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Projected Optimal Sensors from Operator Orbits

Sooryansh Asthana, Yeshma Ibrahim, Norman Tze Wei Koo, Sai Vinjanampathy

Abstract

We unify Ramsey, twist-untwist, and random quantum sensors using operator algebra and account for the Fisher scaling of various sensor designs. We illustrate how the operator orbits associated with state preparation inform the scaling of the sensitivity with the number of subsystems. Using our unified model, we design a novel set of sensors in which a projected ensemble of quantum states exhibits beyond-shot-noise metrological performance. We also show favorable scaling of Fisher information with decoherence models and loss of particles.

Projected Optimal Sensors from Operator Orbits

Abstract

We unify Ramsey, twist-untwist, and random quantum sensors using operator algebra and account for the Fisher scaling of various sensor designs. We illustrate how the operator orbits associated with state preparation inform the scaling of the sensitivity with the number of subsystems. Using our unified model, we design a novel set of sensors in which a projected ensemble of quantum states exhibits beyond-shot-noise metrological performance. We also show favorable scaling of Fisher information with decoherence models and loss of particles.

Paper Structure

This paper contains 11 sections, 1 theorem, 44 equations, 3 figures.

Table of Contents

  1. Time-averaged Quantum Fisher Information for integrable composing Hamiltonians
  2. Haar Ramsey protocol
  3. Average Fisher Information
  4. Equivalence classes
  5. Two-parameter estimation
  6. Metrology with Projected Ensembles
  7. Restoring metrological support via projection and untwirling
  8. Effect of particle loss
  9. (i) Particle loss before phase embedding.---
  10. (ii) Particle loss after phase embedding.---
  11. Limiting behavior.---

Key Result

Theorem 1

For a composing Hamiltonian $H_c$ and a phase-embedding operator $G$, let $\mathfrak{g}=\langle\{-\mathrm{i}H_c,\,-\mathrm{i}G\}\rangle_{\text{Lie}}$ denote their dynamical Lie algebra (DLA). The finite-time averaged variance of $G$ over a time $T$ in the Heisenberg picture depends only on the compo

Figures (3)

  • Figure 1: An edge connects two Pauli strings that are equivalent to each other in the way they anticommute with the generator of phase embedding, thereby defining an equivalence class $\mathcal{C}_k$. Dynamical constraints, either via symmetry (left) or measurement and postselection (right), prevent featureless scrambling within the Hilbert space. This restricts the dynamics to metrologically sensitive operator orbits that have support over macroscopically distant equivalence classes.
  • Figure 2: Schematic illustrating the adaptive quantum sensing protocol using projected ensembles. Alice constructs a projected ensemble and communicates the classical measurement outcome to Bob. Bob is able to implement a beyond shot-noise sensor, but only upon use of the measurement outcome.
  • Figure 3: Probability distribution of Pauli strings as a function of their equivalence class index for the protocol using projected ensembles when $n_e = 1$. The inset shows the scaling of Fisher information with the total system size for different sample sizes.

Theorems & Definitions (1)

  • Theorem 1: Time-averaged QFI and the dynamical Lie algebra