Nonlinear Heisenberg Limit via Uncertainty Principle in Quantum Metrology

TL;DR

This work tackles the quantum metrology precision challenge by reframing the Heisenberg limit in terms of the parameter-space canonical momentum $\hat{\mathcal{K}}$, establishing the bound $\delta g \ge \frac{1}{2\sqrt{\nu}\Delta\bar{\mathcal{K}}}$ and showing that nonlinear scaling can arise when a generating process with indefinite time direction and quantum switch resources increases $\Delta\bar{\mathcal{K}}$ without requiring extra probe energy. The authors present a theoretical framework linking $\hat{\mathcal{K}}$ to the dynamical operator $\hat{V}_{S}=\partial_{g}\hat{H}_{S}$, prove that ancilla-free schemes under bounded $\Delta\bar{V}_{S}$ yield linear scaling, and demonstrate experimentally in quantum optics that a generating process with indefinite time direction yields quadratic scaling in time $T$ and iteration number $N$. Their optical implementation uses photon orbital angular momentum and polarization, with Q-plates and a Dove prism to realize the generating and parameterizing processes, producing projective probabilities $P_{+}=\tfrac{1}{2}[1+\sin(2gT^{2})]$ and $P_{-}=\tfrac{1}{2}[1-\sin(2gT^{2})]$, and showing $\delta g^{(N)}_{\mathrm{exp}}\ge \frac{1}{\sqrt{\nu}(N^{2}+N)}$ for $N$ iterations. The results unify diverse “super-Heisenberg” regimes under a common uncertainty-principle framework and highlight the role of quantum switches and indefinite causal structure in enabling nonlinear- scaling metrology with practical implications for high-precision sensing.

Abstract

The Heisenberg limit is acknowledged as the ultimate precision limit in quantum metrology, traditionally implying that root mean square errors of parameter estimation decrease linearly with the time T of evolution and the number N of quantum gates or probes. However, this conventional perspective fails to interpret recent studies of "super-Heisenberg" scaling, where precision improves faster than linearly with T and N. In this work, we revisit the Heisenberg scaling by leveraging the position-momentum uncertainty relation in parameter space and characterizing precision in terms of the corresponding canonical momentum. This reformulation not only accounts for time and energy resources, but also incorporates underlying resources arising from noncommutativity and quantum superposition. By introducing a generating process with indefinite time direction, which involves noncommutative quantum operations and superposition of time directions, we obtain a quadratic increment in the canonical momentum, thereby achieving a nonlinear-scaling precision limit with respect to T and N. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed probe energy. Our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on enhancing precision in practical quantum metrological and sensing tasks.

Figures (5)

• Figure 1: Schematic of quantum metrological schemes. a Standard quantum metrological scheme with a single-pass parameterizing process. $\hat{H}_{con}$ and $\hat{U}_{con}$ denotes the possible Hamiltonian and unitary controls applied to the probe state. b Standard quantum metrological scheme with the probe passing through the identical parameterizing process $N$ times sequentially. $\hat{H}_{con}^{(i)}$ and $\hat{U}_{con}^{(i)}$ denotes the possible Hamiltonian and unitary controls applied to the probe state during $i$-th parameterizing process. c Quantum metrological scheme with an ITD generating process. The Hamiltonian $\hat{H}_{C}$ of the generating process is conjugate to the characteristic operator $\hat{V}_{S}$ of the parameterizing process. The ancilla is a qubit to implement the quantum switch. d Quantum metrological scheme with the probe passing through the identical ITD generating process and parameterizing process $N$ times sequentially. Every ITD generating process is prior to every single parameterizing process.
• Figure 2: Experimental implementation of a single-shot evolution that includes an ITD generating process and a parameterizing process. A Q-plate and a mirror are used to shift the OAM of the probe in opposite directions for the different spin states $|R\rangle$ and $|L\rangle$. Rotating the Dove prism by an angle of $\alpha/2$ introduces a rotation angle of $\alpha$ on the profile of the photon beam.
• Figure 3: Experimental setup. The system comprises three main parts: Preparation of the probe state and ancilla state, joint evolution of the probe and the ancilla which includes the ITD generating process and the parameterizing process, measurement of the final state. The SPDC process in the PPKTP crystal converts a $405\nm$ photon into an $810\nm$ single-photon pair with orthogonal polarization. Idler photons are separated by a PBS to herald signal photons. The signal photon (zero OAM) serves as the probe, and its polarization state serves as the ancilla. Different Q-plate orders and Dove prism rotation angles implement various evolution times $T$. By adjusting right-angle prism mirrors, photons pass through the Q-plate and Dove prism multiple times, achieving various iteration numbers $N$. Projective measurement is conducted using a PBS, two HWPs, and an $N\times m$-order Q-plate. Photon numbers under orthogonal projections are detected via coincidence counting between SPD 2 and SPD 1, and between SPD 3 and SPD 1.
• Figure 4: Experimental results with different dimensionless evolution time $T$. a Experimental results of the projective probabilities $P_{+}$ and $P_{-}$. The brown line and the cyan line represent the theoretical predictions of $P_{+}$ and $P_{-}$ calculating by Eq. (\ref{['eq:10']}), separately. The brown squares with error bars are the experimental results of $P_{+}$ and the cyan squares with error bar are the experimental results of $P_{-}$. b Experimental results of the RMSE of unknown parameter $g$. The solid blue line represents the nonlinear Heisenberg limit given by Eq. (\ref{['eq:11']}) and the dashed red line represent the linear Heisenberg limit of the standard quantum metrological scheme with the normalized maximum average uncertainty $\Delta\mathcal{V}_{S}=1$. The blue squares with error bars are the experimental results of the RMSE on estimating the unknown parameter $g$.
• Figure 5: Experimental results in sequential strategy with $N$ copies of a single-shot evolution which comprises an ITD generating process and a parameterizing process. a Experimental results of the projective probabilities $P_{+}$ and $P_{-}$. The brown line and the cyan line represent the theoretical predictions of $P_{+}$ and $P_{-}$ calculating by Eq. (\ref{['eq:13']}), separately. The brown squares with error bars are the experimental results of $P_{+}$ and the cyan squares with error bar are the experimental results of $P_{-}$. b Experimental results of the RMSE of unknown parameter $\alpha$. The solid blue line represents the nonlinear Heisenberg limit given by Eq. (\ref{['eq:14']}) and the dashed red line represent the linear Heisenberg limit of the standard quantum metrological scheme with the normalized maximum average uncertainty $\Delta\mathcal{V}_{S}=1$. The blue squares with error bars are the experimental results of the RMSE on estimating the unknown parameter $\alpha$.