Going beyond the standard quantum limit in noisy quantum metrology is an important and challenging task. Here we show how Dicke states can be used to surpass the standard quantum limit and achieve the Heisenberg limit in open quantum systems. The system we study has qubits symmetrically coupled to a resonator, and our objective is to estimate the coupling between the qubits and the resonator. The time-dependent quantum Fisher information with respect to the coupling is studied for this open quantum system where the same decay rates are assumed on all qubits. We show that when the system is initialized to a Dicke state with an optimal excitation number one can go beyond the standard quantum limit and achieve the Heisenberg limit even for finite values of the decays on the qubit and the resonator, particularly when the qubits and resonator are strongly coupled. We compare our results against the highly entangled GHZ state and a completely separable state and show that the GHZ state performs quite poorly, whereas under certain noise conditions the separable state is able to go beyond the standard quantum limit due to subsequent interactions with a resonator.
Going beyond the standard quantum limit in noisy quantum metrology is an important and challenging task. Here we show how Dicke states can be used to surpass the standard quantum limit and achieve the Heisenberg limit in open quantum systems. The system we study has qubits symmetrically coupled to a resonator, and our objective is to estimate the coupling between the qubits and the resonator. The time-dependent quantum Fisher information with respect to the coupling is studied for this open quantum system where the same decay rates are assumed on all qubits. We show that when the system is initialized to a Dicke state with an optimal excitation number one can go beyond the standard quantum limit and achieve the Heisenberg limit even for finite values of the decays on the qubit and the resonator, particularly when the qubits and resonator are strongly coupled. We compare our results against the highly entangled GHZ state and a completely separable state and show that the GHZ state performs quite poorly, whereas under certain noise conditions the separable state is able to go beyond the standard quantum limit due to subsequent interactions with a resonator.
Entanglement is generally considered necessary for achieving the Heisenberg limit in quantum metrology. We construct analogues of Dicke and GHZ states on a single N+1 dimensional qudit that achieve precision equivalent to symmetrically entangled states on N qubits, showing that entanglement is not necessary for going beyond the standard quantum limit. We define a measure of non-classicality based on quantum Fisher information and estimate the achievable precision, suggesting a close relationship between non-classical states and metrological power of qudits. Our work offers an exponential reduction in the physical resources required for quantum-enhanced parameter estimation, making it accessible on any quantum system with a high-dimensional Hilbert space.
We simulate the dynamics of systems with N = 1-20 qubits coupled to a cavity in order to assess their potential for quantum metrology of a parameter in the open systems limit. The qubits and the cavity are both allowed to have losses and the system is studied under various coupling strength regimes. The focus is primarily on the coupling between the qubits using the quantum Fisher information as the measured parameter. Some results on estimating the qubit-cavity detuning parameter are also presented. We investigate the scaling of the uncertainty in the estimate of the qubit-cavity coupling with the number of qubits and for different initial states of the qubits that act as the quantum probe. As initial probe states, we consider Dicke states with varying excitation numbers, the GHZ state, and separable X-polarized states. It is shown that in the strong coupling regime, i.e., when the coupling between the qubits and the cavity is greater than the decay parameters of both the qubits and the cavity, Dicke states with a large excitation number can achieve the Heisenberg limit, with the precision scaling improving as the excitation number increases. A particularly intriguing finding of our study is that in the weak coupling regime, as well as in situations where either the qubit or cavity decay parameters exceed the coupling, the separable X-polarized state is the best in terms of scaling and is even able to achieve the Heisenberg limit in these lossy regimes for the range of N considered.
Phase sensing with entangled multi-qubit states in the presence of noise is a central theme of modern quantum metrology. The present work investigates Dicke state superposition probes for quantum phase sensing under parameter encoding generated by one- and two-body interaction Hamiltonians. Under linear collective-spin encoding, near-optimal Dicke superposition states are shown to exhibit significantly enhanced robustness against phase damping noise compared with Greenberger--Horne--Zeilinger (GHZ), W-superposition, and balanced Dicke states, while maintaining favorable metrological performance under realistic decoherence channels. For two-body interactions, optimal probe states maximizing the quantum Fisher information are identified. Their noise resilience and metrological scaling behaviour under phase damping, amplitude damping, and global depolarizing channels are analyzed. The associated near-optimal Dicke superposition states are found to exhibit improved resilience to phase damping, for the system sizes considered. These results establish tailored near-optimal Dicke state superposition probes as versatile and noise-resilient resources for Heisenberg and super-Heisenberg quantum phase sensing governed by one- and two-body interactions.
Quantum effects in metrology can in principle enhance measurement precision from the so-called standard quantum limit to the Heisenberg limit. Further advancements in quantum metrology largely rely on innovative metrology protocols that can avoid a number of known obstacles, including the challenge of preparing entangled states with sufficient fidelity, the readout noise in measuring highly entangled states, and no-go theorems for quantum metrology under noisy environments. In this Letter, exploiting some peculiar but experimentally feasible dynamical features of a collection of spins with all-to-all time-periodic interactions, we propose a metrology protocol that can circumvent all three mentioned obstacles and yet still make good use of time as a resource for metrology. Specifically, by mapping the dynamics of such a periodically driven spin system to that of a paradigm of quantum chaos but tuned to some high-order quantum resonance, it is shown that a simple SU(2) coherent state can, after evolving to highly entangled states in the ensuing dynamics, be dynamically brought back to the same initial coherent state. The associated quantum Fisher information is found to exhibit quadratic scaling with both the number of spins and the duration of the metrology protocol. The achieved Heisenberg scaling can also largely survive in the presence of Markovian noise. Representing a previously unknown strategy for quantum metrology, the protocol proposed here can be tested on available experimental platforms.
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 summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.