In this talk, we will mainly focus on noisy quantum trees: at each node of a tree, a received qubit unitarily interacts with fresh ancilla qubits, after which each qubit is sent through a noisy channel to a different node in the next level. Therefore, as the tree depth grows, there is a competition between the irreversible effect of noise and the protection against such noise achieved by delocalization of information. Using novel decoders, we demonstrate that quantum information can be indefinitely preserved in infinite noisy quantum trees, without the need for mid-circuit error correction. Furthermore, we establish that for noise above certain thresholds (that depend on properties of the encoding unitary), quantum information decays exponentially with tree depth. Towards the end of the talk, we will briefly consider the optimal distillation of pure coherent states from coherent thermal states in bosonic systems, using time-translation invariant (e.g., phase-insensitive) operations. Remarkably, the lowest achievable error -- as quantified by infidelity (one minus the fidelity) of the output state with the desired coherent state -- is proportional to the inverse of the purity of coherence of the input state, a quantity obtained from the Right-Logarithmic-Derivative (RLD) Fisher information metric, hence revealing an operational interpretation of this quantity.

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