Compressing Quantum Fisher Information

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.

💡 Research Summary

The paper addresses the fundamental problem of efficiently transferring and storing quantum Fisher information (QFI), the key resource that determines the ultimate precision of quantum parameter estimation. While previous work has focused on compressing the full quantum state, the authors ask a more economical question: can the QFI associated with a single‑parameter family of pure states be compressed without preserving the entire state? They answer affirmatively, showing that the complete QFI can be encoded into a single qubit together with a logarithmic amount of classical bits.

The theoretical development starts from a family of pure states |Ψθ⟩ = e^{-iθH}|Ψ⟩, where H is a Hermitian generator and θ is the unknown phase. The QFI of |Ψθ⟩ is given by the standard formula F = 4(⟨∂θΨθ|∂θΨθ⟩ – |⟨Ψθ|∂θΨθ⟩|²). By expanding |Ψθ⟩ in the eigenbasis {|E⟩} of H, the authors rewrite the QFI as a variance of the energy distribution p(E). They then construct a POVM {M_k} whose outcomes k partition the energy distribution into at most two distinct eigenvalues. After measuring this POVM, the post‑measurement state |Ψθ,k⟩ lives in a two‑dimensional subspace and therefore can be faithfully encoded into a single qubit. The number of possible outcomes K is bounded by the dimension d of the subspace spanned by {|E⟩}, giving a classical overhead of at most ⌈log₂(d−1)⌉ bits. Crucially, the average QFI of the conditional states equals the original QFI, so no information about θ is lost.

To illustrate the protocol concretely, the authors consider N identical equatorial qubits |eθ⟩ = (|0⟩ + e^{iθ}|1⟩)/√2, whose collective QFI equals N. They devise a simple “sum/difference” building block: a CNOT gate with one qubit as control, followed by a computational‑basis measurement on the target. The measurement outcome heralds either the addition or subtraction of the two phases, effectively mapping the pair onto a single qubit with phase θ₁±θ₂. When the two input phases are equal, the outcomes are either a doubled phase (2θ) or a null phase (0), and the average QFI remains 2, exactly the QFI of the original pair. By iterating this step N−1 times, all QFI is concentrated in the first qubit, while the sequence of measurement results (a binary string of length ⌈log₂N⌉) provides the necessary classical side information. Thus N copies are compressed into “one qubit + ⌈log₂N⌉ bits”.

Experimentally, the protocol is realized with polarization qubits of photons generated via spontaneous parametric down‑conversion. The CNOT gate is implemented with partially polarizing beam splitters (PPBS) in a linear‑optics architecture, and successful compression is post‑selected by detecting the target photon in the |0⟩ (horizontal) output. The compressed qubit emerges in the state |e2θ⟩, which is verified by measuring in the diagonal basis, yielding a cosine fringe with doubled frequency, as predicted. Imperfections (photon distinguishability, birefringence, wave‑plate errors) are modeled by introducing amplitude, phase, and frequency offset parameters in the fitting function. The experimental data show clear double‑fringe patterns across a 2π phase range.

The authors also implement a non‑deterministic compression scheme based on a Type‑I fusion gate. Here, two equatorial photons are interfered on a polarizing beam splitter; detection of a single photon in one output heralds the other photon as the compressed qubit with doubled phase. Quantum memories are used as buffers to synchronize photons for successive fusion stages, demonstrating the scalability of the approach.

Performance analysis confirms that each successful compression doubles the phase sensitivity, i.e., the compressed qubit carries four times the QFI of a single uncompressed qubit. The measured standard deviation of the phase estimator follows the compressed quantum Cramér‑Rao bound (σ = 1/2 rad⁻¹), and the root‑mean‑square error matches the bound after accounting for a small systematic bias due to drift in photon visibility.

In summary, the work establishes that the quantum Fisher information of any single‑parameter pure‑state family can be compressed into a single qubit plus a logarithmic amount of classical data, without loss of metrological power. The sequential 2→1 compression scheme for equatorial qubits is both theoretically optimal and experimentally demonstrated using linear‑optical components. This result opens new avenues for resource‑efficient quantum sensing, remote metrology, and distributed quantum networks, where transferring only the relevant information (QFI) rather than the full quantum state can dramatically reduce communication and storage overheads.