math-ph — Mathematical Physics
Continuity, Testing Regions and Fragility of Quantum Channel Divergences
Converse bounds on discriminating and estimating quantum channels are evaluated on a characterized model of the channels, which is only known approximately. This book studies when such bounds, and the operational exponents they bound, depend continuously on the channels, and whether their sensitivity to characterization error is a property of the task or only of the bound. The book opens with noisy quantum gates. The supports of the Choi matrices separate genuine fragility, where the operational exponent itself jumps, from spurious fragility, where only single-letter converses jump. Near a gate with an incoherent noise floor, the smallest eigenvalue of the normalized Choi matrix sets the sensitivity. At first order, under errors in the model, this sensitivity belongs to the task. At second order, under a coherent miscalibration, the converses can overshoot the adaptive Stein and Chernoff exponents by factors that grow without bound as the noise decreases, although for small miscalibrations of a depolarized qubit they stay within a factor 3/2. Certifying a converse from tomographic data costs a number of channel uses set by the same floor. The general theory follows. Divergences built from Kubo-Ando means, including the geometric fidelity, the geometric Renyi divergences and the Belavkin-Staszewski relative entropy, are functions of the Arveson-Radon-Nikodym derivative of one channel with respect to the sum of the pair, and in this coordinate they are continuous with dimension-free moduli. In the diamond norm they are continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes, so an ideal unitary gate paired with itself is a point of discontinuity. Divergences built from hypothesis tests, such as the Umegaki relative entropy and the hockey-stick f-divergences, are uniformly continuous in every dimension, and the adaptive