Scalable protocol to coherence estimation from scarce data: Theory and experiment

TL;DR

The paper tackles the challenge of estimating quantum coherence from limited data by reframing the REC estimation into a tractable optimization via a relaxed objective that yields a computable lower bound. The core idea is to replace the NP-hard entropy minimization with a dual, gradient-enabled formulation that produces a lower bound $\beta$ on $C_r(\rho)$, remaining largely insensitive to system size. Numerical results show iteration complexity is size-agnostic, while experiments on two-qubit Werner states demonstrate reliable coherence estimates from scarce measurements. This approach enables scalable, non-tomographic coherence certification for large quantum systems and may extend to other quantum resources like entanglement.

Abstract

Key quantum features like coherence are the fundamental resources enabling quantum advantages and ascertaining their presence in quantum systems is crucial for developing quantum technologies. This task, however, faces severe challenges in the noisy intermediate-scale quantum era. On one hand, experimental data are typically scarce, rendering full state reconstruction infeasible. On the other hand, these features are usually quantified by highly nonlinear functionals that elude efficient estimations via existing methods. In this work, we propose a scalable protocol for estimating coherence from scarce data and further experimentally demonstrate its practical utility. The key innovation here is to relax the potentially NP-hard coherence estimation problem into a computationally efficient optimization. This renders the computational cost in our protocol insensitive to the system size, in sharp contrast to the exponential growth in traditional methods. This work opens a novel route toward estimating coherence of large-scale quantum systems under data-scarce conditions.

Figures (6)

• Figure 1: Iteration number $T$ as a function of the qubit number $N$. The values of $T$ are numerically obtained as the iteration counts required for reaching an accuracy of $10^{-5}$ with a learning rate of $0.05$. Different curves correspond to different numbers of Lagrange multipliers, $\lvert \bm{\lambda} \rvert$, indicated in the legend. Data points in the plot are obtained by averaging over $100$ random instances, with the shaded regions showing one standard deviation.
• Figure 2: Experimental setup. (a) A 405 nm continuous-wave laser pumps a PPKTP crystal placed inside a Sagnac interferometer to generate polarization-entangled photon pairs via type-II spontaneous parametric down-conversion. (b) A family of Werner states are prepared by introducing controlled decoherence into the entangled photon pairs. (c) Polarization measurements are performed in different bases to estimate the expectation values of observables and reconstruct the quantum states via QST. DM: dichroic mirror; SPD: single-photon detector.
• Figure 3: Coherence estimation for Werner states. The blue solid line with a shaded error region shows coherence estimates from QST, with error bars denoting the standard deviation from 1000 repetitions. The red solid line is the theoretical coherence for the ideal state. The purple dotted line marks the lower bound $\beta$ on coherence estimated using the expectation values of observables $ZZ$ and $XX$, while the green dashed line represents $\beta$ obtained with the expectation values of observables $ZZ$, $XX$, and $YY$. Increasing the number of observables notably improves the tightness of the lower bound.
• Figure 4: Numerical results illustrating the closeness of $\beta$ to $\alpha$. Each data point represents a pair $(\alpha,\beta)$, with $\alpha$ on the horizontal axis and $\beta$ on the vertical axis. The values of $\alpha$ and $\beta$ are obtained from the numerical method in our prior work Zhang2018PRL and the algorithm presented in the preceding section, respectively. A total of $500$ data points are produced by randomly choosing the values of $\bm{p}$ and $\bm{q}$. The region is partitioned into two parts respectively with $\alpha\leq\beta$ and $\alpha\geq\beta$, as indicated in the figure legend, separated by the dashed red line.
• Figure 5: Schematic illustration of the setup for generating polarization-entangled photon pair.