PhyslibAlpha.ProbabilisticTheory.CStarAlgebra.QuantumChannel

The type of quantum channels between $C^*$-algebras $A_1$ and $A_2$ is equipped with a structure that allows its elements to be treated as functions. Specifically, for any quantum channel $\Phi \in \text{QuantumChannel}(A_1, A_2)$, this instance defines the application of $\Phi$ to an element $x \in A_1$ (denoted $\Phi(x) \in A_2$) and ensures that the channel is uniquely determined by its values on all elements of $A_1$.

Every quantum channel $\Phi$ between $C^*$-algebras $A_1$ and $A_2$ is a linear map over the complex numbers $\mathbb{C}$. Specifically, for any elements $x, y \in A_1$ and any scalars $a, b \in \mathbb{C}$, the channel satisfies the property $\Phi(ax + by) = a\Phi(x) + b\Phi(y)$.

Let $A_1$ and $A_2$ be $C^*$-algebras. Any quantum channel $\Phi \in \text{QuantumChannel}(A_1, A_2)$ is a completely positive map. This implies that for any $n \in \mathbb{N}$, the induced map $\Phi \otimes \text{id}_n : A_1 \otimes M_n(\mathbb{C}) \to A_2 \otimes M_n(\mathbb{C})$ is a positive map, preserving the cone of positive operators when applied to a subsystem of an entangled state.

Let $A_1$ and $A_2$ be $C^*$-algebras. Every quantum channel $\Phi$ from $A_1$ to $A_2$ preserves the identity element, meaning $\Phi(1_{A_1}) = 1_{A_2}$.

Let $A_1$ and $A_2$ be $C^*$-algebras. For any two quantum channels $f, g \in \text{QuantumChannel}(A_1, A_2)$, if $f(x) = g(x)$ for all $x \in A_1$, then $f = g$.

Given a quantum channel $f: A_1 \to A_2$ between unital $C^*$-algebras, this definition constructs the induced map $f|_{sa}: \text{selfAdjoint } A_1 \to \text{selfAdjoint } A_2$ between their respective spaces of self-adjoint elements. Since $f$ is a unital completely positive map, its restriction to the self-adjoint parts is a unital positive linear map, which constitutes a channel between the order-unit spaces of observables.