 Memoryless quantum channel

A memoryless quantum channel has independent identical noise on successive uses, described by the tensor-power map ${\Phi }^{\otimes n}$. This factorization does not restrict the encoder to product inputs: one can feed an entangled state across uses. A block code with equally likely messages $m$ chooses states ${\rho }_m$ and a decoder POVM $(D_m)$. Its average error is $1-|\mathcal{M}{|}^{-1}{\sum }_m\mathrm{Tr}[D_m{\Phi }^{\otimes n}({\rho }_m)]$, and its rate is $n^{-1}{\log }_2|\mathcal{M}|$. The classical capacity of a quantum channel allows such entangled block inputs; the one-use Holevo capacity gives the product-input capacity and requires regularization in general.