Estimation on Geometric Measure of Quantum Coherence

Jianwei Xu, Lin Zhang, Shao-Ming Fei

Qi-Ming Ding, Xiao‐Xu Fang, He Lu

Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the l1-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.

Daniel F. Walls, Gerard J Milburn

Rhea Alexander, Michalis Skotiniotis, Daniel Manzano

The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state $|ψ\rangle$. Having access to $m$ copies of an unknown pure state $|ψ\rangle$ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success $\ge 2/3$ is $Θ(c(|ψ\rangle)^{-1})$, where $c(|ψ\rangle)$ is the geometric measure of coherence of $|ψ\rangle$. However, assuming access to the unitary $U_ψ$ which prepares the unknown state $|ψ\rangle$, and its inverse $U_ψ^\dagger$, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number $O(c(|ψ\rangle)^{-1/2})$ of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error $\varepsilon$ with a sample complexity that scales as $O(1/\varepsilon)$ as compared to the $O(1/\varepsilon^2)$ sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.

Hai-Jun 海军 Zhang 张, Bin 斌 Chen 陈, Ming 明 Li 李 +2 more

We study the geometric measure of quantum coherence recently proposed in [Phys. Rev. Lett. 115 (2015) 020403]. Both lower and upper bounds of this measure are provided. These bounds are shown to be tight for a class of important coherent states — maximally coherent mixed states. The trade-off relation between quantum coherence and mixedness for this measure is also discussed.

Daniel F. Walls, Gerard J Milburn