Quantum walks: The mean first detected transition time

Liu, Quancheng; Yin, R.; Ziegler, K.; Barkai, Eli

doi:10.1103/physrevresearch.2.033113

Cited by 26 publications

(24 citation statements)

References 41 publications

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“…We now provide general insights from the charge theory, which are used to find the largest eigenvalue of Ŝ in generic situations. The technique here presented follows, and in some cases extends the ideas in [55,67,68], that were developed in the context of the first detection problem. We consider cases where the dark space is empty, simply because a system with a non-empty dark space can be treated exactly with the tools given in Eq.…”

Section: Null Measurements Insights From the Charge Theorymentioning

confidence: 96%

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Driving quantum systems with repeated conditional measurements

Liu,

Ziegler,

Kessler

et al. 2021

Preprint

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We investigate the effect of conditional null measurements on a quantum system and find a rich variety of behaviors. Specifically, quantum dynamics with a time independent H in a finite dimensional Hilbert space are considered with repeated strong null measurements of a specified state. We discuss four generic behaviors that emerge in these monitored systems. The first arises in systems without symmetry, along with their associated degeneracies in the energy spectrum, and hence in the absence of dark states as well. In this case, a unique final state can be found which is determined by the largest eigenvalue of the survival operator, the non-unitary operator encoding both the unitary evolution between measurements and the measurement itself. For a three-level system, this is similar to the well known shelving effect. Secondly, for systems with builtin symmetry and correspondingly a degenerate energy spectrum, the null measurements dynamically select the degenerate energy levels, while the non-degenerate levels are effectively wiped out. Thirdly, in the absence of dark states, and for specific choices of parameters, two or more eigenvalues of the survival operator match in magnitude, and this leads to an oscillatory behavior controlled by the measurement rate and not solely by the energy levels. Finally, when the control parameters are tuned, such that the eigenvalues of the survival operator all coalesce to zero, one has exceptional points that corresponds to situations that violate the null measurement condition, making the conditional measurement process impossible.

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Section: Null Measurements Insights From the Charge Theorymentioning

confidence: 96%

“…The question remains, how to find the eigenvalues satisfying 0<|ξ ι |<1? Here we exploit a beautiful mapping of the problem to a classical charge theory, following the work of Grünbaum et al [55,67,68]. By defining 9), ( 13) and ( 15), we get…”

Section: Classical Charge Picture Mapping and Biorthogonal Eigenstatesmentioning

confidence: 99%

Driving quantum systems with repeated conditional measurements

Liu,

Ziegler,

Kessler

et al. 2021

Preprint

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“…Although the only effect of τ on P det is possible discontinuous changes at resonant τ s, the detection period has a more profound effect on other quantities such as the mean detection time [18,70]. It is an important parameter of the stroboscopic detection protocol.…”

Section: A With Conditional Probabilitymentioning

confidence: 99%

Dark states of quantum search cause imperfect detection

Thiel

Mualem

Meidan

et al. 2020

Phys. Rev. Research

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We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected. This is a quantum version of the first-passage problem. We focus on the total probability P det that the particle is eventually detected in some target state, for example, on a node r d on a graph, after an arbitrary number of detection attempts. Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P det in terms of the energy eigenstates which is generically τ independent. We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection. Specifically, we give a simple upper bound for P det that is controlled by the number of equivalent (with respect to the detection) states in the system. We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.

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“…However, destructive interference may divide the Hilbert space into two components called dark and bright, and this yields an inefficient search and an effect superficially similar to classical non-ergodicity,

. More specifically, an observer performs repeated strong measurements, on a node other than the starting node, in an attempt to detect the particle [ 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ]. In the time intervals between the measurements the dynamics is unitary.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Relation between Detection Probability and Energy Fluctuations

Thiel

Mualem

Kessler

et al. 2021

Entropy

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A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it fully explores space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local measurements, made on a target state, the final detection of the system is not guaranteed since the Hilbert space is split into a bright subspace and an orthogonal dark one. Using this we find an uncertainty relation for the deviations of the detection probability from its classical counterpart, in terms of the energy fluctuations.

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Quantum walks: The mean first detected transition time

Cited by 26 publications

References 41 publications

Driving quantum systems with repeated conditional measurements

Driving quantum systems with repeated conditional measurements

Dark states of quantum search cause imperfect detection

Uncertainty Relation between Detection Probability and Energy Fluctuations

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