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Quantum walks under superposition of causal order

Prateek Chawla, Shrikant Utagi, C. M. Chandrashekar

Abstract

We set the criteria under which superposition of causal order can be incorporated in to quantum walks. In particular, we show that only periodic quantum walks or those with at least one disorder exhibit Superposition of causal order under the action of `quantum switch'. We exemplify our results with a simple example of two-period discrete-time quantum walks. In particular, we observe that periodic quantum walks exhibit causal asymmetry pertaining to the dynamics of the reduced coin state: the dynamics are more non-Markovian for one temporal order than the other. We also note that the non-Markovianity of the reduced coin state due to indefiniteness in causal order tends to match the dynamics of a particular temporal order of the coin state. We substantiate our results with numerical simulations.

Quantum walks under superposition of causal order

Abstract

We set the criteria under which superposition of causal order can be incorporated in to quantum walks. In particular, we show that only periodic quantum walks or those with at least one disorder exhibit Superposition of causal order under the action of `quantum switch'. We exemplify our results with a simple example of two-period discrete-time quantum walks. In particular, we observe that periodic quantum walks exhibit causal asymmetry pertaining to the dynamics of the reduced coin state: the dynamics are more non-Markovian for one temporal order than the other. We also note that the non-Markovianity of the reduced coin state due to indefiniteness in causal order tends to match the dynamics of a particular temporal order of the coin state. We substantiate our results with numerical simulations.
Paper Structure (3 sections, 3 theorems, 20 equations, 3 figures)

This paper contains 3 sections, 3 theorems, 20 equations, 3 figures.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

A quantum walk exhibits nontrivial dynamics under superposition of its forward and reverse causal orders iff at least two of the DTQW steps do not commute. (Here, 'trivial dynamics' implies that the effective dynamics may be written as a quantum walk of the same form as the original DTQW under consi

Figures (3)

  • Figure 1: The plot of the spread of a quantum walker in definite (blue dashed line), reversed (green dashed line), and superposed (orange solid line) causal order of its steps. The walker executes a 2-period discrete-time quantum walk with a localized initial state $\left( \frac{\ket{0} + \ket{1}}{\sqrt{2}} \right) \otimes \ket{x=0}$, for $100$ steps, as defined in Eq. \ref{['eq:QW2pd']}. The two coin angles $\left( \theta_1, \theta_2\right)$ are chosen to be $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively. It is seen that the use of a quantum switch has no effect on the spreading of the quantum walker in its position space. This is true for multi-period DTQWs in general.
  • Figure 2: BLP measure for the cases with definite and equally superposed causal orders of two-period quantum walks. Each walk was executed for $N=100$ steps, with the coin parameters $(\theta_1, \theta_2)$ set to $\left( \frac{\pi}{6}, \frac{\pi}{4} \right)$ and $\left( \frac{\pi}{4}, \frac{\pi}{6} \right)$ in (a) and (b), respectively.
  • Figure 3: Figure illustrating the normalized BLP measure for a DTQW with different periods, under two definite causal orders, and an equal temporal superposition. The walker has $\theta_1 = \frac{\pi}{6}$, and $\theta_i=\frac{\pi}{4}$ for $i > 1$, and the periodic quantum walk is executed for $N=200$ steps. It is seen that the non-Markovianity of the reduced dynamics of the coin state is bounded by the non-Markovianity of the two causal orders.

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • Theorem 2