Entanglement-enhanced quantum metrology in a noisy environment

Quantum metrology overcomes standard precision limits and plays a central role in science and technology. Practically, it is vulnerable to imperfections such as decoherence. Here we demonstrate quantum metrology for noisy channels such that entanglement with ancillary qubits enhances the quantum Fisher information for phase estimation but not otherwise. Our photonic experiment covers a range of noise for various types of channels, including for two randomly alternating channels such that assisted entanglement fails for each noisy channel individually. We simulate noisy channels by implementing space-multiplexed dual interferometers with quantum photonic inputs. We demonstrate the advantage of entanglement-assisted protocols in a phase estimation experiment run with either a single-probe or multiprobe approach. These results establish that entanglement with ancillae is a valuable approach for delivering quantum-enhanced metrology. Our approach to entanglement-assisted quantum metrology via a simple linear-optical interferometric network with easy-to-prepare photonic inputs provides a path towards practical quantum metrology.

Figure 1

Concept of the comparison between the parallel scheme of quantum metrology with and without assisted entanglement. (a) Parallel scheme. Probes go through maps ${\mathrm{\Lambda }}_{\varphi }$ in parallel. (b) Parallel scheme with assisted entanglement. Introducing noiseless ancillae sharing entanglement with probes and implementing joint measurements after the evolution give estimation with an enhanced precision.

Figure 2

Experimental scheme. (a) Setup for entanglement-assisted single-probe approach. Heralded single photons are used to prepare polarization-spatial hyperentangled states for the entanglement-assisted quantum metrology approach. Space-multiplexed noisy channels are realized by the dual interferometric network setup, in which spatial coherence is reduced, and the optical path delay enables the arrival time of the photons passing through different optical paths on the BD (for the amplitude-damping channel) or NBS (for the depolarizing channel) to be different. Random phases are added between photons in different optical paths before recombining them on the BD or NBS. Quantum process tomography is performed via waveplates (WPs), BD, and PBS and enables reconstruction of the process matrices for the channels. (b) Setup for entanglement-assisted two-probe approach. Polarization-entangled photon pairs are used to prepare the four-qubit hyperentangled state. Projective measurements are realized via BDs, WPs, NBSs, and a PBS. Coincidences between paired photons are detected by APDs.

Figure 3

Experimental results of the reconstructed noisy channels. For the entanglement-assisted approach, the reconstructed process matrices are shown for (a) the amplitude-damping channel with $\eta =0.5$ and (d) the depolarizing channel with $p=0.4$ compared with their theoretical predictions in (b) and (e), respectively. Also shown are the fidelities $F$ of the reconstructed process matrices for (c) the amplitude-damping and (f) depolarizing channels as a function of the noise parameters. The red (filled) bars indicate the fidelities for the entanglement-assisted approach and the gray (hatched) ones indicate those for the optimized single-probe approach. Error bars indicate the statistical uncertainty, obtained from Monte Carlo simulations assuming Poissonian photon-counting statistics.

Figure 4

Experimental values of QFI. The QFI vs $\eta$ ($p$) is shown for (a) amplitude-damping and (b) depolarizing channels. Dashed curves show theoretical predictions of QFI for the entanglement-assisted approach, whereas solid curves are for the optimized single-probe approach. Data points are experimental results.

Figure 5

Experimental values of the error $\sqrt{\nu }\delta \varphi$. The error is shown as a function of the channel noise for the single-probe approach in (a) amplitude-damping and (b) depolarizing channels. (c) Result for the two-probe approach in the amplitude-damping channel. Dashed curves show theoretical predictions of the error for the entanglement-assisted approach, whereas solid curves are for approaches without assisted entanglement. The gray shadow denotes the shot-noise limit. Data points are experimental results. Error bars are calculated with the bootstrap method. Interferometric visibilities of the setups are (a) $0.9969\pm 0.0006$, (b) $0.9928\pm 0.0008$, and (c) $0.9699\pm 0.0055$.

Figure 6

(a) Circuit for the single-probe approach. Here $\mathcal{E}$ is a depolarizing channel. (b) Circuit for the entanglement-assisted protocol. (c) Different form of the circuit for the entanglement-assisted protocol for an intuitive understanding of why assisted entanglement helps against noise. Here $\mathcal{F}$ is a specific two-qubit operation. The lower wire in (b) and (c) is for the probe qubit and the upper wire in (b) and (c) is for the ancilla. The double wire on the right corresponds to a bit from a classical measurement.