PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Channel

The instance allows a quantum channel $f$ between $C^*$-algebras $A_1$ and $A_2$ to be treated as a function $f: A_1 \to A_2$. This enables the application of a channel to elements of the algebra and ensures that a channel is uniquely determined by its values on all inputs.

Let $A_1$ and $A_2$ be $C^*$-algebras. Every quantum channel $\Phi: A_1 \to A_2$ is a linear map over the field of complex numbers $\mathbb{C}$.

For $C^*$-algebras $A_1$ and $A_2$, every quantum channel $f: A_1 \to A_2$ is a completely positive map. This means that for any $n \in \mathbb{N}$, the induced map $f^{(n)}: M_n(A_1) \to M_n(A_2)$ on the matrix algebras over $A_1$ and $A_2$ is positive.

For any quantum channel $f$ between $C^*$-algebras $A_1$ and $A_2$, $f$ is a unital map, meaning it preserves the multiplicative identity element: $f(1) = 1$.

Let $A_1$ and $A_2$ be $C^*$-algebras. Given two quantum channels $f, g : A_1 \to A_2$ (modeled as unital completely positive maps), if $f(x) = g(x)$ for all $x \in A_1$, then $f = g$.

Let $f: A_1 \to A_2$ be a quantum channel (modeled as a unital completely positive map) between $C^*$-algebras $A_1$ and $A_2$. This definition constructs the restriction of $f$ to the real linear subspaces of self-adjoint elements, resulting in a unital positive $\mathbb{R}$-linear map from $\text{selfAdjoint } A_1$ to $\text{selfAdjoint } A_2$. This is well-defined because every completely positive map is positive, and positive maps preserve both the identity element and the self-adjointness of elements.