Compressing Quantum Fisher Information

TL;DR

This work shows that the quantum Fisher information (QFI) for a phase parameter encoded in a pure quantum state can be faithfully compressed to a single qubit with only a logarithmic amount of classical information, enabling resource-efficient parameter estimation. The authors provide a general construction that uses a POVM with at most $d-1$ outcomes to encode the QFI into a qubit, preserving the total QFI on average; in the special case of $N$ equatorial qubits, a cascade of $2\to1$ compressions transfers all QFI into one qubit, with the remaining information captured in $\lceil\log_2 N\rceil$ classical bits. They substantiate the theory with two photonic implementations of the basic two-qubit compression block: a CNOT cascade and a Type-I fusion gate, both demonstrating the expected phase-doubling signatures and QCRB-consistent phase estimation, while highlighting practical considerations such as detector efficiency, drift, and probabilistic success. The results provide a pathway to transfer and store QFI in distributed sensing scenarios using far fewer quantum resources, with potential impact on remote metrology and quantum networks. Overall, the paper advances our ability to manipulate and utilize QFI directly, decoupling phase sensitivity from full quantum-state transport.

Abstract

We show that the quantum Fisher information about any phase parameter encoded in a family of pure quantum states can be faithfully compressed into a single qubit, accompanied by a logarithmic amount of classical bits. When the phase is encoded into many identical copies of a qubit state on the equator of the Bloch sphere, we show that the compression can be implemented sequentially, by iteratively compressing pairs of qubits into a single qubit. We experimentally demonstrate this building block in a photonic setup, developing two alternative compression strategies, based on Type-I fusion gate and a postselected implementation of the CNOT gate.

Figures (4)

• Figure 1: a) CNOT cascade compression scheme that compresses the QFI of arbitrary n-qubit inputs down to a single qubit. The qubit in the top path acts as the control qubit for all subsequent CNOT gates. Each CNOT gate performs a two-qubit compression, and measuring the target qubits in the computational basis transfers all of the information to the control qubit. b) Type-I fusion gate compression scheme. Equatorial state $|\psi\rangle_{\theta}$ are input into Type-I fusion gates at the first level. If a single polarization qubit is detected at one of the PBS output ports, the qubit exiting the other port will contain the compressed QFI, with its phase doubled. Non-deterministically compressed qubits with the same phase from the previous level are sent into the next level for further compression. To ensure that two qubits are available at the same time, quantum memories need to be used as buffers to store the qubits until they are ready simultaneously.
• Figure 2: Linear optical Implementation of QFI compression. Two photons generated via spontaneous parametric down-conversion (SPDC) serve as input polarization qubits for the compression schemes (a) and (b), entering through ports 1 and 2. The qubits are prepared in equatorial states with phase $\theta$ using a PBS, a HWP set to $\pi/8-\theta/4$, and a QWP set to $45^\circ$. A CNOT gate is used to interfere the qubits, and post-selection occurs by projecting the qubit in path 4 on $|0\rangle_4$, after which the compressed qubit along path 3 is characterized with a tomography setup.
• Figure 3: Output of successful compression. Sample raw data for projecting compressed qubit $|\psi\rangle_{2\theta}=\frac{1}{\sqrt{2}}(|0\rangle+e^{2i\theta}|1\rangle)$ onto the diagonal basis $|+\rangle$. CNOT plots show detections over a one-second counting interval with phase increments of $2.5^\circ$ in $\theta$, set by rotating the HWPs in the state preparation setup Fig \ref{['fig:optical_setup']}. The single counts corresponds to uncompressed qubits $|\psi\rangle_\theta$ projected onto the diagonal basis with detector dark counts subtracted. The two-photon detection shows oscillations at a doubled frequency with error bars representing $\pm \sigma$ statistical uncertainty, mainly due to laser power fluctuation and Poissonian noise from non-deterministic photon pair generation. In general, fusion gate can add or subtract phases from two different equatorial qubits as shown in tham_experimental_2020. A 2D scan demonstrating this is provided in Appendix. Here, this 1D plot shows the compression scenario where the two input phases are identical.
• Figure 4: The standard deviation and the root-mean-squared error (RMSE) of the estimated phase. The green dashed line represents the Cramér-Rao bound for estimating $\theta$ using and uncompressed qubit $|\psi\rangle_\theta$, while the black dashed line corresponds to the standard deviation for a compressed qubit $|\psi\rangle_{2\theta}$. The squares indicates the experimentally observed variances when estimating a selected set of phases, and the circular dots indicate the per photon RMSEs. The latter are higher due to the bias being scaled up by $\sqrt{N}\approx16.64$ for a mean photon count of $N=277$ for the CNOT data and $N \approx 522$ for Fusion data (see inset). Red markers show the experimental data for phase estimations using compressed qubits from the fusion gate setup, and blue markers represent data from the CNOT compression setup (Detailed descriptions of how we constructed the estimators are provided in the main text). The statistical standard deviation was estimated by performing Monte Carlo Simulations on the phase estimator using measured mean photon numbers with Poissonian noise. Inset: The bias, calculated as $\text{Bias}=\sqrt{\text{RMSE}^2-\sigma^2}$, reveals a systematic error of $0.02 - 0.03$ rad ($1^\circ-2^\circ$), which is attributed to imperfect waveplate retardances and drifting visibility over the course of data collection.

Theorems & Definitions (1)

• Lemma 1