Zeno subspace in quantum-walk dynamics

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We investigate the discrete-time quantum walk evolution under the influence of the periodic measurements in position subspace. The undisturbed survival probability of the particle at the position subspace, P(0, t) is compared with the survival probability after frequent (n) measurements at interval τ= t/n, P(0, τ)ⁿ. We show that P(0, τ)ⁿ > P(0, t) leading to the quantum Zeno effect in the position subspace when a parameter θ in the quantum coin operations and frequency of measurements is greater than the critical value, θ> θc and n>nc. This Zeno effect in the subspace preserves the dynamics in coin Hilbert space of the walk dynamics and has a potential to play a significant role in quantum tasks such as, preserving the quantum state of the particle at any particular position and to understand the Zeno dynamics in a multidimensional system which is highly transient in nature.