PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Automorphism

For a $\star$-automorphism $\beta$ of a $\mathbb{C}$-algebra $A$, denoted as $\beta : A \simeq_{\star a}[\mathbb{C}] A$, this definition constructs the corresponding quantum channel from $A$ to $A$. This channel is a unital completely positive (UCP) map whose underlying action is the $\star$-homomorphism $\beta$.

For a $\star$-automorphism $\beta$ of a unital complex $\star$-algebra $A$, this definition constructs the induced unital positive linear map (often referred to as an order-unit channel) on the real vector space of self-adjoint elements $\text{selfAdjoint}(A)$. The map is defined by the restriction of $\beta$ to the self-adjoint subspace, mapping $a \in \text{selfAdjoint}(A)$ to $\beta(a)$. This map preserves the identity ($\beta(1) = 1$) and maps positive elements to positive elements ($a \ge 0 \implies \beta(a) \ge 0$).

Let $A$ be a unital complex $\star$-algebra. For any $\star$-automorphism $\beta : A \simeq_{\star a}[\mathbb{C}] A$ and any self-adjoint element $a \in \text{selfAdjoint}(A)$, the induced unital positive linear map $\beta.\text{observableUPLM}$ (the order-unit channel on the space of observables) evaluates at $a$ to the same value as the direct action of the $\star$-automorphism on the observable, denoted $\beta.\text{observable}(a)$.

Let $A$ be a unital complex $\star$-algebra and $\beta : A \simeq \text{Aut}_{\star, \mathbb{C}}(A)$ be a $\star$-automorphism. This definition constructs the induced order-unit symmetry on the real vector space of self-adjoint elements $\text{selfAdjoint}(A)$. This symmetry is formed by the unital positive linear map $a \mapsto \beta(a)$ and its inverse $a \mapsto \beta^{-1}(a)$, which preserve the identity and the positivity of observables.

Let $A$ be a unital complex $\star$-algebra and $\beta$ be a $\star$-automorphism of $A$. The order-unit symmetry on the space of self-adjoint elements $\text{selfAdjoint}(A)$ induced by $\beta$, when regarded as a unital positive linear map, is equal to the induced unital positive linear map (also known as an order-unit channel) $\beta.\text{observableUPLM}$ that maps $a \in \text{selfAdjoint}(A)$ to $\beta(a)$.

Let $A$ and $B$ be $\mathbb{C}$-$\star$-algebras. Given a $\mathbb{C}$-$\star$-algebra isomorphism $\beta : A \cong B$, this definition constructs the corresponding quantum channel from $A$ to $B$. This channel is a unital completely positive (UCP) map defined by the action of the $\star$-homomorphism $\beta$.

Let $A$ and $B$ be $\star$-algebras over $\mathbb{C}$. Given a $\mathbb{C}$-$\star$-algebra isomorphism $\beta : A \cong B$, this definition constructs the induced unital positive linear map (UPLM) between the spaces of self-adjoint elements (observables) $\text{selfAdjoint } A$ and $\text{selfAdjoint } B$. For any observable $a \in \text{selfAdjoint } A$, the map is defined by $a \mapsto \beta(a)$. This map is $\mathbb{R}$-linear, preserves the identity element ($1_A \mapsto 1_B$), and preserves positivity (if $a \ge 0$ in $A$, then $\beta(a) \ge 0$ in $B$).

Let $A$ and $B$ be $\mathbb{C}$-$\star$-algebras. For any $\mathbb{C}$-$\star$-algebra isomorphism $\beta : A \cong B$ and any self-adjoint element $a \in \text{selfAdjoint } A$, the induced unital positive linear map $\beta_{\text{obs}} : \text{selfAdjoint } A \to \text{selfAdjoint } B$ (representing the inter-system observable channel) is pointwise equal to the original $\star$-isomorphism, such that $\beta_{\text{obs}}(a) = \beta(a)$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$. Let $\alpha$ and $\beta$ be two one-parameter groups of $\star$-automorphisms of $A$. If for all times $t$ and all elements $a \in A$, the action of $\alpha$ at time $t$ on $a$ is equal to the action of $\beta$ at time $t$ on $a$ (i.e., $\alpha_t(a) = \beta_t(a)$), then $\alpha = \beta$.

For a one-parameter group $\alpha$ of $\star$-automorphisms on a complex $\star$-algebra $A$, this definition constructs the induced one-parameter group of automorphisms on the real vector space of self-adjoint elements $\text{selfAdjoint}(A)$. For each $t \in \mathbb{R}$, the corresponding automorphism is the unital positive linear map (order-unit automorphism) obtained by restricting $\alpha_t$ to the self-adjoint subspace. This allows the algebraic dynamics on $A$ to be viewed as a standard evolution of states and effects within the order-unit space framework.

Let $A$ be a complex $\star$-algebra and $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$. For any time $t \in \mathbb{R}$ and any self-adjoint element $a \in \text{selfAdjoint}(A)$, the action of the induced one-parameter group of observable automorphisms $\alpha.\text{toObservableDynamics}$ at time $t$ on $a$ is equal to the action of the algebraic $\star$-automorphism $\alpha_t$ on $a$: $$\alpha.\text{toObservableDynamics}_t(a) = \alpha_t(a)$$ In other words, the observable dynamics is the restriction of the algebraic dynamics to the subspace of observables at each time $t$.

Given a $\star$-isomorphism $\beta : A \cong B$ between $\mathbb{C}$-algebras and a one-parameter group of $\star$-automorphisms $\alpha$ on $A$ (where each $\alpha_t$ is a $\star$-automorphism of $A$), the transport of $\alpha$ along $\beta$ is the one-parameter group of $\star$-automorphisms on $B$ defined by $t \mapsto \beta \circ \alpha_t \circ \beta^{-1}$.

Let $A$ and $B$ be complex $\star$-algebras. Given a $\star$-isomorphism $\beta : A \cong B$ and a one-parameter group of $\star$-automorphisms $\alpha$ on $A$, the action of the transported dynamics on an element $b \in B$ at time $t \in \mathbb{R}$ is given by: $$(\alpha^\beta)_t(b) = \beta(\alpha_t(\beta^{-1}(b)))$$ where $\alpha_t$ denotes the automorphism on $A$ at time $t$, and $\beta^{-1}$ is the inverse $\star$-isomorphism from $B$ to $A$.

Let $A$ and $B$ be algebras. A $\star$-isomorphism $\beta : A \simeq B$ is said to **intertwine** two dynamics (one-parameter groups of $\star$-automorphisms) $\alpha$ on $A$ and $\gamma$ on $B$ if for every time $t \in \mathbb{R}$ and every element $a \in A$, the following condition holds: $$\beta(\alpha_t(a)) = \gamma_t(\beta(a))$$ This signifies that the action of $\beta$ commutes with the temporal evolution of the system at all times.

Let $A$ and $B$ be algebras, $\beta : A \cong B$ be a $\star$-isomorphism, and $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$. Then $\beta$ intertwines $\alpha$ with the dynamics $\gamma$ on $B$ obtained by transporting $\alpha$ along $\beta$ (where $\gamma_t = \beta \circ \alpha_t \circ \beta^{-1}$). That is, for every $t \in \mathbb{R}$ and $a \in A$, the following condition holds: $$\beta(\alpha_t(a)) = \gamma_t(\beta(a))$$

Let $A$ and $B$ be $\mathbb{C}$-algebras. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$, and $\gamma$ be a one-parameter group of $\star$-automorphisms on $B$. For a $\star$-isomorphism $\beta : A \simeq B$, $\beta$ **intertwines** $\alpha$ and $\gamma$ (meaning $\beta \circ \alpha_t = \gamma_t \circ \beta$ for all $t \in \mathbb{R}$) if and only if $\gamma$ is the **transport** of $\alpha$ along $\beta$ (defined as $\gamma_t = \beta \circ \alpha_t \circ \beta^{-1}$ for all $t \in \mathbb{R}$).

Let $A, B$, and $C$ be algebras. Let $\alpha, \gamma$, and $\delta$ be one-parameter groups of $\star$-automorphisms (dynamics) on $A, B$, and $C$, respectively. If a $\star$-isomorphism $\beta : A \simeq B$ intertwines $\alpha$ and $\gamma$, and a $\star$-isomorphism $\chi : B \simeq C$ intertwines $\gamma$ and $\delta$, then the composed $\star$-isomorphism $\chi \circ \beta : A \simeq C$ intertwines $\alpha$ and $\delta$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$ and let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$. The transport of the flow $\alpha$ along the identity $\star$-isomorphism $\text{id}_A : A \cong A$ is equal to $\alpha$.

Let $A, B,$ and $C$ be $\star$-algebras over $\mathbb{C}$. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$. Given $\star$-isomorphisms $\beta : A \cong B$ and $\gamma : B \cong C$, the transport of $\alpha$ along $\beta$ followed by the transport of the resulting group along $\gamma$ is equal to the transport of $\alpha$ along the composite $\star$-isomorphism $\gamma \circ \beta$.

Let $A$ and $B$ be $\star$-algebras over $\mathbb{C}$. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$ and $\beta: A \cong B$ be a $\star$-isomorphism. If $\alpha$ is transported to $B$ via $\beta$, and the resulting one-parameter group is then transported back to $A$ via the inverse isomorphism $\beta^{-1}$, the result is the original one-parameter group $\alpha$.

For a $\mathbb{C}$-algebra $A$, given a $\star$-automorphism $\beta : A \simeq_{\star} A$ and a one-parameter group of $\star$-automorphisms $\alpha$ on $A$, the conjugation of $\alpha$ by $\beta$ is the one-parameter group of $\star$-automorphisms defined by $t \mapsto \beta \circ \alpha_t \circ \beta^{-1}$. This can be interpreted as viewing the dynamics $\alpha$ through a change of coordinates $\beta$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$. Given a $\star$-automorphism $\beta$ of $A$ and a one-parameter group of $\star$-automorphisms $\alpha$ on $A$, the action of the conjugated group $(\text{conj}_\beta \alpha)$ at time $t \in \mathbb{R}$ on an element $a \in A$ is given by $$(\text{conj}_\beta \alpha)_t(a) = \beta(\alpha_t(\beta^{-1}(a))).$$

For any one-parameter group of $\star$-automorphisms $\alpha$ on a $\mathbb{C}$-algebra $A$, conjugating $\alpha$ by the identity $\star$-automorphism of $A$ results in the original one-parameter group $\alpha$. That is, $\text{conj}(\text{id}_A, \alpha) = \alpha$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$ and let $\alpha$ be a one-parameter group of $\star$-automorphisms of $A$. For any two $\star$-automorphisms $\beta, \gamma$ of $A$, the result of first conjugating the flow $\alpha$ by $\beta$ and then conjugating the resulting flow by $\gamma$ is equal to conjugating the original flow $\alpha$ by the composite $\star$-automorphism $\gamma \circ \beta$.

Let $A$ be a $\mathbb{C}$-algebra and $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$. For any $\star$-automorphism $\beta: A \simeq_{\star} A$, let the conjugation of $\alpha$ by $\beta$ be the one-parameter group defined by $t \mapsto \beta \circ \alpha_t \circ \beta^{-1}$. Then, conjugating the resulting flow by the inverse automorphism $\beta^{-1}$ recovers the original flow $\alpha$. That is, $(\alpha.\text{conj } \beta).\text{conj } \beta^{-1} = \alpha$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$. For any one-parameter group of $\star$-automorphisms $\alpha$ on $A$ and any $\star$-automorphism $\beta : A \simeq_{\star} A$, conjugating $\alpha$ by the inverse $\beta^{-1}$ and then conjugating the result by $\beta$ recovers the original flow $\alpha$. Specifically, if the conjugation of a flow $\alpha$ by an automorphism $\phi$ is defined as $(\text{conj}_\phi \alpha)_t = \phi \circ \alpha_t \circ \phi^{-1}$, then $\text{conj}_\beta (\text{conj}_{\beta^{-1}} \alpha) = \alpha$.