Amplitude-amplified coherence detection and estimation

TL;DR

The paper analyzes how to detect and quantify quantum coherence in unknown pure states under two operational models: access to multiple copies of the state and access to the unitary that prepares the state. It shows that coherence detection with copies requires Θ(1/c(|ψ>) log(1/δ)) samples, while access to the preparing unitary enables a quadratic speedup to Θ(1/√c(|ψ>)) samples, using amplitude amplification; amplitude estimation further yields an O(1/ε) scaling for bounding the geometric coherence. The results provide an operational interpretation of the geometric measure of coherence and establish noise-robust bounds. These advances may improve benchmarking and verification in near-term and fault-tolerant quantum devices and invite extensions to other resource theories.

Abstract

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.

Figures (2)

• Figure 1: (Amplitude-amplified coherence detection). Schematic representing the intuition behind amplitude-amplified coherence detection. We rotate an initial coherent state $\ket{\psi}$ in a 2-dimensional subspace towards an especially chosen subspace which allows us to detect coherence.
• Figure 2: (Complexity of coherence detection). We plot the optimal sample complexity scaling of detecting coherence in the pure state $\ket{\psi}$ as a function of coherence of the state in question, as measured by the geometric measure of coherence $c(\ket{\psi})$. The straight line depicts the lower bound from Theorem \ref{['thm:complexity_state']} supposing access to $m$ copies the state $\ket{\psi}$ on average. In contrast, as shown in Theorem \ref{['thm:coherence_complexity_amp_estimation']}, given access the unitary, $U_\psi$, which prepares $\ket{\psi}$, amplitude-amplified coherence detection provides a quadratic improvement (dashed line). The star and diamond guide the eye to the difference between the two curves at $c(\ket{\psi}) = 10^{-3}$.

Theorems & Definitions (8)

• Theorem 1
• proof
• Proposition 1: Theorem 3 of Brassard2002OGAmpAmp
• Theorem 2
• proof : Upper bound proof
• Proposition 2: Diamond-norm lower bound for unitary channel discrimination Weggemans2025lowerboundsunitary
• Lemma 1
• proof : Proof of lower bound