Completely positive maps

Let $H$ and $K$ be finite-dimensional complex inner-product spaces and write $\mathcal{L}(H)$ and $\mathcal{L}(K)$ for their vector spaces of complex-linear endomorphisms with pointwise operations (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Real and complex inner-product spaces and their induced length, The space $\mathcal{L}(V,W)$ of linear maps with pointwise addition and scalar multiplication, $\mathcal{L}(V,W)$ is a vector space over the common scalar field). Let $\mathrm{\Phi }:\mathcal{L}(H)\to \mathcal{L}(K)$ be complex-linear. For each integer $r\ge 1$, put $R={\mathbb{C}}^r$ with its standard ordered register basis (Finite registers and their ordered computational bases). Write an operator $X\in \mathcal{L}(H\otimes R)$ in its unique $r\times r$ block form $X=[X_{ij}]$ with $X_{ij}\in \mathcal{L}(H)$, and define ${\mathrm{\Phi }}^{(r)}(X)$ to have blocks $[\mathrm{\Phi }(X_{ij})]$ on $K\otimes R$. Lemma Finite ancilla amplification is a well-defined block map proves that this block rule defines a linear map on the full operator space for every finite $r$. For $r=1$, the extension is $\mathrm{\Phi }$ under the canonical unitary identifications $x\mapsto x\otimes e_0$ and $y\mapsto y\otimes e_0$ proved in that lemma.

The map $\mathrm{\Phi }$ is completely positive if, for every finite $r\ge 1$, ${\mathrm{\Phi }}^{(r)}$ sends every nonnegative operator on $H\otimes R$ to a nonnegative operator on $K\otimes R$ (Non-negative and positive operators). The quantifier includes $r=1$; complete positivity therefore implies positivity. No Kraus, Choi, or Stinespring characterization is part of this definition.