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.
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.
We consider the estimation of parameters encoded in the measurement record of a continuously monitored quantum system in the jump unraveling, corresponding to a single-shot scenario, where information is continuously gathered. Here, it is generally difficult to assess the precision of the estimation procedure via the Fisher Information due to intricate temporal correlations and memory effects. In this paper we provide a full set of solutions to this problem. First, for multi-channel renewal processes we relate the Fisher Information to an underlying Markov chain and derive a easily computable expression for it. For non-renewal processes, we introduce a new algorithm that combines two methods: the monitoring operator method for metrology and the Gillespie algorithm which allows for efficient sampling of a stochastic form of the Fisher Information along individual quantum trajectories. We show that this stochastic Fisher Information satisfies useful properties related to estimation on a single run. Finally, we consider the case where some information is lost in data compression/post-selection and provide tools for computing the Fisher Information in this case. All scenarios are illustrated with instructive examples from quantum optics and condensed matter.
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.
The evolution of two-level quantum systems (qubits) in the presence of continuous-time weak measurements can be described by stochastic differential equations, which is dominated by the Hamiltonian and decoherence channels. In more concrete terms, the conventional density matrix evolves according to the stochastic master equation in the Schrödinger picture. Particularly, for a monitored qubit, the deterministic submanifolds concerning the dynamical evolution of the density matrix has been previously obtained. As a complement, We have recently developed deterministic submanifolds for the backward-evolving quantum state for a monitored qubit. In this paper, in order to better understand the behavior of a monitored qubit, by using the Bloch sphere representation and the properties of Pauli matrices, we provide a closed-form expression for probability distribution over the deterministic submanifolds of the backward evolving dynamics in different scenarios. We further analyze the backward-evolving trajectories of the quantum state by taking into account the analytic solutions and the corresponding submanifolds under heterodyne/homodyne detection associated with non-demolition or fluorescence measurements.
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.
Simulation of nonlinear stochastic master equations generally requires trajectory-level evolution of large Hilbert-space density matrices, making strongly nonlinear continuously monitored systems computationally challenging. Here we identify an exactly solvable class centered on polynomial Hamiltonians of commuting quadratures with linear damping and homodyne measurement of those same quadratures. For initial states with a Gaussian measured-quadrature marginal and polynomial conditional-momentum sections, we derive an exact finite stochastic representation through any fixed momentum order. The result extends to interacting multimode systems, arbitrary measurement efficiency, and finite thermal occupation. Its computational cost is linear in trajectory length and independent of Hilbert-space dimension, requiring no Fock-space truncation, Gaussian approximation, or approximate moment closure. As an application, we derive an exact causal filter for monitored cubic-phase gates that uses the homodyne record to determine the variance-minimizing momentum displacement. Numerical simulations validate the representation against direct stochastic-master-equation evolution, demonstrate substantial speedups over converged Fock-space propagation, and show that this record-conditioned recentering can efficiently recover nonlinear squeezing otherwise lost through trajectory averaging.
Generalized master equations (GMEs)---time-local but generally neither trace-preserving nor Hermiticity-preserving---are convenient tools to compute properties of the environment of an open or continuously monitored quantum system. A two-sided master equation yields the fidelity and quantum Fisher information (QFI) of environment states, thereby setting fundamental limits for hypothesis testing and parameter estimation under continuous monitoring. For unmonitored noise or inefficient detection, the QFI of the detectable part of the environment may be obtained from a recently derived GME acting on multiple system replicas. Tilted master equations provide the full counting statistics of quantum jumps and diffusive measurements, enabling, e.g., studies of quantum thermodynamics beyond average values. Here we focus on bosonic linear systems, governed by a quadratic Hamiltonian and linear jump operators, whose dynamics preserves Gaussianity. For Gaussian initial operators, we recast a generic GME as a compact set of ordinary differential equations for the covariance matrix (a Riccati-type equation), first moments, and normalization. These equations can be integrated efficiently without Hilbert-space truncation, and admit analytical results in simple settings. We also provide specialized forms for fidelity/QFI and full counting statistics. We illustrate the formalism with a continuously monitored optical parametric oscillator, using it to determine sensitivity limits for frequency estimation and to benchmark Hasegawa's thermodynamic uncertainty relations.