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Conference on Lasers and Electro-Optics

Probing the measurement process in DTQW via recurrence

Thomas Nitsche, Regina Kruse, Linda Sansoni, M. Štefaňák, T. Kiss, Igor Jex, Sonja Barkhofen, Christine Silberhorn

Published Jan 1, 2017• 0 citations
Publisher

Abstract

The measurement process plays a crucial role in quantum mechanics as it interrupts the unitary evolution of a state. The consequences are apparent when investigating the return probability (Polya-number) of a particle in a Hadamard walk on the line [1, 2]: Depending on whether the evolution is restarted or continued after the measurement, either a transient or a recurrent evolution can be observed. We investigate both cases experimentally for the first time and monitor the evolution of the walker over 39 steps, revealing the fundamental differences of the two cases as predicted by theory: In the restart-regime, the Polya-number will gradually approach 1. In contrast, the continue-regime yields an asymptotic value of 2/π.

DOI
10.1364/cleo_at.2017.jw2a.27

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The measurement process plays a crucial role in quantum mechanics as it interrupts the unitary evolution of a state. The consequences are apparent when investigating the return probability (Polya-number) of a particle in a Hadamard walk on the line [1, 2]: Depending on whether the evolution is restarted or continued after the measurement, either a transient or a recurrent evolution can be observed. We investigate both cases experimentally for the first time and monitor the evolution of the walker over 39 steps, revealing the fundamental differences of the two cases as predicted by theory: In the restart-regime, the Polya-number will gradually approach 1. In contrast, the continue-regime yields an asymptotic value of 2/π.
The measurement process plays a crucial role in quantum mechanics as it interrupts the unitary evolution of a state. The consequences are apparent when investigating the return probability (Polya-number) of a particle in a Hadamard walk on the line [1,2]: Depending on whether the evolution is restarted or continued after the measurement, either a transient or a recurrent evolution can be observed. We investigate both cases experimentally for the first time by introducing sinks in a time-multiplexed quantum walk setup using a fast-switching electro-optic modulator (EOM). Monitoring the evolution of the walker over 39 steps reveals the fundamental differences of the two cases as predicted by theory: In the restart-regime, the Polya-number will gradually approach unity. In contrast, the continue-regime yields an asymptotic value of the Polya-number of 2/π, which is reached in good approximation after only four steps and then remains almost constant.
Measurements on a quantum particle unavoidably affect its state, since the otherwise unitary evolution of the system is interrupted by a nonunitary projection operation. To probe measurement-induced effects in the state dynamics using a quantum simulator, the challenge is to implement controlled measurements on a small subspace of the system and continue the evolution from the complementary subspace. A powerful platform for versatile quantum evolution is represented by photonic quantum walks because of their high control over all relevant parameters. However, measurement-induced dynamics in such a platform have not yet been realized. We implement controlled measurements in a discrete-time quantum walk based on time-multiplexing. This is achieved by adding a deterministic outcoupling of the optical signal to include measurements constrained to specific positions resulting in the projection of the walker's state on the remaining ones. With this platform and coherent input light, we experimentally simulate measurement-induced single-particle quantum dynamics. We demonstrate the difference between dynamics with only a single measurement at the final step and those including measurements during the evolution. To this aim, we study recurrence as a figure of merit, that is, the return probability to the walker's starting position, which is measured in the two cases. We track the development of the return probability over 36 time steps and observe the onset of both recurrent and transient evolution as an effect of the different measurement schemes, a signature which only emerges for quantum systems. Our simulation of the observed one-particle conditional quantum dynamics does not require a genuine quantum particle but is demonstrated with coherent light.
The quantum mechanical generalisation of random walks (called Quantum Walks) present us with a broader spectrum of possibilities compared to their classical counterparts. The aim of the presented study is to explore a new portion of this area by incorporating a new step in the process of the Quantum Walk unique to quantum mechanics: the measurement. Our focus lies in the characterising number of the recurrence behaviour of the walk (Polya-number). We observe the effect of the standard projective measurement, a yes-no measurement on the origin and the effect of different measurement schemes (periodic and random) on the definition and the numeric value of the Polya-number.
Recurrence is an interesting property that describes whether the walker will eventually return to the origin on infinite lattices in classical random walks. Since measurement collapse affects the wave function of the walker, in the quantum case, there are two schemes for studying recurrence properties in quantum walks (QWs) depending on how the measurement process is described. One involves restarting after measurements, known as unmonitored measurements, while the other entails monitoring after each step, referred to as monitored measurements. In this paper, we utilize the dependence of recurrence probabilities on the coin parameter to construct biased-coin QWs on a line and investigate the recurrence properties of both schemes. Based on the bulk-optics framework and using a single-photon source, we experimentally demonstrate the distinct recurrence behaviors resulting from the different evolution processes of the walker in the two schemes. This work showcases diverse recurrence properties in single-particle QW systems, thereby enhancing our understanding of measurement-induced recurrence.
The recurrence time is the time a process first returns to its initial state. Using quantum walks on a graph, the recurrence time is defined through the stroboscopic monitoring of the arrival of the particle to a node of the system. When the time interval between repeated measurements is tuned in such a way that the eigenvalues of the unitary become degenerate, the mean recurrence time exhibits resonances. These resonances imply faster mean recurrence times, which were recorded on quantum computers. The resonance broadening is captured by a restart uncertainty relation [Yin et al., Proc. Natl. Acad. Sci. U.S.A. 122, e2402912121 (2025)]. To ensure a comprehensive analysis, we extend our investigation to include the impact of system size on the widened resonances, showing how the connectivity and energy spectrum structure of a system influence the restart uncertainty relation. Breaking the symmetry of the system, for example time-reversal symmetry breaking with a magnetic flux applied to a ring, removes the degeneracy of the eigenvalues of the unitary, hence modifying the mean recurrence time and the widening of the transitions, and this effect is studied in detail. The width of the resonances studied here is related to the finite time resolution of relevant experiments on quantum computers and to the restart paradigm.
Recurrence in the classical random walk is well known and described by the Pólya number. For quantum walks, recurrence is similarly understood in terms of the probability of a localized quantum walker to return to its origin. Under certain circumstances the quantum walker may also return to an arbitrary initial quantum state in a finite number of steps. Quantum state revivals in quantum walks on cycles using coin operators which are constant in time and uniform across the path have been described before but only incompletely. In this paper we find the general conditions for which full-quantum state revival will occur.