Single-shot quantum state estimation via a continuous measurement in the strong backaction regime

Abstract

We study quantum tomography based on a stochastic continuous-time measurement record obtained from a probe field collectively interacting with an ensemble of identically prepared systems. In comparison to previous studies, we consider here the case in which the measurement-induced backaction has a nonnegligible effect on the dynamical evolution of the ensemble. We formulate a maximum likelihood estimate for the initial quantum state given only a single instance of the continuous diffusive measurement record. We apply our estimator to the simplest problem -state tomography of a single pure qubit, which, during the course of the measurement, is also subjected to dynamical control. We identify a regime where the many-body system is well approximated at all times by a separable pure spin coherent state, whose Bloch vector undergoes a conditional stochastic evolution. We simulate the results of our estimator and show that we can achieve close to the upper bound of fidelity set by the optimal POVM. This estimate is compared to, and significantly outperforms, an equivalent estimator that ignores measurement backaction.

Qubit models of weak continuous measurements: markovian conditional and open-system dynamics

“…General choices for ρ 0 will be incorrect, invalidating some of the properties described above. In particular, the innovation will deviate from a zero-mean random variable, and these deviations observed for a variety of guesses for ρ 0 will yield likelihood ratios that can be used to estimate the state, as was done in [33]. One can also keep track of the trace of the unnormalized state (3.19), which encodes the relative likelihood of the trajectory given the evolution parameters, allowing one to judge different parameter values against one another, as was implemented in [28].…”

Novelty, efficacy, and significance of weak measurements for quantum tomography

“…For example, when doing tomography on ensembles of atoms, weak collective measurements might be compared with nonadaptive separable projective measurements [15,20].…”

Tomography of an optomechanical oscillator via parametrically amplified position measurement

“…If ρ 0 is the initial state of the system, we find a solution having the general form  is a Gaussian completely positive map that depends on only two real-valued functions of the measurement record, X est and Y est as defined in equation (19). Z is a functional of the measurement currents I X and I Y that determines a numerical prefactor in (A.7).…”

“…Although, at any given moment, such a measurement reads out only the position of the oscillator, by monitoring the position over time it is possible to estimate all quadrature observables of the initial state and thereby obtain the required quorum of observables. There is significant prior work on using continuous measurements and time dynamics in this way to perform quantum tomography, particularly the early work by Deutsch and Jessen and collaborators which introduced the idea of continuous measurement tomography in the context of reconstructing the hyperfine state of an ensemble of atoms theoretically [17][18][19] and implemented it in practice [20,21]. To model our measurements, we will apply the standard tools of the trajectory theory of continuous quantum measurements [22] and use it to determine the initial quantum state at the beginning of a measurement.…”