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Error minimization for fidelity estimation of GHZ states with arbitrary noise

Liangzhong Ruan

TL;DR

This work tackles fidelity estimation for $L$-qubit GHZ states shared across a quantum network under arbitrary noise and no prior noise information. It develops a fidelity-preserving diagonalization approach combined with the GHZ Bloch representation and Fisher-information concepts to design a measurement protocol that achieves the minimum mean-squared error using only separable, local Pauli measurements. The authors prove a series of sufficiency results to reduce the problem to an independent-noise setting and derive a tight lower bound on estimation error, then construct Protocol 1 that attains this bound. Numerical experiments on a three-NV-centers platform show substantial error reductions relative to existing fidelity-estimation methods, highlighting practical implementability and robustness to i.i.d. and correlated noise regimes.

Abstract

Fidelity estimation is a crucial component for the quality control of entanglement distribution networks. This work studies a scenario in which multiple nodes share noisy Greenberger-Horne-Zeilinger (GHZ) states. Due to the collapsing nature of quantum measurements, the nodes randomly sample a subset of noisy GHZ states for measurement and then estimate the average fidelity of the unsampled states conditioned on the measurement outcome. By developing a fidelity-preserving diagonalization operation, analyzing the Bloch representation of GHZ states, and maximizing the Fisher information, the proposed estimation protocol achieves the minimum mean squared estimation error in a challenging scenario characterized by arbitrary noise and the absence of prior information. Additionally, this protocol is implementation-friendly as it only uses local Pauli operators according to a predefined sequence. Numerical studies demonstrate that, compared to existing fidelity estimation protocols, the proposed protocol reduces estimation errors in both scenarios involving independent and identically distributed (i.i.d.) noise and correlated noise.

Error minimization for fidelity estimation of GHZ states with arbitrary noise

TL;DR

This work tackles fidelity estimation for -qubit GHZ states shared across a quantum network under arbitrary noise and no prior noise information. It develops a fidelity-preserving diagonalization approach combined with the GHZ Bloch representation and Fisher-information concepts to design a measurement protocol that achieves the minimum mean-squared error using only separable, local Pauli measurements. The authors prove a series of sufficiency results to reduce the problem to an independent-noise setting and derive a tight lower bound on estimation error, then construct Protocol 1 that attains this bound. Numerical experiments on a three-NV-centers platform show substantial error reductions relative to existing fidelity-estimation methods, highlighting practical implementability and robustness to i.i.d. and correlated noise regimes.

Abstract

Fidelity estimation is a crucial component for the quality control of entanglement distribution networks. This work studies a scenario in which multiple nodes share noisy Greenberger-Horne-Zeilinger (GHZ) states. Due to the collapsing nature of quantum measurements, the nodes randomly sample a subset of noisy GHZ states for measurement and then estimate the average fidelity of the unsampled states conditioned on the measurement outcome. By developing a fidelity-preserving diagonalization operation, analyzing the Bloch representation of GHZ states, and maximizing the Fisher information, the proposed estimation protocol achieves the minimum mean squared estimation error in a challenging scenario characterized by arbitrary noise and the absence of prior information. Additionally, this protocol is implementation-friendly as it only uses local Pauli operators according to a predefined sequence. Numerical studies demonstrate that, compared to existing fidelity estimation protocols, the proposed protocol reduces estimation errors in both scenarios involving independent and identically distributed (i.i.d.) noise and correlated noise.
Paper Structure (20 sections, 12 theorems, 144 equations, 2 figures, 1 algorithm)

This paper contains 20 sections, 12 theorems, 144 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Setup and Problem Formulation
  3. Problem Transformation
  4. Sufficiency of considering classical correlated noise
  5. Sufficiency of considering independent noise
  6. Sufficiency of considering the sampled GHZ states
  7. Operation Construction
  8. Lower bound of the measurement error
  9. Achieving the minimum estimation error
  10. Application example
  11. Conclusion
  12. Proof of \ref{['lem:sp2arb']}
  13. Proof of \ref{['lem:id2sp']}
  14. Proof of \ref{['lem:id-sample']}
  15. Proof of \ref{['lem:min']}
  16. ...and 5 more sections

Key Result

Lemma 1

If a measurement operation ${ \cal{O} }^*$ is optimal in $\mathscr{P}$-prob:sp-m, the composite operation $\hat{{ \cal{O} }}^* = { \cal{O} }^* \circ { \cal{T} }$ is optimal in $\mathscr{P}$-prob:1-m.

Figures (2)

  • Figure 1: The variance of the estimated fidelity provided by different protocols. In this figure, $N=2000$ and $M=1000$. In subfigure A, $\delta=0.5$, while in subfigure B, $P_{\mathrm{d}}=0.5$.
  • Figure : Fidelity estimation

Theorems & Definitions (26)

  • Definition 1: Probabilistic multirotation
  • Lemma 1: Equivalence of $\mathscr{P}$-\ref{['prob:1-m']} and $\mathscr{P}$-\ref{['prob:sp-m']}
  • proof
  • Lemma 2: Equivalence of $\mathscr{P}$-\ref{['prob:sp-m']} and $\mathscr{P}$-\ref{['prob:id-m']}
  • proof
  • Lemma 3: Equivalence of Problems \ref{['prob:id-m']} and \ref{['prob:id2-m']}
  • proof
  • Theorem 1: Generality of optimality with independent noise
  • proof
  • Lemma 4: Lower bound for the estimation error
  • ...and 16 more