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Asymptotically optimal joint phase and dephasing strength estimation using spin-squeezed states

Arkadiusz Kobus, Rafał Demkowicz-Dobrzański

Abstract

We show an explicit $N$-qubit protocol involving one-axis-twisted spin squeezed states, that allows for simultaneous phase and dephasing strength estimation with precision that asymptotically matches fundamental quantum metrological bounds. The relevance of the protocol goes beyond this particular model, since any uncorrelated noise quantum metrological model, that allows for at most constant asymptotic quantum enhancement, can be reduced to this problem via an appropriately tailored quantum error-correction procedure.

Asymptotically optimal joint phase and dephasing strength estimation using spin-squeezed states

Abstract

We show an explicit -qubit protocol involving one-axis-twisted spin squeezed states, that allows for simultaneous phase and dephasing strength estimation with precision that asymptotically matches fundamental quantum metrological bounds. The relevance of the protocol goes beyond this particular model, since any uncorrelated noise quantum metrological model, that allows for at most constant asymptotic quantum enhancement, can be reduced to this problem via an appropriately tailored quantum error-correction procedure.

Paper Structure

This paper contains 5 sections, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Metrological model
  3. Asymptotically optimal protocol
  4. Summary
  5. Detailed derivation of moments

Figures (2)

  • Figure 1: Schematic representation of the protocol. First the OAT state is prepared via a squeezing transformation. Then, it is rotated so that the minimal variance direction is along the $y$ axis. Sensing rotates the state by an angle $\varphi$ around the $z$ axis, as well as reduces the length of its angular momentum by approximately a factor of $\eta$. By performing a joint measurement of both $J^2$ and $J_y$ observables one may estimate $\eta$ and $\varphi$ simultaneously. If the squeezing parameter scales as $\chi \propto N^p$, $p \in ]-1,-3/4[$, the protocol asymptotically saturates the fundamental metrological bounds simultaneously for both $\eta$ and $\varphi$ estimation.
  • Figure 2: Variance of estimators $\tilde{\eta},\tilde{\varphi}$, normalised by the corresponding asymptotically optimal quantum Fisher information values, for different scalings of the squeezing strength $\chi$, and dephasing strength $\eta=0.8$. $\chi=N^{-2/3}$ scaling (dash-dotted green line) minimises $\textrm{Var}(J_y)$, $\chi=N^{-3/4}$ scaling (dashed red line) provides the fastest convergence to the asymptotic bound for $\varphi$ estimation, $\chi=0$ (blue dotted line) is optimal for $\eta$ estimation, while $\chi=N^{-5/6}$ (solid black line) allows for optimal simultaneous estimation with precisions converging to the fundamental bounds for both parameters. The gray area is excluded by the bounds.