Permutation Invariant Optimization Problems in Quantum Information Theory: A Framework for Channel Fidelity and Beyond

Exploiting permutation invariance to reduce the exponential scaling of semidefinite programs in quantum information has emerged as a powerful computational technique. In this work, we develop a systematic framework for using this reduction via Schur-Weyl duality for optimization problems, and establish methods that allow one to work fully inside the permutation invariant subspace while performing operations such as (partially) applying channels and taking (partial) traces, or computing expressions like the quantum relative entropy. We then apply our techniques to the problem of computing efficient lower bounds on the channel fidelity over $n$ parallel uses of a quantum channel. The algorithm, which we call symmetric seesaw method, exploits permutation-invariant codes to yield improved lower bounds on the channel fidelity over $n$ uses of the depolarizing and amplitude-damping channel in the regime of tens of channel uses, and was used in [arxiv:2604.27042] to demonstrate non-asymptotic superactivation of quantum capacity for $n = 17$. An implementation of our methods, aimed at being suitable for various quantum information theoretic optimization problems, is also available as an open-source Python package.