Operational quantum, private, and entanglement-assisted capacities

Let $\mathcal{N}:A\to B$ be a finite-dimensional quantum channel and use its memoryless extension ${\mathcal{N}}^{\otimes n}:A^{\otimes n}\to B^{\otimes n}$. Rates use natural logarithms, so they are in nats per channel use. For code sizes $K_n\ge 1$, a rate $R\ge 0$ is achievable when ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{n}^{-1}\ln K_n\ge R$ and the relevant error terms below tend to zero; each capacity is the supremum of its achievable rates.

For quantum transmission, a block code consists of an isometric encoder $W_n:M_n\to A^{\otimes n}$ and a quantum-channel decoder ${\mathcal{D}}_n:B^{\otimes n}\to M_n$, where $\dim M_n=K_n$. Let $R_n\cong M_n$ and let $\mathrm{\mid }{\mathrm{\Phi }}_{K_n}⟩\in R_n\otimes M_n$ be a normalized maximally entangled vector. Its entanglement fidelity is

$F_n:=⟨{\mathrm{\Phi }}_{K_n}\mathrm{\mid }({\mathrm{id}}_{R_n}\otimes {\mathcal{D}}_n{\mathcal{N}}^{\otimes n}{\mathcal{W}}_n)(\mathrm{\mid }{\mathrm{\Phi }}_{K_n}⟩⟨{\mathrm{\Phi }}_{K_n}\mathrm{\mid })\mathrm{\mid }{\mathrm{\Phi }}_{K_n}⟩,$

where ${\mathcal{W}}_n(X)=W_{n}X{W}_n^{\ast }$. A rate is quantum-achievable if $F_n\to 1$; the resulting capacity is $Q(\mathcal{N})$. This is the subspace entanglement-transmission convention.

For private classical transmission, a block code uses equiprobable messages $m\in [K_n]$, input states ${\rho }_{m,n}$ on $A^{\otimes n}$, and a decoding POVM $\{{\mathrm{\Lambda }}_{m,n}{\}}_{m=1}^{K_n}$ on $B^{\otimes n}$. Its average error is

$e_n:=1-\frac{1}{K_n}{\sum }_{m=1}^{K_n}\mathrm{tr}({\mathrm{\Lambda }}_{m,n}{\mathcal{N}}^{\otimes n}({\rho }_{m,n}))\mathrm{.}$

Fix a complementary channel ${\mathcal{N}}^c$ as in Complementary channels and coherent information. Eve's average state is ${\bar{\tau }}_n=K_n^{-1}{\sum }_m({\mathcal{N}}^c{)}^{\otimes n}({\rho }_{m,n})$. Define leakage by its Holevo information

${\mathrm{\ell }}_n:=S({\bar{\tau }}_n)-\frac{1}{K_n}{\sum }_{m=1}^{K_n}S(({\mathcal{N}}^c{)}^{\otimes n}({\rho }_{m,n}))\mathrm{.}$

A rate is privately achievable when $e_n\to 0$ and ${\mathrm{\ell }}_n\to 0$; this is strong secrecy in mutual-information form. The corresponding capacity is $P(\mathcal{N})$. The value is independent of the chosen complementary dilation, as follows from Private information is an average coherent-information difference.

For classical transmission with prior shared entanglement, for each block length $n$ the sender and receiver begin with a finite-dimensional shared state ${\sigma }_{S_A{S}_B}$ independent of the equiprobable message. For each $m\in [K_n]$, the sender applies a quantum channel ${\mathcal{E}}_{m,n}:\mathcal{L}(S_A)\to \mathcal{L}(A^{\otimes n})$. The receiver measures $B^{\otimes n}\otimes S_B$ with a POVM $\{{\mathrm{\Gamma }}_{m,n}{\}}_{m=1}^{K_n}$. Its average error is

$e_n^{EA}:=1-\frac{1}{K_n}{\sum }_{m=1}^{K_n}\mathrm{tr}\text{\hspace{0.17em}⁣}\left[{\mathrm{\Gamma }}_{m,n}\right(({\mathcal{N}}^{\otimes n}\circ {\mathcal{E}}_{m,n})\otimes {\mathrm{id}}_{S_B})({\sigma }_{S_A{S}_B})]\mathrm{.}$ A rate is achievable when this error tends to zero; there is no constraint on the dimension or consumption of the shared entanglement. The resulting capacity is $C_E(\mathcal{N})$.

The input, output, and reference registers are finite-dimensional in each block. No single-letter formula is included in these operational definitions.