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Standard Quantum Limit and Heisenberg Limit in Function Estimation.

Naoto Kura1, Masahito Ueda1,2,3

  • 1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyou-ku, Tokyo 113-0033, Japan.

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Summary
This summary is machine-generated.

We established fundamental error bounds for quantum metrology function estimation. These bounds are theoretically optimal, even with quantum measurements, aligning with the Nyquist-Shannon sampling theorem.

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Area of Science:

  • Quantum Metrology
  • Information Theory

Background:

  • Parameter estimation is well-established in quantum metrology.
  • Function estimation presents unique conceptual and mathematical challenges.
  • Despite difficulties, function estimation holds significant utility.

Purpose of the Study:

  • To establish fundamental error bounds for function estimation in quantum metrology.
  • To analyze error bounds for spatially varying phase operators with varying function smoothness.
  • To investigate the impact of interparticle entanglement on these error bounds.

Main Methods:

  • Theoretical analysis of error bounds for function estimation.
  • Consideration of both absence and presence of interparticle entanglement.
  • Evaluation of probe states, including position-localized and wave-number-localized states.

Main Results:

  • Identified fundamental error bounds for function estimation.
  • Error bounds correspond to the standard quantum limit (no entanglement) and Heisenberg limit (with entanglement).
  • Demonstrated that these bounds are achievable with specific probe states and are theoretically optimal for any probe state.

Conclusions:

  • Quantum metrology for function estimation is fundamentally limited by the Nyquist-Shannon sampling theorem.
  • Optimal error bounds are achievable regardless of probe state type.
  • Quantum measurement does not circumvent classical information limits in function estimation.