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arXiv:2401.06729 · Physical Review A

Quantum sensing of even- versus odd-body interactions

Aparajita Bhattacharyya, Debarupa Saha, Ujjwal Sen

Published Sep 10, 2025• 4 citations
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Abstract

We analyze the scaling of quantum Fisher information with the number of system particles in the limit of large number of particles, as a function of the number of parties interacting with each other, for encoding Hamiltonians having arbitrary-body interactions. We find that estimation of coupling strength of such arbitrary-body encoding Hamiltonians provide a super-Heisenberg scaling that increases monotonically with an increase in the number of interacting particles, in the limit of large number of system particles. Moreover, we also find that the optimal probes corresponding to Hamiltonians that contain even-body interaction terms, may be entangled, but certainly not so in all bipartitions, and particularly, it is possible to attain optimal precision using asymmetric probes. Thereby we find a complementarity in the requirement of asymmetry and genuine entanglement in optimal probes for estimating strength of odd- and even-body interactions respectively. Additionally, we provide an upper bound on the number of parties up to which one can always obtain an asymmetric product state that gives the best metrological precision for even-body interactions. En route, we find the quantum Fisher information in closed form for two- and three-body interactions for arbitrary number of parties. We also provide an analysis of the case when the Hamiltonian contains local fields and up to k-body interaction terms, where the strength of interaction gradually decreases with an increase in the number of parties interacting with each other. Interestingly, we find a similar dichotomy in the nature of the optimal probe in this case as well. Further, we identify conditions on the local component of the Hamiltonian, for which this dichotomy is still shown to exist for two- and three-body encoding Hamiltonians with arbitrary local dimensions.

DOI
10.1103/3f8n-b1kp

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We analyze the scaling of quantum Fisher information with the number of system particles in the limit of large number of particles, as a function of the number of parties interacting with each other, for encoding Hamiltonians having arbitrary-body interactions. We find that estimation of coupling strength of such arbitrary-body encoding Hamiltonians provide a super-Heisenberg scaling that increases monotonically with an increase in the number of interacting particles, in the limit of large number of system particles. Moreover, we also find that the optimal probes corresponding to Hamiltonians that contain even-body interaction terms, may be entangled, but certainly not so in all bipartitions, and particularly, it is possible to attain optimal precision using asymmetric probes. Thereby we find a complementarity in the requirement of asymmetry and genuine entanglement in optimal probes for estimating strength of odd- and even-body interactions respectively. Additionally, we provide an upper bound on the number of parties up to which one can always obtain an asymmetric product state that gives the best metrological precision for even-body interactions. En route, we find the quantum Fisher information in closed form for two- and three-body interactions for arbitrary number of parties. We also provide an analysis of the case when the Hamiltonian contains local fields and up to k-body interaction terms, where the strength of interaction gradually decreases with an increase in the number of parties interacting with each other. Interestingly, we find a similar dichotomy in the nature of the optimal probe in this case as well. Further, we identify conditions on the local component of the Hamiltonian, for which this dichotomy is still shown to exist for two- and three-body encoding Hamiltonians with arbitrary local dimensions.
We consider the estimation of an unknown parameter θ via a many-body probe. The probe is initially prepared in a product state and many-body time-independent interactions enhance its θ sensitivity during the dynamics and/or in the steady state. We present bounds on the quantum Fisher information, and corresponding optimal interacting Hamiltonians, for two paradigmatic scenarios for encoding θ : (i) via unitary Hamiltonian dynamics (dynamical metrology), and (ii) in the Gibbs and diagonal ensembles (time-averaged dephased state), two ubiquitous steady states of many-body open dynamics. We then move to the specific problem of estimating the strength of a magnetic field via interacting spins and derive two-body interacting Hamiltonians that can approach the fundamental precision bounds. In this case, we additionally analyze the transient regime leading to the steady states and characterize trade-offs between equilibration times and measurement precision. Overall, our results provide a comprehensive picture of the potential of many-body control in quantum sensing.
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.
We investigate the sensing capacity of non-equilibrium dynamics in quantum systems exhibiting Bloch oscillations. By focusing on the resource efficiency of the probe, quantified by quantum Fisher information, we find different scaling behaviors in two different phases, namely localized and extended. Our results provide a quantitative ansatz for quantum Fisher information in terms of time, probe size, and the number of excitations. In the long-time regime, the quantum Fisher information is a quadratic function of time, touching the Heisenberg limit. The system size scaling drastically depends on the phase changing from quantum-enhanced scaling in the extended phase to size-independent behavior in the localized phase. Furthermore, increasing the number of excitations always enhances the precision of the probe, although, in the interacting systems the enhancement becomes less eminent than the non-interacting probes. This is due to the induced localization by increasing the interaction between the excitations. We show that a simple particle configuration measurement together with a maximum likelihood estimation can closely reach the ultimate precision limit in both single- and multi-particle probes.
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.
With optimal control theory, we compute the maximum possible quantum Fisher information about the interaction parameter for a Kitaev chain with tunable long-range interactions in the many-particle Hilbert space. We consider a wide class of decay laws for the long-range interaction and develop rigorous asymptotic analysis for the scaling of the quantum Fisher information with respect to the number of lattice sites. In quantum metrology nonlinear many-body interactions can enhance the precision of quantum parameter estimation to surpass the Heisenberg scaling, which is quadratic in the number of lattice sites. Here for the estimation of the long-range interaction strength, we observe the Heisenberg to super-Heisenberg transition in such a linearlinearlinear model, related to the slow decaying long-range correlations in the model. Finally, we show that quantum control is able to improve the prefactor rather than the scaling exponent of the quantum Fisher information. This is in contrast with the case where quantum control has been shown to improve the scaling of quantum Fisher information with the probe time. Our results clarify the role of quantum controls and long-range interactions in many-body quantum metrology.
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.