Superresolving collective quantum measurements

TL;DR

The paper tackles super-resolution of two incoherent bosonic signals encoded in a finite mixture by exploiting permutation symmetry across multiple copies of the state. It develops a two-stage, collective measurement protocol based on weak Schur-sampling (spectrum measurement) to estimate the mixture's purity, followed by a tailored refinement to extract the relative intensity $q$ and separation $\epsilon$, achieving simultaneous estimation that saturates the multi-parameter quantum Cramér–Rao bound. Key results include analytic quantum Fisher information analyses for the single-boson case, the construction of a Schur-Weyl-based measurement that saturates the bound in the large-$N$ limit, and robustness to misalignment in centroid, with extensions to both known and unknown centroid scenarios. The work also outlines concrete experimental pathways using quantum memories, Schur transforms, and multi-port optical or atomic platforms, highlighting practical routes to implement super-resolving collective measurements in imaging and communication tasks.

Abstract

We demonstrate a method for super-resolving signals encoded as finite mixtures of bosonic modes using collective measurements that exploit permutation symmetry. Specifically, we use multiple copies of the state $ρ$ of the finite mixture to extract an estimate for the purity of $ρ$ via a spectrum measurement, the weak Schur-sampling measurement. Depending on the outcome we then further fine-grain the measurement to optimally extract an estimate of the relative intensity between the two incoherent mixtures. Our protocol furnishes simultaneous estimates for both the relative intensity and the separation of incoherent signals saturating the multi-parameter Cramér-Rao bound, and is robust against misalignment errors. We also provide viable experimental avenues for implementing such collective measurements in different set-ups.

Figures (5)

• Figure 1: The distribution of intensity as a function of the one-dimensional spatial coordinate for bosons described by the state $\rho$ of Eq. (\ref{['eq:finite_mixture']}) with $\ket{\psi},\,\ket{\phi}\in L^2(\mathbb{R})$ given by Eq. (\ref{['eq:Gaussian']}). The separation between the two signals is $\epsilon=x_1-x_0$. The geometric centre of the two sources is $x_g=x_0+\frac{\epsilon}{2}$ whilst the centre of intensity is $x_c=x_0+\epsilon(1-q)$. For $\frac{\epsilon}{\sigma}\geq 1$ the sources can be resolved by measuring the intensity along the $x$-axis (direct imaging). Sub-Rayleigh and super-resolving measurements deal with the regime $\sigma\gg\epsilon$, i.e., for distributions with very high overlap.
• Figure 2: Relative error difference between the matrix elements of the Fisher information for the measurement in Eq. (\ref{['eq:AngMommeas']}) and the QFI of Eq. (\ref{['eq:QFI_epsilon']}) as a function of the number of copies. The overlap and relative intensity are set to $c = 0.97$ and $q = 0.8$ respectively.
• Figure 3: Relative error between the matrix elements of the Fisher information for the measurement in Eq. (\ref{['eq:AngMommeasavg']}) and the QFI of Eq. (\ref{['eq:QFI_epsilon']}) as a function of the number of copies. The overlap and relative intensity are set to $c = 0.97$ and $q = 0.8$ respectively.
• Figure 4: A circuit representation of the protocol implementing our super resolving collective measurement strategy for estimating the purity and relative intensity of two incoherent bosonic sources. The Schur transform $U_{\mathrm{Schur}}$ is applied to $N$ bosons each of which in the state $\rho$ of Eq. (\ref{['eq:finite_mixture']}) (black lines) and $\log_2(N)$ qubits initialized in the state $\ket{0}$ (blue lines). The Schur transform imprints the value of the total angular momentum $J$ in binary form on the ancilla qubits. Upon measuring the latter (small triangle), the classical value $J$ is used to implement the required symmetrized $\mathrm{SU}(2)$ rotation $D^{(J)\dagger}(\gamma)$ (Eq. (\ref{['eq:AngMommeas']})), where for a known intensity centroid $\gamma=\theta-\phi$ after which the bosons are measured in the $\{\ket{\psi_0(x_c)}, \ket{\psi_0^\perp(x_c)}\}$ basis, whereas for the arbitrary intensity centroid $\gamma=\theta-\phi_{\mathrm{avg}}$ followed by a Haar-measurement in different values of $y$$\{\ket{\psi_0(y)}, \ket{\psi_0^\perp(y)}\}$.
• Figure 5: The Young frames of partitioning $5$ into at most two parts. The vectors $\hat{\bm \lambda}\in (0,1)^2;\, \hat{\bm\lambda}=\frac{Y}{N}$ are the corresponding estimates of the eigenvalues of $\rho\in\mathcal{B}(\mathcal{H}_2)$.