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Heisenberg-Limited Quantum Metrology without Ancillae

Qiushi Liu, Yuxiang Yang

Abstract

Extensive research has been dedicated to the asymptotic theory of quantum metrology, where the goal is to determine the ultimate precision limit of quantum channel estimation when many accesses to the channel are allowed. The ultimate limit has been well established, but in general a noiseless and controllable ancilla is required for attaining it. Little is known about the metrological performance without noiseless ancillae, which is more relevant in practical circumstances. In this Letter, we present a novel theoretical framework to address this problem, bridging quantum metrology and the asymptotic theory of quantum channels. Leveraging this framework, we prove sufficient conditions for achieving the Heisenberg limit with repeated applications of the channel to be estimated, both with and without applying interleaved unitary control operations. For the latter case, we design an algorithm to identify the control operation explicitly.

Heisenberg-Limited Quantum Metrology without Ancillae

Abstract

Extensive research has been dedicated to the asymptotic theory of quantum metrology, where the goal is to determine the ultimate precision limit of quantum channel estimation when many accesses to the channel are allowed. The ultimate limit has been well established, but in general a noiseless and controllable ancilla is required for attaining it. Little is known about the metrological performance without noiseless ancillae, which is more relevant in practical circumstances. In this Letter, we present a novel theoretical framework to address this problem, bridging quantum metrology and the asymptotic theory of quantum channels. Leveraging this framework, we prove sufficient conditions for achieving the Heisenberg limit with repeated applications of the channel to be estimated, both with and without applying interleaved unitary control operations. For the latter case, we design an algorithm to identify the control operation explicitly.
Paper Structure (12 sections, 5 theorems, 50 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 5 theorems, 50 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Vectorization
  3. Spectral properties of quantum channels
  4. Proof of Lemma \ref{['lem:associated QFI']}
  5. Proof of Lemma \ref{['lem:asymptotic associated QFI']}
  6. Proof of Theorem \ref{['thm:condition for HL']}
  7. Proof of Theorem \ref{['thm:HL with unitary control']}
  8. Algorithm for identification of the unitary control
  9. Proof of Theorem \ref{['thm:HL Pauli noise']}
  10. Supplemental examples
  11. The singularity of HNKS condition
  12. HL without decoherence-free subspace

Key Result

Lemma 1

The associated QFI of the associated state $\tilde{\rho}_\theta=\lvert\rho_\theta\rrangle\llangle\rho_\theta\rvert/\Tr \left(\rho_\theta^2\right)$ is given by

Figures (3)

  • Figure 1: Control-enhanced sequential strategy for estimating $\theta$ from $N$ queries to $\mathcal{E}_\theta$ ($N=2$ as an illustrated example). Trivial $\mathcal{U}_c=\mathcal{I}$ corresponds to a control-free strategy.
  • Figure 2: The normalized output QFI $F^Q(\rho_\theta)/N$ versus the number of queries $N$. The (expectation value of the) QFI is computed over $1000$ trials with random control errors. The gray lines represent different SPAM error rates ($p_{\mathrm{SPAM}}=0,0.01,0.1$) under ideal control ($\sigma_{\mathrm{control}}=0$). The red ($\sigma_{\mathrm{control}}=0.01$) and blue ($\sigma_{\mathrm{control}}=0.04$) lines show how the HL deviates as control error increases, with dashed lines corresponding to the presence of SPAM errors and solid lines indicating the absence of SPAM errors. The solid black line represents the performance of a control-free strategy.
  • Figure 3: Comparison between the actual normalized QFI $F^Q(\rho_\theta)/N^2$ and the bound Eq. (\ref{['eq:oscillating QFI bound']}). The input state is a fixed state $\rho_0=\mathrm{diag}\{1/4,1/4,1/2\}+\alpha \mathrm{diag}\{1/4,1/4,-1/2\}$ for $\alpha=0.9$.

Theorems & Definitions (11)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 1 more