Experimental realization of a photonic weighted graph state for quantum metrology

Abstract

Quantum metrology seeks to push the boundaries of measurement precision by harnessing quantum phenomena. Conventional methods often rely on maximally entangled resources, with states that are usually challenging to produce and sustain in practical setups. Here, we show that the maximally entangled constraint can be lifted by experimentally realizing a photonic two-qubit weighted graph state with an arbitrarily tunable graph weight. We use the generated state as a resource for quantum-enhanced phase sensing. We experimentally characterize the state and study its minimum estimator variance for two distinct local measurement bases as the graph weight varies from the maximally entangled to weakly entangled limit. We find excellent quantitative agreement with theoretical predictions, and observe a gain in precision beyond the classically attainable precision limit for graph weights substantially below the maximally entangled limit. This confirms that considerably less entanglement is required to achieve a quantum advantage. Albeit non-scalable in our test setup, this work represents the first experimental realization of weighted graph states with a tunable graph weight using linear optics. We expect more scalable versions of the model to be possible in an on-chip photonic platform.

Figures (11)

• Figure 1: Experimental setup for creating the two-qubit weighted graph $\ket*{\Gamma_{\phi_{12}}}$ with a tunable graph weight, and for performing a quantum metrology protocol. A 405 nm continuous wave (CW) diode laser pump beam (vertically polarized) passes through a half-wave plate (HWP) set to $22.5^{\circ}$, a quarter-wave plate (QWP) set to $10^{\circ}$, and two cascaded $\beta$-barium borate crystals (5mm by 5mm by 0.5mm each) with mutually perpendicular optic axes, to create the polarization-entangled $\ket*{\Phi^{(-)}}$ state. The generated photons are separated: photon 1 goes to a sensing stage. Photon 2 is split by a PBS into paths $r_2$ and $l_2$, forming a Mach-Zehnder interferometer (MZI). The $R_z(\phi'_{12})$ rotation in path $l_2$ (implemented by a QWP-HWP-QWP configuration) is the core tuning mechanism for the graph weight $\phi_{12}$. The paths recombine at a non-polarizing beam splitter (NPBS), where port $p_2$ is selected. Both photons then enter the quantum sensing stage, where the phase $\theta$ is encoded by the unitary $U(\theta) = R^{(1)}_x(\theta) \otimes R^{(2)}_z(\theta)$. Finally, both photons are detected after passing through an arbitrary polarization projective measurement (QWP-HWP-PBS), interference filters (800 nm $\pm$ 40 nm), and coupled into single-mode fibers for detection by avalanche photodiode (APD) single-photon detectors and a coincidence counting module. The detector output for photon 1 is delayed by 1.5 ns before it enters the coincidence counting module.
• Figure 2: Normalized photon counts and fit as a function of the phase difference between the two optical paths $\varphi^{\prime}$, varied by translating the mirror on a stage in steps of 60 nm. Each data point is an average of 10 measurements, where each measurement is from accumulated photon counts over a 10-second interval. The normalized photon counts are fitted to $f(x) = a\cos\left(b x + c\right) + d$, where the parameters of the fit are $a, b, c$ and $d$. The error bars represent the standard deviation of the mean. The discontinuity around the 6th step was due to a double jog. See Appendix \ref{['sec:fringe_visibility']} for more details.
• Figure 3: Real and imaginary parts of the reconstructed density matrices for the state $(\ket*{H_1, H_2} -e^{i\varphi^{\prime}}\ket*{V_1, V_2})/\sqrt{2}$ where $\varphi^{\prime}$ takes on the values (a)$\varphi^{\prime}=0$, (b)$\varphi^{\prime}=\pi/4$, (c)$\varphi^{\prime}=\pi/2$, (d)$\varphi^{\prime}=3\pi/4$, and (e)$\varphi^{\prime}=\pi$, respectively. All the density matrices are obtained using a maximum likelihood estimation (MLE) that takes the photon counts collected over a 10-second interval for each of the 16 projective measurements in Tab. \ref{['tab:2q_tomo_measurements']}.
• Figure 4: Real and imaginary parts of the reconstructed density matrices for the weighted graph states $(\ket*{H_1, D_2} + \ket*{V_1, D_{2}^{\phi_{12}}})/\sqrt{2}$ where $\phi_{12}$ takes on the values (a)$\phi_{12}=0$, (b)$\phi_{12}=\pi/4$, (c)$\phi_{12}=\pi/2$, (d)$\phi_{12}=3\pi/4$, and (e)$\phi_{12}=\pi$, respectively. All the density matrices are obtained using a maximum likelihood estimation (MLE) that takes the photon counts collected over 10 seconds interval for each of the 16 projective measurements in Tab. \ref{['tab:2q_tomo_measurements']}.
• Figure 5: (a) Single-shot observable variance $(\Delta A)^2$, and (b) magnitude of the observable derivative $\abs*{\partial_{\theta}\expval*{\hat{A}}}$, measured via a 2-point finite difference method for the optimal local Pauli measurements as a function of the graph weight $\phi_{12}$ for the sensing phase $\theta=0$. The errors represent 95% confidence intervals for the respective quantities.