Finite-dimensional quantum erasure channel

Let $A$ be a $d$-dimensional complex inner-product space with $d\ge 1$. Form the orthogonal direct sum $B:=A\oplus \mathbb{C}e$, where $e$ is a unit erasure vector orthogonal to every vector in $A$; the inner product on pairs is $⟨(x,a),(y,b){⟩}_B=⟨x,y{⟩}_A+a\bar{b}$. Thus $\dim B=d+1$ by If $V={⨁}_{i<n}{U}_i$ with every $U_i$ finite-dimensional, then $V$ is finite-dimensional and ${\dim }_{F}V={\sum }_{i<n}{\dim }_F{U}_i$; in particular ${\dim }_F(U\oplus W)={\dim }_{F}U+{\dim }_{F}W$. Let $J:A\to B$ be the isometric inclusion into the data summand.

For an erasure probability $ϵ\in [0,1]$, the $d$-dimensional quantum erasure channel is the linear map

${\mathcal{E}}_{d,ϵ}(X):=(1-ϵ)JX{J}^{\ast }+ϵ\mathrm{tr}(X)\mathrm{\mid }e⟩⟨e\mathrm{\mid },X\in \mathcal{L}(A)\mathrm{.}$

For a density operator $\rho$, this outputs the data state with probability $1-ϵ$ and the orthogonal erasure flag with probability $ϵ$. The definition is on the full operator space, and its channel property is established in The finite-dimensional erasure map is a quantum channel ↗.