Quantum total detection probability from repeated measurements II. Exploiting symmetry

Thiel, Felix; Mualem, Itay; Kessler, David A.; Barkai, Eli

doi:10.48550/arxiv.1909.02114

Cited by 5 publications

(10 citation statements)

References 52 publications

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“…The TBM (and related systems) when subject to projective measurements has in recent years been extensively studied in the context of detection problem corresponding to a quantum particle evolving under the dynamics to arrive at a chosen set of sites [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. In the context of the present work, we briefly summarize the contribution of [59] that also considers projective measurements at random times.…”

Section: The Model Subject To Projective Measurements At Random Timesmentioning

confidence: 99%

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

2022

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In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. The formalism is very general in that it applies to any quantum system, to any form of non-unitary interaction, and to any probability distribution for the random times. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, e.g. in modelling nano wires. Considering the case of one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, owing to spontaneous quantum fluctuations tunnels between the nearest-neighbour sites. We consider two representative forms of interactions, one that implements a stochastic reset of quantum dynamics in which the density operator is at random times reset to its initial form, and one in which projective measurements are performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement (MSD) of the particle about its initial location. This stands in stark contrast to the behavior in the absence of interactions, when the particle has an unbounded growth of the MSD in time, with no signatures of localization. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which it does not require us to perform measurements at exactly regular intervals that are hallmarks of such scenarios.

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Section: The Model Subject To Projective Measurements At Random Timesmentioning

confidence: 99%

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

2022

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“…Another difference between the return and the transition problem is that, for instance, the total detection probability P det for the return is always unity, while the transition probability to another state is sensitive, e.g. to the geometric symmetry of the underlying graph [40]. The qualitative difference between return and transition properties originates in the fact that the return properties are based on the amplitude u n alone, whereas the transition properties depend on both ampli-tudes u n and v n .…”

Section: Discussionmentioning

confidence: 99%

“…Unlike classical random walks on finite graphs, here one can find the total detection probability less than unity. The quantum particle will go to some "dark states", where they will never be detected [36,37,40].…”

Section: Pseudo Degeneracymentioning

confidence: 99%

Quantum walks: the first detected transition time

Liu,

Yin,

Ziegler

et al. 2020

Preprint

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We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate 1/τ . A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state |ψin of the walker is orthogonal to the detected state |ψ d . We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value, by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameter of the model, which exhibits a blow-up of the mean transition time, we get simple expressions for the mean transition time. Using previous results on the fluctuations of the return time, corresponding to |ψin = |ψ d , we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation.

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“…The TBM and related systems when subject to projective measurements with time evolution following scheme 2 of dynamics have in recent years been studied to address the issue of when does a quantum particle evolving under the dynamics arrive at a chosen set of sites [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. A crucial difference is that, barring [27], these contributions almost exclusively implement measurements at regular intervals of times, unlike the scheme considered in this work in which sequence of measurements at random times is implemented.…”

Section: J Stat Mech (2022) 033212mentioning

confidence: 99%

Quantum random walk and tight-binding model subject to projective measurements at random times

Das¹,

Gupta²

2022

J. Stat. Mech.

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What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: how does the survival probability S m , namely, the probability that an initial state survives even after m number of measurements, behave as a function of m? We address these issues in the context of two paradigmatic quantum systems, one, the quantum random walk evolving in discrete time, and the other, the tight-binding model evolving in continuous time, with both defined on a one-dimensional periodic lattice with a finite number of sites N. For these two models, we present several numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in m for large m. By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability exhibits two characteristic m values, namely, m 1 ⋆ ( N ) ∼ N and m 2 ⋆ ( N ) ∼ N δ with δ > 1. These scales are such that (i) for m large and satisfying

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Quantum total detection probability from repeated measurements II. Exploiting symmetry

Cited by 5 publications

References 52 publications

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

Quantum walks: the first detected transition time

Quantum random walk and tight-binding model subject to projective measurements at random times

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