Certification of quantum properties with imperfect measurements

TL;DR

The paper tackles the problem of certifying convex functions of quantum states under realistic noise, including shot noise and imperfect measurements. It extends the confidence-region tomography framework by incorporating an operational-distance bound for measurement imperfections, yielding an expanded feasible region defined by $\sum_{k=1}^m |\mathrm{tr}(E_k \rho) - \hat f_k| \le \epsilon_1 + \epsilon_2$, where $\epsilon_1 = \sqrt{ \frac{2}{N} \ln \frac{2^m}{\delta} }$ and $\epsilon_2$ bounds $d_{\mathrm{op}}(E,F)$. Robust bounds for any convex $\mathcal{F}$ are obtained by solving $\mathcal{F}_{\mathrm{LB}} = \min_\rho \mathcal{F}(\rho)$ or $\mathcal{F}_{\mathrm{UB}} = \max_\rho \mathcal{F}(\rho)$ subject to the above constraints, without requiring informationally complete measurements. The method is demonstrated on three numerical applications—minimum preparation fidelity, magnetization, and entanglement witnesses—showing reliable certification even when calibration errors are present. This framework provides a practical, general, and efficient tool for robust quantum-state certification in near-term experiments.

Abstract

The accurate characterization of quantum systems is essential for the advancement of quantum technologies. In particular, certifying convex functions of quantum states plays a central role in many applications. We present a certification method for experimentally prepared quantum states that accounts for both shot noise and measurement imperfections in the data-acquisition stage. Building upon previous work, our method extends confidence regions to accommodate imperfect control over measurements. The values of the functions can then be bounded using convex optimization techniques. We provide explicit prescriptions for quantifying the noise contribution from finite statistics and for estimating the effect of measurement imperfections. By jointly incorporating statistical and systematic errors, the method yields a robust certification framework for quantum experiments.

Figures (3)

• Figure 1: Minimum two-qubit state fidelity as a function of the total number of shots $N_{shots}$, for a confidence level of $0.997$. From top to bottom: $Y$-axis rotation shift, amplitude damping, phase damping and depolarizing channels. The operational distance is bounded by Eq. \ref{['eq:fid_bound']}. Solid lines represent the median value obtained from 100 Haar-random pure states, while the shaded areas depict the interquartile ranges. Results are displayed for different values $\gamma$ of the corresponding channel (Appendix \ref{['app:noise_model']}).
• Figure 2: Maximum spin magnetization as a function of the noise rotation angle $\gamma$, for $5$ qubits and a confidence level of $0.997$. Solid lines represent the median value obtained from 100 Haar-random pure states, while the shaded areas depict the interquartile ranges. The results that account for measurement noise are shown in blue, while the red curves correspond to results that ignore operator noise, with varying $N_{shots}$.
• Figure 3: Minimum value of the witness as a function of the total number of shots $N_{shots}$, for $2$ qubits and a confidence level of $0.997$. From top to bottom: entangled state considering measurement noise, separable state ignoring noise, and separable state considering measurement noise. The operational distance is bounded by Eq. \ref{['eq:fid_bound']}. Solid lines represent the median value obtained from 100 Haar-random pure states, while the shaded areas depict the interquartile ranges.