Abstract
One of the hallmarks of quantum mechanics is the impossibility of simultaneous measurement of non-commuting observables with projective measurements. This, however, can be circumvented by using continuous quantum measurements. Here we investigate the temporal correlations of the output signals of detectors continuously and simultaneously measuring the qubit observables σz and σz cosφ + σx sinφ, for various angles φ. Using the quantum Bayesian formalism, we obtain analytical expressions for the correlators, which we find to be in good agreement with those obtained from experimentally measured output signals. The agreement is particularly good for cross-correlators, even at times shorter than the cavity modes decay time. We further discuss how the correlators can be applied for parameter estimation, and use them to infer a small residual qubit Hamiltonian arising from calibration inaccuracy in the experimental data. Our work opens up new possibilities to perform quantum metrology based on temporal correlations of measured data.
Highlights
Continuous quantum measurements (CQMs) have become a unique platform to explore fundamental aspects of quantum phenomena and have potential applications to quantum information science
While a simultaneous measurement of non-commuting observables is impossible with instantaneous projective measurements, nothing theoretically forbids such a measurement using CQMs. (This is so because a CQM can be regarded as a series of infinitesimally weak quantum measurements, and partial measurements of non-commuting observables become commuting with each other in the limit of infinitesimally weak strength.) Aside from new physics, such a protocol may open up new areas of applications, inaccessible to projective measurements
We focus on the temporal correlations of the output signals from two linear detectors measuring continuously and simultaneously the qubit observables σz and σφ ≡ σz cosφ + σx sinφ, where σx and σz are the Pauli matrices and φ is an angle between the two measurement directions on the Bloch sphere (Fig. 1)
Summary
Continuous quantum measurements (CQMs) have become a unique platform to explore fundamental aspects of quantum phenomena and have potential applications to quantum information science They have been discussed theoretically in various contexts (e.g.,1–10), and in the past decade superconducting qubits have become the main experimental system for the realization of CQMs.[11,12,13,14,15,16,17] CQMs are shedding new light on our understanding of the quantum measurement process, and there is a growing interest in CQM applications, including quantum feedback,[13,16,18,19,20] rapid state purification,[21] preparation of entangled states,[22,23,24] and continuous quantum error correction.[25,26]. The theoretical discussion of a simultaneous measurement of incompatible observables has a long history.[27,28,29,30] For the measurement of noncommuting observables of a qubit, statistics of time-integrated detector outputs and fidelity of state monitoring directly via timeintegrated outputs has been analyzed in ref
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