Markus Plautz: Sudden irretrievable loss of a quantum trajectory

In each step of a quantum trajectory, a system undergoes unitary evolution followed by a measurement update. Inevitably, realistic measurements introduce some degree of information loss, leading to uncertainty in the resulting state. This work addresses the question: under what conditions is a quantum trajectory irretrievably lost to this uncertainty? We define a trajectory as irretrievably lost if even a quantum principal component algorithm cannot recover it from the mixed state. To explore this, we numerically study a model where the entanglement between measured and unmeasured qubits can be tuned from volume to area law. Our results reveal that the trajectory remains recoverable when entanglement production is limited. However, we observe a sudden loss of retrievability for extensive entanglement, manifesting as a sharp second-order phase transition. We construct an exactly solvable model using tools from random matrix theory and free probability to gain analytical insights. This model reproduces the same phenomenology, which allows us to study the nature of the phase transition (or its absence) in detail. These findings have implications for quantum feedback control and error correction. They highlight potential challenges posed in the limit of low measurement fidelities and the role of entanglement in preserving trajectory information.