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We consider the temporal correlations of the quantum state of a qubit subjectto simultaneous continuous measurement of two non-commuting qubit observables.Such qubit state correlators are defined for an ensemble of qubit trajectories,which has the same fixed initial state and can also be optionally constrainedby a fixed final state. We develop a stochastic path integral description forthe continuous quantum measurement and use it to calculate the consideredcorrelators. Exact analytic results are possible in the case of idealmeasurements of equal strength and are also shown to agree with solutionsobtained using the Fokker-Planck equation. For a more general case withdecoherence effects and inefficiency, we use a diagrammatic approach to findthe correlators perturbatively in the quantum efficiency. We also calculate thestate correlators for the quantum trajectories which are extracted from readoutsignals measured in a transmon qubit experiment, by means of the quantumBayesian state update. We find an excellent agreement between the correlatorsbased on the experimental data and those obtained from our analytical andnumerical results.

Simultaneous continuous measurement of non-commuting observables: quantum state correlations