We analyze the estimation of a time-dependent perturbation acting on a continuously monitored quantum system. We describe the temporal fluctuations of the perturbation by a hidden Markov model, and we combine quantum measurement theory and classical filter theory into a time-evolving hybrid quantum and classical trajectory. The forward-backward analysis that permits smoothed estimates of classical hidden Markov models has a counterpart in the theory of retrodiction and past quantum states. As a specific example, we apply our hybrid trajectory and past-quantum-state theory to the sensing of a fluctuating magnetic field by microwave interrogation of a single quantum spin.
We analyze the estimation of a time-dependent perturbation acting on a continuously monitored quantum system. We describe the temporal fluctuations of the perturbation by a hidden Markov model, and we combine quantum measurement theory and classical filter theory into a time-evolving hybrid quantum and classical trajectory. The forward-backward analysis that permits smoothed estimates of classical hidden Markov models has a counterpart in the theory of retrodiction and past quantum states. As a specific example, we apply our hybrid trajectory and past-quantum-state theory to the sensing of a fluctuating magnetic field by microwave interrogation of a single quantum spin.
Smoothing is an estimation technique that takes into account both past and future observations and can be more accurate than filtering alone. In this Letter, a quantum theory of smoothing is constructed using a time-symmetric formalism, thereby generalizing prior work on classical and quantum filtering, retrodiction, and smoothing. The proposed theory solves the important problem of optimally estimating classical Markov processes coupled to a quantum system under continuous measurements, and is thus expected to find major applications in future quantum sensing systems, such as gravitational wave detectors and atomic magnetometers.
We analyze the problem of estimating past quantum states of a monitored system from a mathematical perspective in order to ensure self-consistency with the principle of quantum nondemolition. Despite several claims of “measuring noncommuting observables” in the physics literature, we show that we are always measuring commuting processes. Our main interest is in the notion of quantum smoothing, or retrodiction. In particular, we examine proposals to estimate the result of an external measurement made on an open quantum systems during a period where it is also undergoing continuous monitoring. A full analysis shows that the nondemolition principle is in place, and so a well-posed statistical inference problem can be formulated. We extend the formalism to consider multiple independent external measurements made on the system over the course of a continual period of monitoring.
We consider a quantum system continuously monitored in time which in turn is coupled to an arbitrary dissipative classical system (diagonal reduced density matrix). The quantum and classical dynamics can modify each other, being described by an arbitrary time-irreversible hybrid Lindblad equation. Given a measurement trajectory, a conditional bipartite stochastic state can be inferred by taking into account all previous recording information (filtering). Here, we demonstrate that the joint quantum-classical state can also be inferred by taking into account both past and future measurement results (smoothing). The smoothed hybrid state is estimated without involving information from unobserved measurement channels. Its average over recording realizations recovers the joint time-irreversible behavior. As an application we consider a fluorescent system monitored by an inefficient photon detector. This feature is taken into account through a fictitious classical two-level system. The average purity of the smoothed quantum state increases over that of the (mixed) state obtained from the standard quantum jump approach.
State smoothing is a technique to estimate a state at a particular time, conditioned on information obtained both before (past) and after (future) that time.For a classical system, the smoothed state is a normalized product of the filtered state (a state conditioned only on the past measurement information and the initial preparation) and the retrofiltered effect (depending only on the future measurement information).For the quantum case, while there are well-established analogues of the filtered state (ρ F ) and the retrofiltered effect ( ÊR ), their product does not, in general, provide a valid quantum state for smoothing.However, this procedure does seem to work when ρ F and ÊR are mutually diagonalizable.This fact has been used to obtain smoothed quantum states-purer than the filtered states-in a number of experiments on continuously monitored quantum systems, in cavity QED and atomic systems.In this paper we show that there is an implicit assumption underlying this technique: that if all the information were known to the observer, the true system state would be one of the diagonal basis states.This assumption does not necessarily hold, as the missing information is quantum information.It could be known to the observer only if it were turned into a classical measurement record, but then its nature would depend on the choice of measurement.We show by a simple model that, depending on that measurement choice, the smoothed quantum state can: agree with that from the classical method, disagree with it but still be co-diagonal with it, or not even be co-diagonal with it.That is, just because filtering and retrofiltering appear classical does not mean classical smoothing theory is applicable in quantum experiments.
A density matrix ρ(t) yields probabilistic information about the outcome of measurements on a quantum system. We introduce here the past quantum state, which, at time T, accounts for the state of a quantum system at earlier times t<T. The past quantum state Ξ(t) is composed of two objects, ρ(t) and E(t), conditioned on the dynamics and the probing of the system until t and in the time interval [t, T], respectively. The past quantum state is characterized by its ability to make better predictions for the unknown outcome of any measurement at t than the conventional quantum state at that time. On the one hand, our formalism shows how smoothing procedures for estimation of past classical signals by a quantum probe [M. Tsang, Phys. Rev. Lett. 102, 250403 (2009)] apply also to describe the past state of the quantum system itself. On the other hand, it generalizes theories of pre- and postselected quantum states [Y. Aharonov and L. Vaidman, J. Phys. A 24, 2315 (1991)] to systems subject to any quantum measurement scenario, any coherent evolution, and any Markovian dissipation processes.
Smoothing is an estimation method whereby a classical state (probability distribution for classical variables) at a given time is conditioned on all-time (both earlier and later) observations. Here we define a smoothed quantum state for a partially monitored open quantum system, conditioned on an all-time monitoring-derived record. We calculate the smoothed distribution for a hypothetical unobserved record which, when added to the real record, would complete the monitoring, yielding a pure-state "quantum trajectory." Averaging the pure state over this smoothed distribution yields the (mixed) smoothed quantum state. We study how the choice of actual unraveling affects the purity increase over that of the conventional (filtered) state conditioned only on the past record.