Compressing Quantum Fisher Information

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.

• The paper proves that the quantum Fisher information of any one-parameter pure-state family can be compressed into one qubit plus at most ⌈log₂(d−1)⌉ classical bits while preserving it on average.
• The authors develop a sequential two-to-one protocol for equatorial qubits that retains the full QFI of N copies using one qubit and ⌈log₂N⌉ classical bits, with outcomes tracked through measurement results.
• Photonic experiments using postselected CNOT and fusion gates confirm the predicted fourfold phase-sensitivity enhancement, while highlighting probabilistic success and systematic measurement bias as practical limitations.

The paper "Compressing Quantum Fisher Information" (2602.09358) establishes that the quantum Fisher information (QFI) about a single phase parameter, encoded in an arbitrary pure quantum state, can be faithfully compressed into the state of a single qubit together with only a logarithmic number of classical bits. The authors—Tang, Guenza Marcus, Lupu-Gladstein, Pang, Dobney, Chiribella, Steinberg, and Yilmaz—prove this result in full generality for one-parameter families of pure states of finite-dimensional systems, and demonstrate its core two-qubit building block experimentally on a photonic platform using two distinct linear-optical architectures.

Motivation and relation to prior compression results

Prior work on multi-copy quantum state compression showed that $N$ copies of an arbitrary pure qubit state can be compressed into $\log_2(N+1)$ qubits via the Schur–Weyl transform [plesch_efcient_2010], with experimental proof-of-principle by Rozema et al. [rozema_quantum_2014], and that the general asymptotic limit for $f$-parameter families is $\frac{f}{2}\log_2(N+1)$ hybrid (quantum plus classical) memory bits [yang_compression_2018-1]. These protocols preserve the state itself. The present work addresses a weaker but operationally distinct requirement: preserving only the QFI about one specific parameter. Since faithful state preservation is not necessary for high-precision estimation of that parameter, the relevant question is how little quantum memory suffices to transfer or store the metrological content. Earlier work on storing time in quantum memory used a different accuracy measure than the QFI [yang2018quantum], leaving the minimum-qubit question for QFI open; this paper closes it with a strong answer: one qubit, independent of system dimension or copy number.

The equatorial-qubit protocol

For $N$ identical copies of the equatorial state $$|e_\theta\rangle = (|0\rangle + e^{i\theta}|1\rangle)/\sqrt{2}$$, whose joint QFI is $N$, the protocol iterates a simple $2\to 1$ building block: apply a CNOT gate with the first qubit as control, then measure the target in the computational basis. For two identical inputs, the control collapses to either $|e_{2\theta}\rangle$ (QFI 4) or $|e_0\rangle$ (QFI 0), each with probability 1/2, so the average QFI is exactly 2—the original value. Cascading this block $\log_2(N+1)$0 times transfers all QFI to a single control qubit carrying phase $\log_2(N+1)$1. For identical phases, the resulting phase $\log_2(N+1)$2 has QFI $\log_2(N+1)$3, and averaging over the binomially distributed outcome count $\log_2(N+1)$4 recovers exactly $\log_2(N+1)$5. Because $\log_2(N+1)$6 takes at most $\log_2(N+1)$7 values, the classical side information required is $\log_2(N+1)$8 bits. This establishes that the entire metrological resource of an $\log_2(N+1)$9-qubit GHZ-like probe can be held in one physical qubit plus logarithmic classical memory—a result directly relevant to remote sensing and sensor networks, where sensing and measurement occur at different locations.

General pure-state construction

The general case treats states $f$0 decomposed in the eigenbasis of the generator $f$1, so that the QFI equals the variance of the energy distribution $f$2. The authors construct a measurement with operators $f$3 whose post-measurement states have average QFI equal to the original if and only if the conditional means satisfy $f$4. Using Carathéodory's theorem on the compact convex set of distributions with fixed mean, they prove (Appendix) that each extreme point has support on at most two energy eigenvalues, so every post-measurement state lives in a two-dimensional subspace and encodes faithfully into a single qubit via an isometry $f$5. The number of outcomes satisfies $f$6, giving a total compressed memory of one qubit plus $f$7 classical bits. Notably, the construction preserves the QFI on average over heralded outcomes rather than deterministically; individual outcomes may carry more or less QFI than the input.

Experimental demonstration

Both photonic implementations use polarization qubits from Type-I SPDC pairs (808 nm photons from a BBO crystal pumped at 404 nm), detecting roughly $f$8 pairs/sec, with postselection on two-photon coincidences within a 4 ns window.

CNOT scheme: A partially polarizing beam splitter realization of the CNOT gate [ralph_linear_2002] succeeds with probability 1/9; projecting the target onto $f$9 prepares the control in $\frac{f}{2}\log_2(N+1)$0. Scanning $\frac{f}{2}\log_2(N+1)$1 from $\frac{f}{2}\log_2(N+1)$2 to $\frac{f}{2}\log_2(N+1)$3 in $\frac{f}{2}\log_2(N+1)$4 steps yields the expected doubled-frequency fringe, $\frac{f}{2}\log_2(N+1)$5, with imperfection parameters accounting for imperfect photon indistinguishability, PPBS birefringence, and waveplate calibration errors.

Fusion-gate scheme: As an alternative with higher success probability (1/2 per pair versus 1/9), a Type-I fusion gate implemented with a single PBS post-selects the branch with photons on different paths; a half-wave plate at 22.5° followed by computational-basis measurement steers the surviving photon to $\frac{f}{2}\log_2(N+1)$6 or $\frac{f}{2}\log_2(N+1)$7. Iterating this scheme compresses $\frac{f}{2}\log_2(N+1)$8 qubits to between 1 and $\frac{f}{2}\log_2(N+1)$9 qubits, with one classical bit per remaining qubit tracking spurious $N$0 shifts.

The key quantitative validation: measuring $N$1 successfully compressed qubits in the optimal basis should yield estimator standard deviation $N$2 rad against the compressed QCRB, half the uncompressed bound of 1 rad—i.e., a fourfold QFI enhancement per photon. The measured standard deviations track the compressed limit closely, confirming successful QFI compression. However, RMSEs consistently exceed standard deviations due to a systematic bias of approximately 0.02–0.03 rad ($N$3–$N$4), attributed to drifting fringe visibility during sequential data acquisition and imperfect waveplate retardances. Unlike statistical noise, this bias does not scale as $N$5 and dominates at high photon numbers; the authors note that randomizing acquisition order would mitigate it.

Limitations and open questions

Several limitations are stated plainly in the paper. Both optical schemes are probabilistic: the CNOT gate succeeds with probability 1/9 and the fusion gate with probability 1/2, so QFI is preserved only conditioned on successful heralding events; deterministic operation in optics would require auxiliary qubits and quantum memories (as in the KLM teleportation-based gate), while platforms such as trapped atoms and superconducting circuits offer deterministic CNOTs natively. The fusion-gate cascade is nondeterministic even in principle, yielding up to $N$6 output qubits in the worst case—matching the symmetric-subspace state-compression rate rather than the single-qubit optimum. The general construction requires knowledge of the generator $N$7 and the energy distribution $N$8 to design the measurement operators $N$9, raising the question of whether blind or adaptive versions exist. Finally, the result is restricted to single-parameter families of pure states; extension to multiparameter families (where the QFI becomes a matrix and incompatible generators introduce trade-offs) and to mixed states remains unaddressed.

Conclusion

This paper proves that the QFI of any one-parameter family of pure states compresses into a single qubit plus $$|e_\theta\rangle = (|0\rangle + e^{i\theta}|1\rangle)/\sqrt{2}$$0 classical bits, with an explicit sequential protocol for multi-copy equatorial qubits achieving one qubit plus $$|e_\theta\rangle = (|0\rangle + e^{i\theta}|1\rangle)/\sqrt{2}$$1 bits. Photonic experiments verify the $$|e_\theta\rangle = (|0\rangle + e^{i\theta}|1\rangle)/\sqrt{2}$$2 compression through both a postselected CNOT and a Type-I fusion gate, observing the predicted fourfold phase-sensitivity enhancement consistent with the compressed quantum Cramér–Rao bound, modulo a small systematic bias from visibility drift. The result reframes quantum-state compression for metrology: what must be transmitted or stored in distributed sensing scenarios is not the state but its Fisher information, and that quantity fits in one qubit.