The accuracy and precision of Cd isotope measurement at different injecting sample sizes.

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...

Abstract-This note describes the stability problems of uncertain systems with arbitrarily time-varying and severe time-delay. Using new Lyapunov-Krasovskii functionals, less conservative stability conditions are obtained for such systems. The results are illustrated using the numerical examples based on simple linear matrix inequalities.

We use a near quantum limited detector to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit. For each measured quantum coherent trajectory, we separately identify energy changes of the qubit as heat and work, and verify the first law of thermodynamics for an open quantum system. We further establish the consistency of these results by comparison with the master equation approach and the two-projective-measurement scheme, both for open and closed dynamics, with the help of a quantum feedback loop that compensates for the exchanged heat and effectively isolates the qubit. arXiv:1703.05885v3 [quant-ph]