Splitting Probability in Monitored Quantum Dynamics

Abstract

We consider a continuous-time quantum walk stroboscopically subjected to projective measurements on two distinct targets with sampling time period τ. We construct a general renewal framework for the first-detection amplitude at each target, from which we obtain the associated splitting probabilities. For Hamiltonians with reflection symmetry, we further show that each dual-target first-detection amplitude can be viewed as an interference effect between a pair of single-target first-detection amplitudes. This formulation yields exact expressions for the splitting probabilities and gives special sampling times at which they undergo pointwise discontinuous jumps due to a form of ergodicity breaking associated with dark states. As an illustrative example, we apply the framework to a one-dimensional tight-binding lattice chain with nearest-neighbor hopping, and study the splitting probabilities as functions of the initial position and sampling time. Below a critical sampling time, 0<τ< τ_c where τ_c = 2 π/ ΔE with ΔE being the energy bandwidth, the splitting probabilities for large system sizes become equal to 1/2, independent of the initial position, as long as this position does not coincide with target sites. Above the critical sampling time, τ≥ τ_c, the splitting probabilities deviate from 1/2 and develop oscillations as a function of initial position. The wavenumber associated with these oscillations is Q=2sin⁻¹(rτ_c/τ), where r is a positive integer. We also explore the effects of relative phase and spatial separation between different components of a coherent initial state on the splitting probabilities. Finally, we study how a finite number of measurements modifies the splitting probabilities and broadens the resonances at special sampling times.

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BibTeX RIS

BibTeX

@misc{singh2026splitting,
  title = {{Splitting Probability in Monitored Quantum Dynamics}},
  author = {Prashant Singh and Sabine Tornow and David A Kessler and Eli Barkai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29547},
  eprint = {2610.29547},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29547},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Splitting Probability in Monitored Quantum Dynamics
AU  - Prashant Singh
AU  - Sabine Tornow
AU  - David A Kessler
AU  - Eli Barkai
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29547
UR  - https://arxiv.org/abs/2610.29547
AB  - We consider a continuous-time quantum walk stroboscopically subjected to projective measurements on two distinct targets with sampling time period τ. We construct a general renewal framework for the first-detection amplitude at each target, from which we obtain the associated splitting probabilities. For Hamiltonians with reflection symmetry, we further show that each dual-target first-detection amplitude can be viewed as an interference effect between a pair of single-target first-detection amplitudes. This formulation yields exact expressions for the splitting probabilities and gives special sampling times at which they undergo pointwise discontinuous jumps due to a form of ergodicity breaking associated with dark states. As an illustrative example, we apply the framework to a one-dimensional tight-binding lattice chain with nearest-neighbor hopping, and study the splitting probabilities as functions of the initial position and sampling time. Below a critical sampling time, 0<τ< τ_c where τ_c = 2 π/ ΔE with ΔE being the energy bandwidth, the splitting probabilities for large system sizes become equal to 1/2, independent of the initial position, as long as this position does not coincide with target sites. Above the critical sampling time, τ≥ τ_c, the splitting probabilities deviate from 1/2 and develop oscillations as a function of initial position. The wavenumber associated with these oscillations is Q=2sin^−1(rτ_c/τ), where r is a positive integer. We also explore the effects of relative phase and spatial separation between different components of a coherent initial state on the splitting probabilities. Finally, we study how a finite number of measurements modifies the splitting probabilities and broadens the resonances at special sampling times.
ER  -

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