Strong Converse for Quantum Capacity via a Fully Quantum Blowing-Up Lemma

Abstract

We prove an exponential strong converse for quantum communication through every finite-dimensional memoryless channel: at any fixed rate above the quantum capacity, the entanglement-transmission fidelity of every code decays exponentially with the number of channel uses. The proof has two main ingredients. First, a fully quantum blowing-up lemma converts a low-fidelity code into a high-fidelity code, with a loss in the number of transmitted qubits controlled by the projective tensor norm of the orthogonal projection on the image of the Stinespring dilation of the channel across the receiver--environment bipartition. Second, a low-degree polynomial construction approximates the tensor power of this projector with exponential accuracy while controlling its projective norm, providing the approximation needed for the blowing-up argument.