27 Oct 2025
Abstract
Weak quantum measurements enable real-time tracking and control of dynamical quantum systems, producing quantum trajectories — evolutions of the quantum state of the system conditioned
on measurement outcomes. For classical systems, the accuracy of trajectories can be improved by
incorporating future information, a procedure known as smoothing. Here we apply this concept
to quantum systems, generalising a formalism of quantum state smoothing for an observer monitoring a quantum system exposed to environmental decoherence, a scenario important for many
quantum information protocols. This allows future data to be incorporated when reconstructing the
trajectories of quantum states. We experimentally demonstrate that smoothing improves accuracy
using a continuously measured nanomechanical resonator, showing that the method compensates for
both gaps in the measurement record and inaccessible environments. We further observe a key predicted departure from classical smoothing: quantum noise renders the trajectories nondifferentiable.
These results establish that future information can enhance quantum trajectory reconstruction, with
potential applications across quantum sensing, control, and error correction.
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