Finite completely positive trace-preserving maps

Let $d,e\ge 1$ be integers, let $H_{\mathrm{i}\mathrm{n}}={\mathbb{C}}^d$ and $H_{\mathrm{o}\mathrm{u}\mathrm{t}}={\mathbb{C}}^e$ with their standard inner products, and write $M_d(\mathbb{C})$ and $M_e(\mathbb{C})$ for the complex linear maps on these spaces. A matrix $X$ is positive semidefinite, written $X\ge 0$, when it is Hermitian and $⟨Xv,v⟩\ge 0$ for every vector $v$. The finite spectral convention in Finite matrix spectral and state calculus gives positive matrices their orthonormal eigenbases and positive square roots.

A positive map is a complex-linear map $\mathrm{\Phi }:M_d(\mathbb{C})\to M_e(\mathbb{C})$ such that $\mathrm{\Phi }(X)\ge 0$ whenever $X\ge 0$. For each integer $r\ge 1$, identify $M_r(\mathbb{C})\otimes M_d(\mathbb{C})$ with matrices on ${\mathbb{C}}^r\otimes {\mathbb{C}}^d$ in the ordered product basis. The map $\mathrm{\Phi }$ is completely positive when ${\mathrm{id}}_r\otimes \mathrm{\Phi }$ is positive for every such $r$, where ${\mathrm{id}}_r$ is the identity map on $M_r(\mathbb{C})$.

The map $\mathrm{\Phi }$ is trace-preserving when $\mathrm{Tr}(\mathrm{\Phi }(X))=\mathrm{Tr}(X)$ for every $X\in M_d(\mathbb{C})$. A finite quantum channel is a complex-linear map that is both completely positive and trace-preserving, abbreviated CPTP. Unitality, $\mathrm{\Phi }(I_d)=I_e$, is a separate condition and is not part of this definition.

This is a mathematical definition. A physical preparation, measurement, or open-system model must separately identify when one of these maps represents a physical transformation.