-E_n)}$), the stationary state is Gibbsian for arbitrary spectra; violation leads to nonthermal stationary distributions, confirmed numerically.

Figure 2: Thermal distance $d_{\mathrm{th}}(t)$ (red) and dephased counterpart (blue) demonstrating that only genuine Gibbsian stationary distributions attain vanishing thermal distance; level-dependent rates induce persistent nonthermal deviation.

Collision-Assisted Quantum Walk and Structural Decoupling

Embedding the classical process into a unitary, collision-assisted quantum walk is achieved via coupling the energy ladder to three-level ancillas (channels for upward, downward, lazy transitions). The quantum map $p_-$0 interpolates between incoherent classical ($p_-$1) and coherent ($p_-$2) regimes. The key result is structural decoupling: for initial states diagonal in the energy basis, population evolution is entirely classical and independent of $p_-$3. Coherence is generated locally by boundary transitions (between ground and first excited states) and subsequently transported, but has no back-action on populations.

Explicitly, the reduced dynamics is:

$p_-$4

where $p_-$5 is the classical birth-death-lazy map, and the $p_-$6-term injects coherence off-diagonally.

Equilibration at the population level proceeds as in the classical model, achieving Gibbs statistics when detailed balance holds. However, the full quantum state, driven by persistent, locally-generated coherence, does not converge to the Gibbs state unless $p_-$7. The structural blockwise decoupling ensures the diagonal and off-diagonal sectors evolve independently.

Quantitative Analysis: Perturbative Bounds and Numerical Results

A perturbative expansion in $p_-$8 quantitatively bounds the coherence-induced deviation:

$p_-$9

where $p_0$0 is the purely classical thermal distance, and the cumulative boundary occupation governs the quantum correction. As $p_0$1, $p_0$2, so the asymptotic deviation $p_0$3 is $p_0$4 for small $p_0$5.

Numerical simulations confirm:

• Population relaxation and Gibbsian equilibrium are unaffected by coherence parameter $p_0$6.
• The thermal distance for the full state saturates at a finite value for $p_0$7, verifying nonconvergence.
• For small $p_0$8, $p_0$9 scales linearly with $\Phi$0, matching perturbative prediction. At larger $\Phi$1, nonperturbative effects emerge, but bounds remain tight.

  Figure 3: Time evolution of the thermal distance $\Phi$2 across coherence parameter $\Phi$3 values, with classical ($\Phi$4) relaxation matching Gibbsian convergence and coherent cases exhibiting persistent deviation.

  

  Figure 4: Asymptotic thermal distance $\Phi$5 scaling linearly with $\Phi$6, confirming perturbative analysis and analytical bounds; deviation from linearity signals nonperturbative regime.

Implications, Extensions, and Outlook

The structural decoupling identified has substantive implications:

• Practical: Thermal equilibrium of populations can be achieved independent of quantum state convergence, with coherence serving as an explicit nonthermal correction. This is critical for quantum simulation and algorithmic frameworks where population statistics suffice for thermodynamic predictions but coherence persists.
• Theoretical: The minimal framework exposes the distinction between classical transport-induced equilibration (energy-space mixing) and genuine quantum thermalization (full state convergence), emphasizing the centrality of structure and locality in decoherence and irreversibility.
• Future Directions: Possible extensions include continuous-time mapping, energy-conserving embeddings with explicit thermodynamic interpretation, and generalization to interacting graphs or ladders. The phenomena detailed may extend naturally to coin-based quantum walks and other discrete-time models, offering experimental accessibility.

Conclusion

This work establishes energy-space quantum walks as a rigorous and transparent paradigm to probe thermalization mechanisms in quantum systems. It demonstrates—both analytically and numerically—that classical population dynamics and quantum coherence can be categorically decoupled in the dynamics, yielding thermalization of populations without full state convergence. The persistence of nonequilibrium coherence is controllable and quantitatively bounded, illuminating the landscape wherein classical transport and quantum corrections coexist. This result invites a reevaluation of thermalization protocols, supports the modular understanding of quantum irreversibility, and opens fertile avenues for quantum thermodynamics and transport in synthetic dimensions.