Optimal finite-dimensional probe states for quantum phase estimation

TL;DR

The paper addresses the problem of optimizing quantum phase estimation when restricting to finite-dimensional, two-mode Fock states with fixed mean photon number $\bar{n}$ and dimension $N$. It derives explicit optimal finite-dimensional probe states (OFPS) for both linear and nonlinear phase shifts via Theorems 1–4, detailing their form across regimes and the corresponding quantum Fisher information (QFI). It shows that increasing the Fock dimension can provide metrological advantages beyond the fixed particle number, and it compares OFPS against entangled coherent states, noting that OFPS can outperform continuous-variable benchmarks for large $N$. The authors also propose optimal measurements, including adaptive schemes to render parity and photon-counting measurements universally optimal, and they assess performance under realistic noise (particle loss), outlining practical implications for weak-light, satellite, and on-chip quantum metrology, as well as open questions for future work.

Abstract

Phase estimation is a major mission in quantum metrology, especially in quantum interferometry. A full phase estimation scheme usually includes the optimal probe state and measurement. For the finite-dimensional states in Fock basis, the N00N state ceases to be optimal when the average particle number is fixed yet not equal to the Fock dimension (Fock number of the highest occupied Fock state of one mode), and what is the true optimal finite-dimensional probe state in this case is still undiscovered. Hereby we present several theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice. These optimal finite-dimensional probe states reveal an important fact that the Fock dimension could be treated as a metrological resource, and the given scheme is particularly useful in scenarios where weak light or limited particle number is demanded.

Figures (17)

• Figure 1: Comparison of the QFI between the $\mathrm{\bar{n}00\bar{n}}$ state (red line) and OFPS (blue stars) for (a) linear phase shifts and (b) nonlinear phase shifts. The Fock dimension of the OFPSs is $N=10$ in the plots.
• Figure 2: Preparation of the nonlinear OFPS given in Eq. (\ref{['eq:non_optstate2']}) with the one-axis twisting model. (a) Optimal control amplitude for the preparation process. (b) The evolved constraint value during the dynamics. (c) The evolved fidelity during the dynamics. (d) The tomography of the target and prepared states in the basis $\{\ket{j,m}\}$ . In the plots $\bar{n}=6$ and $N=5$.
• Figure 3: Comparison of the QFI between the entangled coherent state (red line) and the OFPS (blue stars) for (a) linear phase shifts and (b) nonlinear phase shifts. The average particle number $\bar{n}=4$ in the plots.
• Figure 4: Performance comparison between the adaptive schemes realized by the sharpness (dashed-blue line) and mutual information (dash-dotted-green line), and Bayesian estimations (solid-red line) in [(a1)-(a2), (b1)-(b2)] noiseless and [(c1)-(c2), (d1)-(d2)] noisy scenarios. 2000 rounds of experiments are numerically simulated and all results in the plots are the average performance of them. The performance of all simulations are given in Appendix \ref{['sec:adapt']}. In the figure $N=10$ and the true value of $\phi$ is taken as 0.2. In the noisy case the transmission rates $T_1=T_2=0.9$.
• Figure 5: Performance comparison between the OFPS and $\mathrm{\bar{n}00\bar{n}}$ state in the noiseless scenario with $\bar{n}=8$ and $N=10$. The solid-red and dashed-blue lines represent the Bayesian estimation for $\mathrm{\bar{n}00\bar{n}}$ state and adaptive estimation for the OFPS, respectively.