Abstract Phase transitions represent a compelling tool for classical and quantum sensing applications. It has been demonstrated that quantum sensors can in principle saturate the Heisenberg scaling, the ultimate precision bound allowed by quantum mechanics, in the limit of large probe number and long measurement time. Due to the critical slowing down, the protocol duration time is of utmost relevance in critical quantum metrology. However, how the long-time limit is reached remains in general an open question. So far, only two dichotomic approaches have been considered, based on either static or dynamical properties of critical quantum systems. Here, we provide a comprehensive analysis of the scaling of the quantum Fisher information for different families of protocols that create a continuous connection between static and dynamical approaches. In particular, we consider fully-connected models, a broad class of quantum critical systems of high experimental relevance. Our analysis unveils the existence of universal precision-scaling regimes. These regimes remain valid even for finite-time protocols and finite-size systems. We also frame these results in a general theoretical perspective, by deriving a precision bound for arbitrary time-dependent quadratic Hamiltonians.
Abstract Phase transitions represent a compelling tool for classical and quantum sensing applications. It has been demonstrated that quantum sensors can in principle saturate the Heisenberg scaling, the ultimate precision bound allowed by quantum mechanics, in the limit of large probe number and long measurement time. Due to the critical slowing down, the protocol duration time is of utmost relevance in critical quantum metrology. However, how the long-time limit is reached remains in general an open question. So far, only two dichotomic approaches have been considered, based on either static or dynamical properties of critical quantum systems. Here, we provide a comprehensive analysis of the scaling of the quantum Fisher information for different families of protocols that create a continuous connection between static and dynamical approaches. In particular, we consider fully-connected models, a broad class of quantum critical systems of high experimental relevance. Our analysis unveils the existence of universal precision-scaling regimes. These regimes remain valid even for finite-time protocols and finite-size systems. We also frame these results in a general theoretical perspective, by deriving a precision bound for arbitrary time-dependent quadratic Hamiltonians.
Phase transitions represent a compelling tool for classical and quantum sensing applications. It has been demonstrated that quantum sensors can in principle saturate the Heisenberg scaling, the ultimate precision bound allowed by quantum mechanics, in the limit of large probe number and long measurement time. Due to the critical slowing down, the protocol duration time is of utmost relevance in critical quantum metrology. However, how the long-time limit is reached remains in general an open question. So far, only two dichotomic approaches have been considered, based on either static or dynamical properties of critical quantum systems. Here, we provide a comprehensive analysis of the scaling of the quantum Fisher information for different families of protocols that create a continuous connection between static and dynamical approaches. In particular, we consider fully-connected models, a broad class of quantum critical systems of high experimental relevance. Our analysis unveils the existence of universal precision-scaling regimes. These regimes remain valid even for finite-time protocols and finite-size systems. We also frame these results in a general theoretical perspective, by deriving a precision bound for arbitrary time-dependent quadratic Hamiltonians.
Quantum metrology shows that by exploiting nonclassical resources it is possible to overcome the fundamental limit of precision found for classical parameter-estimation protocols. The scaling of the quantum Fisher information -- which provides an upper bound to the achievable precision -- with respect to the protocol duration is then of primarily importance to assess its performances. In classical protocols the quantum Fisher information scales linearly with time, while typical quantum-enhanced strategies achieve a quadratic (Heisenberg) or even higher-order polynomial scalings. Here we report a protocol that is capable of surpassing the polynomial scaling, and yields an exponential advantage. Such exponential advantage is achieved by approaching, but without crossing, the critical point of a quantum phase transition of a fully-connected model in the thermodynamic limit. The exponential advantage stems from the breakdown of the adiabatic condition close to a critical point. As we demonstrate, this exponential scaling is well captured by the new bound derived in arXiv:2110.04144, which in turn allows us to obtain approximate analytical expressions for the quantum Fisher information that agree with exact numerical simulations. In addition, we discuss the limitations to the exponential scaling when considering a finite-size system as well as its robustness against decoherence effects. Hence, our findings unveil a novel quantum metrological protocol whose precision scaling goes beyond the paradigmatic Heisenberg limit with respect to the protocol duration.
Critical quantum metrology aims to harness critical properties near quantum phase transitions to enhance parameter estimation precision. However, critical slowing down inherently limits the achievable precision within a finite evolution time. To address this challenge, we establish a fundamental scaling limit of critical quantum metrology with respect to the total evolution time. We find that the effective winding number of the system's phase-space trajectory determines the scaling bound of quantum Fisher information. Furthermore, we demonstrate that the exponential scaling of the quantum Fisher information can be obtained, and for this, it is necessary to increase the effective winding number by the total evolution time. Based on the topological nature of the effective winding number, we explicitly derive an optimal time-dependent control whose form is, surprisingly, a simple on-off scheme. We highlight that such an exponential scaling of quantum Fisher information remains valid even without reaching the critical point and in the presence of thermal dissipation, albeit with a decreased exponent.
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4 and T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.
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 quench dynamics of an open quantum system involving a quantum phase transition. In the isolated case, the quench dynamics involving the phase transition exhibits a number of scaling relations with the quench rate as predicted by the celebrated Kibble-Zurek mechanism. In contact with an environment however, these scaling laws break down and one may observe an anti-Kibble-Zurek behavior: slower ramps lead to less adiabatic dynamics, increasing thus nonadiabatic effects with the quench time. In contrast to previous works, we show here that such anti-Kibble-Zurek scaling can acquire a universal form in the sense that it is determined by the equilibrium critical exponents of the phase transition, provided the excited states of the system exhibit singular behavior, as observed in fully connected models. This demonstrates novel universal scaling laws granted by a system-environment interaction in a critical system. We illustrate these findings in two fully connected models, namely, the quantum Rabi and the Lipkin-Meshkov-Glick models. In addition, we discuss the impact of nonlinear ramps and finite-size systems.