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arXiv:2610.26204

Wave-Particle Complementarity in Quantum Walk from a Quantum Mechanical Perspective

Norio Konno, Shigeki Matsutani, Etsuo Segawa, Tomoyuki Terada

math-phquant-phSubmitted Oct 19, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

A quantum walk is a mathematical model and a quantum version of a random walk. For the past two decades, it has been studied from mathematical and physical viewpoints. It has many desirable properties. This paper reevaluates quantum walk from a quantum mechanical perspective, as Feynman essentially proposed it as a toy model in his path integral method. Then we demonstrate that the quantum walk model exhibits both particle-like and wave-like properties by showing reflection and partial reflection as a particle and interference as a wave. In particular, for an initial state supported at two sites $\pm\ell$, we explicitly derive the weak limit distribution of the Hadamard walk and show that the Konno distribution is modulated by the interference factor $2\cos^2(\ell k(x))$. The resulting interference profile is reminiscent of far-field (Fraunhofer-type) diffraction.

Comments
27 pages, 14 figures

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A quantum walk is a mathematical model and a quantum version of a random walk. For the past two decades, it has been studied from mathematical and physical viewpoints. It has many desirable properties. This paper reevaluates quantum walk from a quantum mechanical perspective, as Feynman essentially proposed it as a toy model in his path integral method. Then we demonstrate that the quantum walk model exhibits both particle-like and wave-like properties by showing reflection and partial reflection as a particle and interference as a wave. In particular, for an initial state supported at two sites ±ℓ\pm\ell±ℓ, we explicitly derive the weak limit distribution of the Hadamard walk and show that the Konno distribution is modulated by the interference factor 2cos⁡2(ℓk(x))2\cos^2(\ell k(x))2cos2(ℓk(x)). The resulting interference profile is reminiscent of far-field (Fraunhofer-type) diffraction.
We show that quantum walks interpolate between a coherent ``wave walk'' and a random walk depending on how strongly the walker's coin state is measured; i.e., the quantum walk exhibits the quintessentially quantum property of complementarity, which is manifested as a tradeoff between knowledge of which path the walker takes vs the sharpness of the interference pattern. A physical implementation of a quantum walk (the quantum quincunx) should thus have an identifiable walker and the capacity to demonstrate the interpolation between wave walk and random walk depending on the strength of measurement.
Interference and diffraction phenomena are the basic properties shown by waves. De-Broglie concept of matter waves brought in the extended version of wave properties of material particles. Wave properties of quantum particles are important to understand various phenomena in physics. Matter waves are a core part of quantum theory. Now wave-particle duality is exploited for many applications such as imaging microscopic particles using electron microscopes etc. Here we deduce the interference and diffraction of quantum particles on the basis of the propagation amplitude according to the path integral formulation of quantum mechanics. The intensity distribution due to interference and diffraction of quantum particles are compared with those obtained from the formal methods. The results agree with the traditional approach assuming a deBroglie wave associated with particles.
Abstract The phenomenon of quantum erasure has shed light on the nature of wave-particle duality and quantum complementarity. A number of quantum erasers have been realized using the far-field diffraction of photons from a Young double-slit apparatus. By marking the path of a photon using an additional degree of freedom, the usual Young interference pattern is destroyed. An appropriate measurement of the system marking the photon’s path allows one to recover the interference pattern. Here quantum erasure is considered in the context of near-field diffraction. To observe interference in the near-field requires the use of two periodic wave functions, so that the usual ‘which way’ marker then becomes a ‘which-wave function’ marker. We determine the propagation distances for which quantum erasure, or more generally the observation of interference between the two periodic wave functions, can be observed. The meaning of wave and particle-like properties in this scenario is discussed. These results could lead to quantum eraser experiments with material particles, for which interference effects are more readily observed in the near-field rather than the far-field.
We investigate the two-component quantum walk in one-dimensional lattice. We show that the inter-component interaction strength together with the hopping imbalance between the components exhibit distinct features in the quantum walk for different initial states. When the walkers are initially on the same site, both the slow and fast particles perform independent particle quantum walks when the interaction between them is weak. However, stronger inter-particle interactions result in quantum walks by the repulsively bound pair formed between the two particles. For different initial states when the walkers are on different sites initially, the quantum walk performed by the slow particle is almost independent of that of the fast particle, which exhibits reflected and transmitted components across the particle with large hopping strength for weak interactions. Beyond a critical value of the interaction strength, the wave function of the fast particle ceases to penetrate through the slow particle signalling a spatial phase separation. However, when the two particles are initially at the two opposite edges of the lattice, then the interaction facilitates the complete reflection of both of them from each other. We analyze the above mentioned features by examining various physical quantities such as the on-site density evolution, two-particle correlation functions and transmission coefficients.
With the help of quantum-scattering-theory methods and the approximation of the stationary phase, we propose a one-dimensional quantum-random-walk (QRW) model, which describes for both tunneling and scattering above the potential, the coherent diffusion of independent particles described by wave packets in a periodic one-dimensional lattice. The QRW model describes for each lattice cell the time evolution of modulating amplitudes of two opposite-moving wave packets as they are scattered by periodic potential barriers. Since the QRW model is a coherent process, interference contributions in the probabilities bring about strong departures from classical results. For many identical free particles we obtain the theoretical and graphical Bose and Fermi two-body QRW probability distribution. The result is generalized to N identical free particles and we obtain the N-body Bose and Fermi QRW probability distribution.