Approximate pushforward designs and image bounds on approximations

Abstract

We extend the framework of quantum pushforward designs to the approximate setting, in which moment operators agree only up to finite precision. We first formulate a general transfer theorem for maps that act linearly at the level of moments. This separates direct applications, controlled by standard Schatten-norm estimates, from refinements that use additional tensorial structure. Dephasing is shown to be contractive and therefore sends an approximate projective design to a simplex design without increasing its error. Ordinary partial-trace estimates give immediate bounds for mixed-state and channel designs. We then prove a sharp Schatten-norm bound for the partial trace restricted to the totally symmetric subspace. This replaces the full-environment norm coefficient by one governed by a symmetric-subspace dimension and yields asymptotically tighter estimates for both mixed-state and channel designs. Numerical simulations for induced mixed-state designs are consistent with the resulting hierarchy of bounds.

26 pages, 2 figures, comments welcome!