Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit

Abstract

The quantum state of a superconducting transmon qubit inside a three-dimensional cavity is monitored by transmission of a microwave field through the cavity. The information inferred from the measurement record is incorporated in a density matrix ρt, which is conditioned on probe results until t, and in an auxiliary matrix Et, which is conditioned on probe results obtained after t. Here, we obtain these matrices from experimental data and we illustrate their application to predict and retrodict the outcome of weak and strong qubit measurements.In quantum mechanics, predictions about the outcome of experiments are given by Born's rule which for a state vector |ψ i provides the probability P (a) = | a|ψ i | 2 that a measurement of an observable with eigenstates |a yields one of the eigenvalues a. As a consequence of the measurement, the quantum state is projected into the state |a . Yet, after this measurement, further probing of the system is possible, and the probability that the quantum system yields outcome a and is subsequently detected in a final state |ψ f factors into the product | ψ f |a | 2 | a|ψ i | 2 . Considering initial and final states raises the issue of post-selection in quantum measurements: What is the probability that the result of the measurement of was a, if we consider only the selected measurement events where the initial state was |ψ i and the final state was |ψ f ? The answer is known as the Aharonov-Bergmann-Lebowitz rule [1],and it differs from Born's rule, which takes into account only knowledge about the state prior to the measurement. While it is natural that full measurement records reveal more information about the state of a physical system at a given time t than data obtained only until that time, the interpretation of the time symmetric influences from the future and from the past measurement events on P ABL has stimulated some debate, see for example [1][2][3][4][5][6]. Meanwhile, probabilistic state assignments and correlations observed in atomic, optical and solid state experiments have been conveniently understood in relation to post-selection [7][8][9][10][11], and precision probing theories [12][13][14][15][16][17] have incorporated full measurement records.In this letter, we consider a superconducting qubit that is subject to continuous monitoring and driven unitary evolution. We make use of the full measurement record and examine how measurements before time t can be used to make predictions, while measurements after time t can be used to make retrodictions about measurements at time t. We then consider a recent generalization [2] of Eq.(1) to the case of continuously monitored and evolving mixed states. Our experiments verify the predictions of both projective and weak (weak value) measurements conditioned on full measurement records. These predictions are more confident and nontrivially different from predictions based only on the measurement record up to time t.To analyze non-pure states and partial measurements, we represent our system by a density matrix ρ, and me...

Past observable dynamics of a continuously monitored qubit

“…Notably, this last result improves upon a recent proposal [46] that constructs a "smoothed quantum state" to estimate the observations made by an unknown second observer, since that method can never outperform the most pure causal state that uses all collected data. The conclusions of our study are consistent with prior work concerned with time-symmetric quantum state estimates, such as the two-state-vector formalism [37,47,48], quantum smoothing [49][50][51][52], bidirectional quantum states [53], and past quantum states [54][55][56][57]. However, we emphasize here the practical consequence for ongoing research into continuous quantum measurements: Applying optimal classical signal processing techniques to a single realization of collected data from a continuous quantum measurement produces results that do not correspond to the causal quantum state.…”

Cramér-Rao bound for time-continuous measurements in linear Gaussian quantum systems

“…On the other hand, the information leaking into the environment can be in principle used for parameter estimation as well, in particular via time-continuous monitoring of the environment itself [5,6]. While several strategies based on time-continuous measurements and feedback have been proposed for quantum state engineering, in particular with the main goal of generating steady-state squeezing and entanglement [5,[7][8][9][10][11][12][13][14][15] or to study and exploit trajectories of superconducting qubits [16,17], less attention has been devoted to parameter estimation. Notable exceptions are the estimation of a magnetic field via a continuously monitored atomic ensemble [18], the tracking of a varying phase [19][20][21], the estimation of Hamiltonian and environmental parameters [22][23][24][25][26][27][28][29], and optimal state estimation for a cavity optomechanical system [30].…”

Probability distributions of continuous measurement results for conditioned quantum evolution

“…The average signals have been experimentally studied in [13,14,16]. The technical level of these experiments permits the characterization of the complete statistics of the measurement outputs.…”

“…Recent experimental advances have made possible the efficient continuous measurement and monitoring of elementary quantum systems (qubits) giving the information on individual quantum trajectories [10][11][12]. The individual traces of quantum evolution can be postselected by a projective measurement at the end of evolution, thus enabling the experimental investigation of conditioned quantum evolution where both initial and final states are known [13][14][15][16]. For experimentally relevant illustrations, we concentrate in this paper on a setup of resonance fluorescence [13].…”