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arXiv:2606.26167

Criterion for qubit-assisted quantum metrology approaching Heisenberg scaling

Peng Chen, Jun Jing

quant-phSubmitted Jun 24, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

We study in this work the metrology precision of a probe system coupled to an ancillary qubit. Restricting the probe-qubit coupling along only one or two directions is found to be a sufficient criterion for the effective dynamical generator to achieve the Heisenberg limit in precision. Under the criterion, the quantum Fisher information (QFI) about the to-be-estimated parameter becomes the expectation value of mean square of the effective generator with respect to the initial state of the composite system. Our criterion is justified in two distinct systems. For a bosonic probe, QFI about the displacement estimation is found to be proportional to the mean excitation number of probe. It renders a counterintuitive result that quantum metrology sensitivity can be enhanced by increasing the temperature of the probe system. For a spin-ensemble probe, QFI about both rotation-phase and magnetic-field estimation exhibit a quadratic dependence on the probe-spin number. It is found that even when the spin-ensemble is prepared as a finite-temperature state, faraway from the so-called resource states, e.g., the squeezed state or the Greenberger-Horne-Zeilinger state, QFI can still manifest a Heisenberg scaling behavior.

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We study in this work the metrology precision of a probe system coupled to an ancillary qubit. Restricting the probe-qubit coupling along only one or two directions is found to be a sufficient criterion for the effective dynamical generator to achieve the Heisenberg limit in precision. Under the criterion, the quantum Fisher information (QFI) about the to-be-estimated parameter becomes the expectation value of mean square of the effective generator with respect to the initial state of the composite system. Our criterion is justified in two distinct systems. For a bosonic probe, QFI about the displacement estimation is found to be proportional to the mean excitation number of probe. It renders a counterintuitive result that quantum metrology sensitivity can be enhanced by increasing the temperature of the probe system. For a spin-ensemble probe, QFI about both rotation-phase and magnetic-field estimation exhibit a quadratic dependence on the probe-spin number. It is found that even when the spin-ensemble is prepared as a finite-temperature state, faraway from the so-called resource states, e.g., the squeezed state or the Greenberger-Horne-Zeilinger state, QFI can still manifest a Heisenberg scaling behavior.
We propose a quantum metrology protocol based on a two-step joint evolution of the probe system and an ancillary qubit and quantum measurement. With a proper initial state of the ancillary qubit and an optimized evolution time, the quantum Fisher information (QFI) about the phase parameter encoded in the probe system is found to be determined by the expectation value of the square of a phase generator, irrespective of the probe initial state. Consequently, even if the probe is prepared as a finite-temperature state, faraway from the so-called resource state, e.g., the squeezed spin state or the Greenberger-Horne-Zeilinger state in atomic systems, the QFI in our protocol can approach the Heisenberg scaling N2{N}^{2}N2 with respect to the probe size NNN. This quadratic scaling behavior shows robustness against the imperfections about the initial state of the ancillary qubit and the optimized evolution time of the whole system. Also it is not sensitive to the deviation in system parameters and the qubit decoherence. Using the time-reversal strategy and a single-shot projective measurement, the classical Fisher information (CFI) in our metrology protocol is saturated with its quantum counterpart. Our work thus provides an economical method to reach the Heisenberg limit in metrology precision with no input of entanglement or squeezing.
We propose a measurement-based quantum metrology protocol in a composite model, where the probe system (a spin ensemble) is coupled to an ancillary two-level system (qubit) with a general Heisenberg XXZ interaction. With optimized probe-ancilla coupling strengths and duration of joint evolution, the two parallel evolution paths of the probe system induced by the unconditional measurement on qubit can transform an eigenstate of the collective angular momentum operator of spin ensemble into a two-component state with a large distance in eigenspace. The quantum Fisher information about the phase encoded in the probe system of polarized states or their superposition, that could be relaxed to mixed states, can therefore manifest an exact or asymptotic quadratic scaling with respect to the probe size (spin number) NNN. The quadratic scaling behavior is found to be insensitive to the imperfect encoding operator, polarized direction of probe, and coupling strength. The phase sensitivity can approach the Heisenberg limit by virtue of the parity detection on either ancillary qubit or probe system. This work justifies that the unconditional measurement on a weakly coupled qubit could be an efficient resource to replace Greenberger-Horne-Zeilinger-type states and squeezing Hamiltonian for exceeding the standard quantum limit in metrology precision.
In the scenario of the probe-ancilla interaction, we propose a quantum metrology protocol by the unconditional measurement on the ancillary qubit after an optimized period of joint evolution from a product state. Its key element is the construction of two parallel evolution paths by the measurement that can transform the probe system (a spin ensemble) from an eigenstate of a collective angular momentum operator $|j,m\ensuremath{\rangle}$ to a superposed state $(|j,m\ensuremath{\rangle}+|j,\ensuremath{-}m\ensuremath{\rangle})/\sqrt{2}$. With synchronous parametric encoding and qubit measurement, the quantum Fisher information about the phase encoded in the probe system with optimized initial states can exactly attain the Heisenberg scaling N2{N}^{2}N2 with respect to the probe size (spin number) NNN. The quadratic scaling behavior is not sensitive to the imprecise control over the joint evolution time, the time delay between encoding and measurement, and the coherence in the probe ensemble or the ancillary system that would be degraded by local dephasing. The classical Fisher information of the spin ensemble is found to saturate with its quantum counterpart, irrespective of the idle joint evolution after the parametric encoding. We suggest that both Greenberger-Horne-Zeilinger (GHZ)-like states and nonlinear Hamiltonian are not necessary resources for exceeding the standard quantum limit in metrology precision since in our protocol even thermal states can hold an asymptotic quadratic scaling.
The quantum Fisher information of a quantum state with respect to a certain parameter quantifies the sensitivity of the quantum state to changes in that parameter. Maximizing the quantum Fisher information is essential for achieving the optimal estimation precision of quantum sensors. A typical quantum sensor involves a qubit(e.g. a spin-1/2) probe undergoing an unknown rotation, here the unknown rotation angle is the parameter to be estimated. A well known limitation is that if the rotation axis is unknown, the maximal quantum Fisher information is impossible to attain. This limitation has been lifted recently by leveraging entanglement between the probe qubit and an ancilla qubit. Namely, through measurement of the ancilla after the axis is revealed, one can prepare the probe that is optimal for any unknown rotation axis. This proposal, however, works only for a spin-1/2. Considering large spin probes can achieve a larger quantum Fisher information, offering enhanced metrological advantage, we here utilize the entanglement between a large spin probe and an ancilla to achieve optimal quantum Fisher information for estimating the rotation angle, without prior knowledge of the rotation axis. Different from the previous spin-1/2 case, achieving the optimal precision with large spins generally requires post-selection, resulting in a success probability dependent on the dimension of the Hilbert space. Furthermore, we extend the encoding state from the maximally entangled case to general entangled states, showing that optimal metrology can still be achieved with a certain success probability.
Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-12\frac{1}{2}21​ particle and compare the results for estimating the amplitude and the direction of the magnetic field.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.