A Correspondence Between Qubit- and Boson-Based Sensing Protocols

Abstract

Recent years have seen a surge of interest in adapting non-classical states to search for physics beyond the Standard Model, including both qubit- and boson-based systems. These two classes of systems are closely related through the Holstein--Primakoff transformation (HPT). In this work, we show that the HPT offers novel insights into inherently quantum-mechanical detection of new physics. We begin by discussing how the HPT can be used to unify certain existing qubit- and boson-based proposals in a single framework. We then show how it naturally reveals a novel generalization of GHZ states which offer superior performance for the detection of rare interactions. We demonstrate that these novel states can be readily prepared, and combine the coherent advantage of GHZ states with superior error rejection. Finally, we show that the HPT also reveals a natural extension of the Hamiltonian-Not-in-Lindblad-Span theorem for collections of bosonic modes.

The paper asks how qubit-ensemble and bosonic sensing protocols, particularly searches for wave-like dark matter, can be related through the Holstein–Primakoff transformation. It develops correspondences between Dicke/Fock, spin-coherent/coherent, and GHZ/cat states and their rotation/displacement sensing protocols. The Dicke–Fock mapping is exact within a truncated bosonic space; several other correspondences use latitude-dependent approximations or preparation-based analogies rather than exact state mappings. A proposed qubit protocol projects an equatorial spin-coherent state onto excitation number modulo $k$, producing “GHZ-$k$” states. For $k\geq3$ and $N\gg k^2$, transverse-rotation QFI approaches $(N^2+N)/2$, versus $N^2$ for ordinary GHZ states. Modulo-$k$ readout distinguishes an excitation from up to $k-2$ decays. With mean decay count $\lambda\ll1$ per window, the estimated false-signal probability is $\lambda^{k-1}/(k-1)!+\lambda e^{-\omega/T}+\epsilon N(1+\log_2 k)$, including thermal excitations and gate errors. Preparation and readout use ancilla-assisted modular measurements; realistic gate-level performance is left open. For $m$ bosonic modes with time-averaged total occupation $n$, linear signals and identical local linear noise, an energy-constrained HNLS analysis bounds inverse estimation variance by $O(m(n+1)T^2)$ outside the Lindblad span, and $O(m\min\{(n+1)T^2,T/\kappa\})$ inside it. Purification and channel-QFI bounds establish the limits; error-corrected coherent-state codes and collective-mode squeezing establish achievability for $n\geq1$. Ordinary photon loss places displacement signals inside the span, eliminating asymptotic occupation-number enhancement. The results assume ideal control and, where needed, noiseless ancillas; achievability for $n\ll1$ and detailed higher-order extensions are not established.