Memoryless quantum channel (${\Phi }^{\otimes n}$)

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.

A quantum erasure channel transmits an input state unchanged with probability $1-p$ and replaces it by an orthogonal, recognizable erasure flag with probability $p$. Its output space has one more dimension than the input. This is a quantum channel for $0\le p\le 1$ and preserves all input coherences on the nonerased branch.

For input dimension $D$, an input of entropy $S(\rho )$ gives output entropy $h_2(p)+(1-p)S(\rho )$ and reference-output entropy $h_2(p)+pS(\rho )$. These follow from entropy of an orthogonal quantum mixture and purity of the input purification. Thus the quantum mutual information is $2(1-p)S(\rho )$, maximized at $\rho =I_D/D$. The entanglement-assisted classical capacity is consequently $2(1-p){\log }_{2}D$ bits per use; for a binary input it is $2(1-p)$.

This is the supremum of reliable classical bits per use of a memoryless quantum channel when sender and receiver share unlimited prior quantum entanglement. If $|{\psi }_{\rho }{⟩}_{RA}$ purifies the input, let ${\omega }_{RB}=({\mathrm{id}}_R\otimes \mathcal{N})(|{\psi }_{\rho }⟩⟨{\psi }_{\rho }|)$. The single-letter capacity formula is ${\max }_{\rho }[S(\rho )+S(\mathcal{N}(\rho ))-S({\omega }_{RB})]$, the maximum quantum mutual information between reference and output. The preshared resource is independent of the message.