Error exponent of activated non-signaling assisted classical-quantum channel coding

We provide a tight asymptotic characterization of the error exponent for classical-quantum channel coding assisted by activated non-signaling correlations. Namely, we find that the optimal exponent\, -- \,also called reliability function\, -- \,is equal to the well-known sphere packing bound, which can be written as a single-letter formula optimized over Petz-R\'enyi divergences. Remarkably, there is no critical rate and as such our characterization remains tight for arbitrarily low rates below the capacity. On the achievability side, we further extend our results to fully quantum channels. Our proofs rely on semi-definite program duality and a dual representation of the Petz-R\'enyi divergences via Young inequalities. As a result of independent interest, we find that the Petz-R\'enyi divergences of order $\alpha\in[0,2]$ are upper bounded by the sandwiched R\'enyi divergences of order $1/(2-\alpha)\in[1/2,\infty]$.