Resource-optimal quantum mode parameter estimation with multimode Gaussian states

Abstract

Quantum mode parameter estimation determines parameters governing the shape of electromagnetic modes occupied by a quantum state of radiation. Canonical examples, time delays and frequency shifts, underpin radar, lidar, and optical clocks. A comprehensive framework recently established that broad families of quantum states can attain the Heisenberg limit, surpassing any classical strategy. This raises a fundamental question: among all quantum-enhanced strategies, which is truly optimal? Answering this requires identifying physically meaningful resources governing each estimation task, so quantum states can be compared on equal footing. We show these resources are connected to the eigenmode basis of the generator of the relevant mode transformation. For time-shift estimation, whose generator is diagonal in the frequency domain, the pertinent resources are the mean frequency and bandwidth; analogous quantities emerge for other transformations. Our framework unifies two historically separate perspectives: the particle-number aspect and the mode-structure of quantum light, providing a coherent picture of quantum-enhanced sensing with multimode radiation. Within this unified framework, we derive a tight upper bound on the quantum Fisher information for multimode Gaussian states, expressed in terms of these natural resources, and analytically identify the optimal Gaussian states saturating it. These optimal states take a particularly transparent form in the generator eigenbasis, a structural simplicity reflecting the deep connection between the geometry of the mode transformation and the architecture of the optimal probe. We further demonstrate that multimode homodyne detection constitutes the optimal measurement, achieving this bound and completing the end-to-end characterization of optimal quantum metrology strategies for mode parameter estimation.

Figures (5)

• Figure 1: (Top) General mode parameter estimation protocol. An input state $\ket{\psi}$ passes through an optical medium that induces a unitary mode transformation $\hat{U}_\lambda$, encoding the parameter of interest $\lambda$ into the state. $M$ being the number of modes. (Bottom) The relevant resources for the mode parameter estimation task are the signal photon number $N_S$, and the two modal resource parameters $\bar{g}$ and $\Delta g$, corresponding to the mean and variance of the normalized generator intensity distribution, as defined in the main text.
• Figure 2: Quantum circuit representing the optimal mode parameter estimation protocol. Two single-mode squeezed states are prepared in modes $\hat{a}_i,\hat{a}_j$ with squeezing angles $\phi_i=\phi_j=0$ (for illustrative clarity) after which the passive Gaussian unitary $\hat{V} =\hat{B}$ transfers this squeezing to modes $\hat{b}_i,\hat{b}_j$. In that mode basis, the mode parameter transform $\hat{U}_\lambda$ acts as a simple phase shift on each individual mode. After crossing the sample, the state is measured, with either phase-sensitive or insensitive detection.
• Figure 3: Real part of the coherent states signal $\Psi(t) = e^{-(t-t_{0})^{2}/\sigma_{t}^{2}}\,e^{i\omega_{0}t}$ (left) and its Fourier transform $\tilde{\Psi}(\omega)$ (right). This figure illustrates a convenient representation of the time and frequency resources for classical and coherent states. The resources are the temporal and spectral bandwidths $\sigma_{t}$ and $\sigma_{\omega}$, and the central frequency and mean arrival time $\omega_{0}$ and $t_{0}$, here shown for a Fourier-limited pulse.
• Figure 4: (a), (b) JSI and JTI for variance optimal states for time estimation, with equal squeezing parameters $r_{i}=r_{j}$, central time-of-arrival $t_i=t_j$, and central frequency such as $\vert\omega_j-\omega_i\vert\gg \sigma_\omega$. We remind that the resource $\Delta\omega$ is related to other parameters such as $\frac{\vert \omega_j-\omega_i\vert^2}{2} = \Delta\omega^2- \sigma_\omega^2\approx \Delta\omega^2$. (c), (d) JSI and JTI for simultaneous variance optimal state for time and frequency estimation, that corresponds to the parameters: equal squeezing parameter $r_{i}=r_{j}$, with $\left\lvert t_{i}-t_{j}\right\rvert\gg \sigma_{t}$, and $\vert\omega_j-\omega_i\vert\gg \sigma_\omega$. We have chosen arbitrary central frequency and time $\overline{\omega},\overline{t}$ for the two distributions.
• Figure 5: Two mode transforms are illustrated: $a)$ beam displacement, where the beam is shifted by a distance $d$ in direction transverse to the detection plane, and $b)$, beam tilt, where the beam is rotated by an angle $\vartheta$ around an axis perpendicular to the reference axis. This schematic is widely inspired from delaubert2006quantum.

Theorems & Definitions (10)

• Theorem 3.1
• Definition 3.2: Optimality classes of states
• Definition 3.3: Optimal measurements
• Lemma D.1
• Lemma D.2
• proof
• proof
• proof
• Lemma K.1
• proof