Finite Kraus representation

Let $H$ and $K$ be finite-dimensional (have finite bases by Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis) complex inner-product spaces (Real and complex inner-product spaces and their induced length), with the root convention that each inner product is linear in its first argument. Write $\mathcal{L}(H)$ for the complex-linear endomorphisms of $H$ and $\mathcal{L}(H,K)$ for the complex-linear maps from $H$ to $K$ (Linear map between vector spaces over the same field). A finite Kraus representation of a complex-linear map $\mathrm{\Phi }:\mathcal{L}(H)\to \mathcal{L}(K)$ is a finite family $(K_a{)}_{a\in A}\subseteq \mathcal{L}(H,K)$ such that

$\mathrm{\Phi }(X)={\sum }_{a\in A}{K}_{a}X{K}_a^{†}\text{for every }X\in \mathcal{L}(H),$

where $K_a^{†}:K\to H$ is the adjoint (The adjoint $T^{\ast }:W\to V$ is characterised by $⟨Tv,w{⟩}_W=⟨v,T^{\ast }w{⟩}_V$), which exists uniquely in finite dimensions (Every linear map between finite-dimensional inner product spaces has a unique adjoint). The composition $K_a\circ X\circ K_a^{†}$ maps $K$ to $K$, so the sum defines an output endomorphism for each input endomorphism $X$. The index set $A$ may be empty; its sum is then the zero map. Zero operators are allowed in a nonempty family.