Experimental demonstration of a multi-particle collective measurement for optimal quantum state estimation

TL;DR

This work demonstrates an experimental realization of a two-particle collective measurement for optimal quantum state estimation, implementing the Massar–Popescu strategy with non-maximally entangled MP_i projections enabled by Hong–Ou–Mandel interference. Compared to the optimal LOCC benchmark, the collective measurement performs at least as well and shows a fidelity advantage when systematic errors are accounted for, with significant gains in the tetrahedron-state scenario. The authors apply the same collective approach to quantum-state tomography and observe near-Gill–Massar scaling, with infidelity decreasing approximately as $1/N_{ ext{ens}}$, indicating practical viability for efficient state characterization. These results establish the feasibility and benefits of multi-particle collective measurements and lay groundwork for extending such techniques to larger ensembles and more complex quantum systems.

Abstract

We experimentally demonstrate a two-particle collective measurement proposed as the optimal solution to a quantum state estimation game. Our results suggest that, in practice, the collective measurement strategy is at least as good as the best local approach, and it achieves a higher average fidelity when accounting for systematic errors. This photonic implementation uses a recently developed universal two-photon projective measurement based on Hong-Ou-Mandel interference, polarization-dependent loss, and unitary operations. We compare the performance to the case where the entangling component of the measurement is suppressed. We further apply the collective measurement to quantum state tomography, observing a near-optimal scaling of the infidelity with the total number of samples.

Figures (4)

• Figure 1: Scheme and experimental setup used for the Massar and Popescu game. The setup consists of half-wave plates (HWP), quarter-wave plates (QWP), a partial polarizer, a 50:50 beamsplitter (BS), and single-photon avalanche photodiode detectors (APD). See the main text for details.
• Figure 2: Scatter plots for the fidelity of the Massar and Popescu two-copy game: (a) using the collective approach; (b) using the LOCC approach; over the $\{\theta, \phi\}$ state-space. The left-side of each circle represents the experimentally measured fidelity, while the right-side represents the theoretically expected fidelity.
• Figure 3: Fidelity of the Massar and Popescu two-copy game for the rotated tetrahedron states $\ket{\uparrow_{n_i}}'=\hat{R}\ket{\uparrow_{n_i}}$, where $\hat{R}$ is a fixed rotation in the Bloch sphere. These states are equivalent to Eq. \ref{['eq:tetras']} for the optimal collective measurement (see Supplemental document for details). The green dashed-line indicates the expected fidelity of $\bar{\mathcal{F}}^{\text{Coll}}_\text{Tetra} = 5/6 \approx 0.8333$. The experimental collective measurement results are presented in blue, while the experimental measurement results with suppressed entanglement are in gray. The dashed lines represent the average for each data set. The error bars are obtained by assuming Poisson statistics and using Gaussian error propagation.
• Figure 4: Tomographic infidelity with the number of samples $N_\text{ens}$ for the randomly-chosen states (a) $\ket{\vec{n}} = 0.883 \ket{H} + (-0.216 + 0.416 i) \ket{V}$ and (b) $\ket{\vec{n}} = 0.308 \ket{H} + (-0.916 + 0.256 i) \ket{V}$ shown on a log-log plot. The average values over multiple experimental trials for the collective measurement tomography are shown in orange, with the blue line showing the power-law fits to the first eight data points: (a) $1-\bar{F}=(1.5\pm0.4)N_\text{ens}^{-0.95\pm 0.04}$; (b) $1-\bar{F}=(1.6\pm0.4)N_\text{ens}^{-1.04\pm 0.05}$. The plotted error bars are calculated as the standard error of the mean over multiple trials. The green line indicates the Gill-Massar bound $1-\bar{F}=1/N_\text{ens}$, while the grey line indicates the theoretical average infidelity of Pauli tomography for pure states, given by $1-\bar{F}=(13/12)N^{-1}_\text{ens}$.