One-Shot Quantum Darwinism: Entropic Redundancy Thresholds and the Emergence of Classicality via Measured Rényi Mutual Information, with Applications to Trapped Ion Quantum Processors

We introduce a one-shot information-theoretic approach to quantum Darwinism, the process by which classical objectivity arises from quantum mechanics through the redundant encoding of information about a quantum system in a set of independent fragments of an environment. The core information-theoretic object is the measured smooth Renyi-2 mutual information Iₑpsilon, M₂ (S: F) ᵣho, given by the supremum over the set of local product measurements of the measured smooth Renyi-2 divergence Dₑpsilon, T₂. This object is given a semidefinite programming (SDP) variational representation analogous to the Regula-Tomamichel formulation of the measured smooth Renyi divergence. Using this object, we prove matching achievability and converse results for the one-shot redundancy function Rₑpsilon, delta, as well as a classicality criterion. We show that for the i. i. d. fragment model, the condition Iₑpsilon, M₂ (S: F1) >= delta * Hₑpsilon, Mₘin (S) is both sufficient for (eps, delta) -objectivity and asymptotically necessary up to a log (1/eps) correction term, which vanishes for eps -> 0. We also prove a second-order asymptotic expansion of the redundancy function, a strong converse exponent for the rate of decay of objectivity below the classicality threshold, and a phase diagram of the quantum Darwinism transition. We apply this formalism to 171Yb+ ion quantum processors, obtaining explicit bounds on the redundancy function for the phonon-mediated environment model in terms of the Lamb-Dicke parameter, decoherence rates, and chain length. All of the results are found to be consistent with the orthonormality of the normal mode eigenvectors.