A Random Coding Based Proof for the Quantum Coding Theorem

We present a proof for the quantum channel coding theorem which relies on the fact that a randomly chosen code space typically is highly suitable for quantum error correction. In this sense, the proof is close to Shannon’s original treatment of information transmission via a noisy classical channel. 1 Preliminaries 1.1 Quantum channel In the theory of information transmission the information is ascribed to the configuration of a physical system, and the transmission is ascribed to the dynamical evolution of that configuration under the influence of an in general noisy environment. It is therefore customary to characterize an information carrying system solely by its configuration space, and to consider its intrinsic dynamics as part of the transmission. In a quantum setting we identify a system Q with its Hilbert space, denoted by the same symbol Q. Its dimension |Q| will be always assumed to be finite. The system’s configuration is a quantum state described by a density operator ρ in B(Q), the set of bounded operators on Q. The process of information transmission can be any dynamics of an open quantum system Q according to which an initial input state ρ evolves to a final output state ρ′, defining in this way the operation of a quantum channel N 1. Mathematically, N is a completely positive mapping of B(Q) onto itself, or, when we admit that the Email address: [email protected] For an introduction into the theory of quantum information see e.g. [1, 2].