Enhanced quantum phase estimation with $q$-deformed nonideal nonclassical light

TL;DR

By modeling the probe with $q$-deformed nonlinear photon states, the authors address quantum phase estimation in a Mach-Zehnder interferometer under realistic nonideal light statistics. Using the Jordan–Schwinger representation, they derive closed-form photon-count distributions $p(n_1,n_2|\phi)$ and show that the quantum Fisher information $F_Q=4\,\Delta^2 \hat{J}_y$ coincides with the classical Fisher information $F_C$ for photon-counting measurements, confirming optimal readout. They demonstrate that $F_Q$ increases as the deformation parameter $q$ decreases, with $q$-deformed cat states yielding the strongest metrological gains and surpassing the Heisenberg limit at low mean photon numbers. Bayesian phase estimation saturates the QCRB across tested deformations, validating photon counting as an effective, nonadaptive strategy for these nonideal states. Overall, the work establishes $q$-deformed states as tunable nonclassical resources for quantum sensing, motivating experimental realization and extensions to multi-parameter and adaptive metrology.

Abstract

We investigate quantum phase estimation in a Mach-Zehnder interferometer using q-deformed photon states, including q-coherent and q-cat states, which model realistic deviations from ideal light sources. By deriving closed-form photon count likelihoods via the Jordan-Schwinger mapping, we compute the quantum and classical Fisher information and perform Bayesian inference on simulated detector data. Our results show that photon counting remains an optimal measurement strategy even for deformed states, with classical and quantum Fisher information in exact agreement. Furthermore, the phase sensitivity improves with increasing q-deformation, indicating enhanced metrological performance driven by nonclassical photon statistics. These findings highlight the utility of q-deformed states in quantum sensing.

Figures (4)

• Figure 1: (Color online). A schematic illustration of a Mach-Zehnder interferometric setup. Two pairs of input states $\ket{\text{cat}}_q^\pm$, $\ket{\alpha}_q$ are applied to the input modes of the first BS. A relative phase shift is applied by adjusting $\text{PS}_1$ and $\text{PS}_2$. Finally the output states of $\text{BS}_2$ are detected using the detectors $\text{D}_1$ and $\text{D}_2$.
• Figure 2: (Color online). QFI ($F_Q$) versus the average input photon number $\langle \hat{n}\rangle$ for two different values of the deformation parameter $q$ represented by the red (dashed) and green (dashed-dot) lines. The blue (dotted) line corresponds to the undeformed (usual) cat and coherent state input, while the black (solid) line represents the Heisenberg-limited case. \ref{['fig2a']}$q$-deformed even cat state \ref{['fig2b']}$q$-deformed odd cat state. $n_{max} = 30$ was chosen for all computations.
• Figure 3: (Color online). Variation of QFI with the deformation parameter $q$ for two different pairs of input states. The blue (dashed) line represents the behavior of the QFI for a $q$-coherent and an $q$-odd-cat state input, while the red (solid) line corresponds to a $q$-coherent and a $q$-even-cat input. In both cases, the expected photon number of the input is fixed at $\langle n \rangle = 20$. The QFI increases as the deformation parameter $q$ decreases, indicating that greater deformation leads to enhanced QFI and quantum sensitivity. At higher levels of deformation, the even-cat state outperforms the odd-cat state in terms of QFI. $n_{max} = 30$ was chosen for all computations.
• Figure 4: (Color online). Blue (solid) line: QCRB on the phase‐uncertainty $\Delta\phi$, computed from the QFI at $q = 0.9$ with a cutoff of $n_{max} = 30$. Black (solid) line: Cramér-Rao lower bound given by the HL. Data points: Bayesian estimates of $\Delta \phi$ obtained from simulated detector counts based on $\nu=30$ independent measurements at deformation parameter $q=0.9$ and true phase $\phi=\pi/2$. For each value of the total mean photon number $\langle n\rangle_T$, 100 simulation runs were performed. Red (circular) markers represent the sample mode of $\Delta\phi$, with error bars indicating the central 68$\%$ credible interval. As $\langle n\rangle_T$ increases, the Bayesian estimates converge toward the Cramér-Rao limit, asymptotically saturating the bound.