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Reference-frame-independent quantum metrology

Satoya Imai, Otfried Gühne, Géza Tóth

Abstract

How can we perform a metrological task if only limited control over a quantum system is given? Here, we present systematic methods for conducting nonlinear quantum metrology in scenarios lacking a common reference frame. Our approach involves preparing multiple copies of quantum systems and then performing local measurements with randomized observables. First, we derive the metrological precision using an error propagation formula based solely on local unitary invariants, which are independent of the chosen basis. Next, we provide analytical expressions for the precision scaling in various examples of nonlinear metrology involving two-body interactions, like the one-axis twisting Hamiltonian. Finally, we analyze our results in the context of local decoherence and discuss its influences on the observed scaling.

Reference-frame-independent quantum metrology

Abstract

How can we perform a metrological task if only limited control over a quantum system is given? Here, we present systematic methods for conducting nonlinear quantum metrology in scenarios lacking a common reference frame. Our approach involves preparing multiple copies of quantum systems and then performing local measurements with randomized observables. First, we derive the metrological precision using an error propagation formula based solely on local unitary invariants, which are independent of the chosen basis. Next, we provide analytical expressions for the precision scaling in various examples of nonlinear metrology involving two-body interactions, like the one-axis twisting Hamiltonian. Finally, we analyze our results in the context of local decoherence and discuss its influences on the observed scaling.

Paper Structure

This paper contains 14 sections, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A no-go theorem
  3. Parameter estimation and error propagation
  4. Main method
  5. Nonlinear Hamiltonian dynamics
  6. Decoherence
  7. Collective randomization
  8. Experimental implications
  9. Conclusion
  10. Acknowledgments
  11. Proof of Observation \ref{['ob:fourcopy']}
  12. Derivation of Observation \ref{['ob:jx2twobodyhamscl']}
  13. Proof of Observation \ref{['ob:collective']}
  14. Exponential scaling

Figures (3)

  • Figure 1: Sketch of the quantum metrology scheme presented in this paper for two copies of an $N$-particle state with randomized measurements. The parameter $\theta$ is encoded in the 1st and 2nd copy via $\Lambda_\theta \otimes \Lambda_\theta$ highlighted by gray color. Subsequently, a randomized measurement $M_2 = \sum_{i=1}^N M_i^{(2)}$ is performed on the two-copy state, where each local observable $M_i^{(2)}$ acts on the 1st and 2nd copy (highlighted by a vertical box). This paper introduces systematic methods for evaluating the precision, denoted as $(\Delta\theta)^2$, in a reference-frame-independent manner.
  • Figure 2: Sensitivity of the metrological gain defined in Eq. (\ref{['eq:gainkcopy']}) to parameter shifts based on Observation \ref{['ob:jx2twobodyhamscl']} in $N=100$, where $p$ denotes the noise parameter in the local depolarizing channel in Eq. (\ref{['eq:locladepola']}).
  • Figure 3: Growth in the metrological gain defined in Eq. (\ref{['eq:gainkcopy']}) for an increasing number of particles based on Observation \ref{['ob:jx2twobodyhamscl']} with a fixed $\theta=1/N$, where $p$ denotes the noise parameter in the local depolarizing channel in Eq. (\ref{['eq:locladepola']}).

Theorems & Definitions (5)

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