Completely Positive Quantum Trajectories with Applications to Quantum State Smoothing

Abstract

Here, we are concerned with comparing estimation schemes for the quantum state under continuous measurement (quantum trajectories), namely quantum state filtering and, as introduced by us [Phys. Rev. Lett. 115, 180407 (2015)], quantum state smoothing. Unfortunately, the cumulative errors in the most typical simulations of quantum trajectories with a total time of simulation $T$ can reach orders of $T\Delta t$. Moreover, these errors may correspond to deviations from valid quantum evolution as described by a completely positive map. Here we introduce a higher-order method that reduces the cumulative errors in the complete positivity of the evolution to of order $T\Delta t^2$, whether for linear (unnormalised) or nonlinear (normalised) quantum trajectories. Our method also guarantees that the discrepancy in the average evolution between different detection methods (different `unravellings', such as quantum jumps or quantum diffusion) is similarly small. This equivalence is essential for comparing quantum state filtering to quantum state smoothing, as the latter assumes that all irreversible evolution is unravelled, although the estimator only has direct knowledge of some records. In particular, here we compare, for the first time, the average difference between filtering and smoothing conditioned on an event of which the estimator lacks direct knowledge: a photon detection within a certain time window. We find that the smoothed state is actually {\em less pure}, both before and after the time of the jump. Similarly, the fidelity of the smoothed state with the `true' (maximal knowledge) state is also lower than that of the filtered state before the jump. However, after the jump, the fidelity of the smoothed state is higher.

arXiv comment14 pages, 10 figures

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• Entanglement of a pair of quantum emitters via continuous fluorescence measurements: a tutorial
• Generalized Rate Operator Quantum Jumps via Realization-Dependent Transformations
• Time-delayed quantum feedback and incomplete decoherence suppression with no-knowledge measurement
• Quantum state smoothing cannot be assumed classical even when the filtering and retrofiltering are classical
• Quantum state-preparation control in noisy environment via most-likely paths
• Completely positive trace-preserving maps for higher-order unraveling of Lindblad master equations
• Quantum state smoothing when Alice assumes the wrong type of monitoring by Bob