We discuss the problem of estimating a frequency via [Formula: see text]-qubit probes undergoing independent dephasing channels that can be continuously monitored via homodyne or photo-detection. We derive the corresponding analytical solutions for the conditional states, for generic initial states and for arbitrary efficiency of the continuous monitoring. For the detection strategies considered, we show that: (i) in the case of perfect continuous detection, the quantum Fisher information (QFI) of the conditional states is equal to the one obtained in the noiseless dynamics; (ii) for smaller detection efficiencies, the QFI of the conditional state is equal to the QFI of a state undergoing the (unconditional) dephasing dynamics, but with an effectively reduced noise parameter.
We discuss the problem of estimating a frequency via [Formula: see text]-qubit probes undergoing independent dephasing channels that can be continuously monitored via homodyne or photo-detection. We derive the corresponding analytical solutions for the conditional states, for generic initial states and for arbitrary efficiency of the continuous monitoring. For the detection strategies considered, we show that: (i) in the case of perfect continuous detection, the quantum Fisher information (QFI) of the conditional states is equal to the one obtained in the noiseless dynamics; (ii) for smaller detection efficiencies, the QFI of the conditional state is equal to the QFI of a state undergoing the (unconditional) dephasing dynamics, but with an effectively reduced noise parameter.
We study quantum frequency estimation for N qubits subjected to independent Markovian noise, via strategies based on time-continuous monitoring of the environment. Both physical intuition and an extended convexity property of the quantum Fisher information (QFI) suggest that these strategies are more effective than the standard ones based on the measurement of the unconditional state after the noisy evolution. Here we focus on initial GHZ states and on parallel or transverse noise. For parallel noise, i.e. dephasing, we show that perfectly efficient time-continuous photo-detection allows to recover the unitary (noiseless) QFI, and thus to obtain a Heisenberg scaling for every value of the monitoring time. For finite detection efficiency, one falls back to the noisy standard quantum limit scaling, but with a constant enhancement due to an effective reduced dephasing. Also in the transverse noise case we obtain that the Heisenberg scaling is recovered for perfectly efficient detectors, and we find that both homodyne and photo-detection based strategies are optimal. For finite detectors efficiency, our numerical simulations show that, as expected, an enhancement can be observed, but we cannot give any conclusive statement regarding the scaling. We finally describe in detail the stable and compact numerical algorithm that we have developed in order to evaluate the precision of such time-continuous estimation strategies, and that may find application in other quantum metrology schemes.
Frequency estimation, a cornerstone of quantum metrology, has been significantly enhanced by advanced quantum sensing strategies. However, most protocols rely either on static or time-independent encoding mechanisms, inherently limiting their achievable precision scaling, or on control strategies requiring changing the Hamiltonian and/or implementing feedback mechanisms. To overcome this, we investigate a simpler dynamical encoding protocol where the quantum oscillator is driven by a general continuous temporal frequency modulation Ω(t)=ω0f(t). We analytically demonstrate that for a given modulation profile f(t) and its corresponding time-integral F(t), the quantum Fisher information (QFI) scales as O(F(t)2). This enhancement stems from the fact that temporal encoding fundamentally alters the mechanism of dynamical phase accumulation. Crucially, when evaluated under the energy and evolution-time constraints, this framework reveals a genuine precision enhancement over the conventional time-independent baseline. By analyzing explicit polynomial and exponential modulations, we establish that arbitrary precision scaling can be deterministically engineered, with ultimate bounds that are asymptotically saturable via optimal homodyne detection. Our framework provides a universal paradigm for exploiting time-dependent quantum control in next-generation sensors.
We consider frequency estimation in a noisy environment with noisy probes. This builds on previous studies, most of which assume that the initial probe state is pure, while the encoding process is noisy, or that the initial probe state is mixed, while the encoding process is noiseless. Our work is more representative of reality, where noise is unavoidable in both the initial state of the probe and the estimation process itself. We prepare the probe in a GHZ diagonal state, starting from n + 1 qubits in an arbitrary uncorrelated mixed state, and subject it to parameter encoding under dephasing noise. For this scheme, we derive a simple formula for the (quantum and classical) Fisher information, and show that quantum enhancements do not depend on the initial mixedness of the qubits. That is, we show that the so-called 'Zeno' scaling is attainable when the noise present in the encoding process is time inhomogeneous. This scaling does not depend on the mixedness of the initial probe state, and it is retained even for highly mixed states that can never be entangled. We then show that the sensitivity of the probe in our protocol is invariant under permutations of qubits, and monotonic in purity of the initial state of the probe. Finally, we discuss two limiting cases, where purity is either distributed evenly among the probes or concentrated in a single probe.
Continuous weak measurements of quantum systems are of great relevance in quantum foundations and applications. They can be achieved by probing the quantum system of interest by repeatedly measuring an auxiliary system weakly coupled to it. Here we study two qubits coupled in different points to a common one-dimensional electromagnetic field and simultaneously monitored in the right- and left-propagating output channels by homodyne detection. Using a collision-model description, we derive an analytical Stochastic Master Equation (SME) governing the resulting diffusive quantum trajectories, including the interference between the two measurement channels. For nonlinear functions of the quantum state, such as entropies, averages over quantum trajectories generally differ from the corresponding quantities evaluated on the unconditional state. Through this mechanism, we show that continuous monitoring generates entanglement, absent in the unconditional dynamics, and enhances quantum magic in the qubit pair during the decay. Both resources can be tuned through the optical phase accumulated between the qubits and the phases of homodyne local oscillators. Our results establish continuous homodyne monitoring of multiple emitters as a tunable mechanism for generating quantum resources.
We derive an alternative formulation of the stochastic master equation describing the evolution of quantum systems continuously measured via homodyne detection in a squeezed-thermal environment. This formulation allows us to identify two distinct collapse operators, corresponding respectively to the monitored and unmonitored channels, and to express the quantum state evolution directly in terms of Kraus-like operators. As a consequence, it facilitates the application of advanced numerical methods that account for higher-order corrections and discrete-time monitoring, and it also enables the efficient numerical evaluation of Cramér-Rao bounds in parameter estimation problems. To demonstrate the latter, we evaluate the Fisher information of the continuous homodyne monitoring for qubit frequency estimation in the presence of a squeezed-thermal bath. Our results show that exponential scaling of the steady-state Fisher information rate with squeezing is retained even at non-zero temperature.
Continuous monitoring is a powerful tool for analyzing quantum systems and is increasingly used as a non-demolition technique in quantum metrology. In this context, noisy data acquired through continuous monitoring can be used to precisely pinpoint unknown parameters of the system. However, extending the theoretical framework that connects these data to the underlying parameters beyond Markovian dynamics is notoriously difficult, primarily because such dynamics cannot be expressed as completely positive divisible maps. To overcome this, we propose a method based on the reaction coordinate mapping to extend these parameter-estimation techniques beyond the Markovian regime. Our approach specifically targets linear systems with non-Markovian dynamics undergoing Gaussian continuous measurements, such as homodyne detection. Within this framework, we analyze Bayesian estimation and provide an analytical expression for the Fisher information, alongside the asymptotic scaling of the estimation precision for arbitrary parameters. Finally, we demonstrate the efficacy of our method through the example of thermometry of a bosonic bath.