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Uncertainty Relation between Detection Probability and Energy Fluctuations.

Felix Thiel1, Itay Mualem1, David Kessler1

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Summary
This summary is machine-generated.

Quantum walks may not reach all nodes due to destructive interference, unlike classical random walks. Measurements can split the quantum system, affecting detection probability and leading to an uncertainty relation tied to energy fluctuations.

Keywords:
Zeno subspacesdark and bright statesergodicityquantum search

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Area of Science:

  • Quantum physics
  • Graph theory
  • Quantum computation

Background:

  • Classical random walkers on finite graphs are ergodic, guaranteeing arrival at any node with probability one.
  • Quantum walks can exhibit non-ergodic behavior due to destructive interference, impacting exploration of the graph space.

Purpose of the Study:

  • To investigate the impact of destructive interference and projective measurements on quantum walk search processes.
  • To establish an uncertainty relation for detection probability deviations in quantum walks compared to classical random walks.

Main Methods:

  • Simulating quantum walks on finite graphs.
  • Applying repeated projective local measurements on a target state.
  • Analyzing the splitting of Hilbert space into bright and dark subspaces.
  • Deriving an uncertainty relation based on detection probability and energy fluctuations.

Main Results:

  • Destructive interference can lead to effectively non-ergodic quantum walks.
  • Projective measurements can result in the system being localized in a dark subspace, preventing detection.
  • An uncertainty relation is found connecting deviations in detection probability to energy fluctuations.

Conclusions:

  • Quantum walks exhibit fundamentally different search dynamics compared to classical random walks.
  • The interplay between interference, measurement, and Hilbert space structure governs quantum search efficiency.
  • Energy fluctuations play a crucial role in the uncertainty of detection probabilities in measured quantum walks.