PhyslibAlpha.ProbabilisticTheory.CStarAlgebra.Automorphism

Given a $\star$-algebra isomorphism $\beta : A \cong_{\star} B$ between complex $\star$-algebras $A$ and $B$, this definition constructs the corresponding quantum channel from $A$ to $B$. The channel is defined by the underlying $\star$-algebra homomorphism of $\beta$, which maps the identity of $A$ to the identity of $B$.

Given a $\star$-isomorphism $\beta : A \cong B$ between complex $\star$-algebras $A$ and $B$, this definition constructs the induced quantum channel between their spaces of self-adjoint observables, $\text{selfAdjoint } A$ and $\text{selfAdjoint } B$. The channel is a unital positive linear map that sends each self-adjoint element $a \in \text{selfAdjoint } A$ to $\beta(a) \in \text{selfAdjoint } B$.

Let $A$ and $B$ be complex $\star$-algebras and let $\beta : A \cong B$ be a $\star$-isomorphism. Let $\beta_{\text{obs}} : \text{selfAdjoint } A \to \text{selfAdjoint } B$ be the quantum channel between the spaces of self-adjoint observables induced by $\beta$. For any self-adjoint element $a \in \text{selfAdjoint } A$, applying the observable channel yields the same result as the $\star$-isomorphism itself, i.e., $\beta_{\text{obs}}(a) = \beta(a)$.

Let $A$ be a unital complex $C^*$-algebra. For any $\star$-automorphism $\beta: A \cong A$ and any self-adjoint element $a \in \text{selfAdjoint } A$, the value of the induced quantum channel $\beta.\text{toObservableChannel}$ applied to $a$ is equal to the action of the $\star$-automorphism on $a$ as an observable, denoted by $\beta.\text{observable}(a)$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$. Let $\alpha$ and $\beta$ be two one-parameter groups of $\star$-automorphisms on $A$. If for every time $t$ and every element $a \in A$, the action of $\alpha$ matches the action of $\beta$ (i.e., $\alpha(t)(a) = \beta(t)(a)$), then the groups themselves are equal, $\alpha = \beta$.

Given a one-parameter group of $\star$-automorphisms $\alpha$ on a $\star$-algebra $A$, this definition constructs the induced one-parameter group of automorphisms on the space of self-adjoint observables $\text{selfAdjoint } A$. For each time $t \in \mathbb{R}$, the corresponding automorphism on the observables is the restriction of the $\star$-automorphism $\alpha(t)$ to the self-adjoint part of $A$.

Let $A$ be a unital $\star$-algebra over $\mathbb{C}$ and $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$. For any time $t \in \mathbb{R}$ and any self-adjoint observable $a \in \text{selfAdjoint } A$, the induced one-parameter group of automorphisms on the observables, $\alpha.\text{toObservableDynamics}$, evaluated at time $t$ and applied to $a$, is equal to the action of the $\star$-automorphism $\alpha(t)$ on the observable $a$.

Given a $\star$-isomorphism $\beta : A \cong B$ between two algebras and a one-parameter $\star$-automorphism group $\alpha$ on $A$ (where $\alpha_t$ denotes the automorphism at time $t \in \mathbb{R}$), this definition constructs a corresponding one-parameter $\star$-automorphism group on $B$. For any $t \in \mathbb{R}$ and $b \in B$, the transported dynamics is given by the composition $\beta \circ \alpha_t \circ \beta^{-1}(b)$.

Let $A$ and $B$ be algebras and $\beta : A \cong B$ be a $\star$-isomorphism. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$, where $\alpha_t$ denotes the automorphism at time $t \in \mathbb{R}$. The transport of the dynamics $\alpha$ along $\beta$ (denoted here as $\alpha \cdot \beta$) defines a one-parameter group of $\star$-automorphisms on $B$. For any $t \in \mathbb{R}$ and $b \in B$, the action of this transported dynamics is given by $$(\alpha \cdot \beta)_t(b) = \beta(\alpha_t(\beta^{-1}(b))),$$ where $\beta^{-1}$ is the inverse $\star$-isomorphism of $\beta$.

A $\star$-isomorphism $\beta: A \simeq\star_a[\mathbb{C}] B$ intertwines two dynamics $\alpha$ (on $A$) and $\gamma$ (on $B$), where $\alpha$ and $\gamma$ are one-parameter groups of $\star$-automorphisms, if for every time $t \in \mathbb{R}$, the following diagram commutes: $\beta \circ \alpha_t = \gamma_t \circ \beta$.

Let $A$ and $B$ be $\star$-algebras over $\mathbb{C}$. For any $\star$-isomorphism $\beta : A \simeq B$ and any one-parameter group of $\star$-automorphisms $\alpha$ on $A$, the $\star$-isomorphism $\beta$ intertwines $\alpha$ with the dynamics $\alpha$ transported along $\beta$ to $B$. That is, for all $t \in \mathbb{R}$, the following diagram commutes: $\beta \circ \alpha_t = (\alpha.\text{transport } \beta)_t \circ \beta$.

Let $A$ and $B$ be $\star$-algebras over $\mathbb{C}$. Let $\alpha$ and $\gamma$ be one-parameter groups of $\star$-automorphisms acting on $A$ and $B$ respectively, and let $\beta: A \cong B$ be a $\star$-isomorphism. Then $\beta$ intertwines $\alpha$ and $\gamma$ (that is, $\beta \circ \alpha_t = \gamma_t \circ \beta$ for all $t \in \mathbb{R}$) if and only if $\gamma$ is the dynamics transported from $\alpha$ along $\beta$ (defined as $\gamma_t = \beta \circ \alpha_t \circ \beta^{-1}$).

Let $A$, $B$, and $C$ be algebras over $\mathbb{C}$. Let $\alpha$, $\gamma$, and $\delta$ be one-parameter groups of $\star$-automorphisms acting on $A$, $B$, and $C$ respectively. Suppose $\beta: A \cong B$ is a $\star$-isomorphism that intertwines $\alpha$ and $\gamma$, meaning $\beta \circ \alpha_t = \gamma_t \circ \beta$ for all $t \in \mathbb{R}$, and $\chi: B \cong C$ is a $\star$-isomorphism that intertwines $\gamma$ and $\delta$, meaning $\chi \circ \gamma_t = \delta_t \circ \chi$ for all $t \in \mathbb{R}$. Then the composed $\star$-isomorphism $\chi \circ \beta: A \cong C$ intertwines $\alpha$ and $\delta$, such that $(\chi \circ \beta) \circ \alpha_t = \delta_t \circ (\chi \circ \beta)$ for all $t \in \mathbb{R}$.

Let $\alpha$ be a one-parameter group of $\star$-automorphisms on a $\mathbb{C}$-algebra $A$. Transporting the group $\alpha$ along the identity $\star$-isomorphism $\text{id}_A: A \cong A$ results in the original group $\alpha$.

Let $A, B,$ and $C$ be $\mathbb{C}$-algebras. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$ (representing the dynamics or flow of a quantum system). Given $\star$-isomorphisms $\beta : A \cong B$ and $\gamma : B \cong C$, successive changes of algebraic coordinates transport the flow $\alpha$ by their composite. That is, transporting $\alpha$ from $A$ to $B$ via $\beta$, and subsequently transporting that result from $B$ to $C$ via $\gamma$, is equivalent to transporting $\alpha$ directly from $A$ to $C$ via the composition $\gamma \circ \beta$.

Let $A$ and $B$ be $\mathbb{C}$-algebras. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$ (representing a reversible quantum flow), and let $\beta : A \cong B$ be a $\star$-isomorphism. Transporting the flow $\alpha$ from $A$ to $B$ via $\beta$, and then transporting the resulting flow back from $B$ to $A$ via the inverse $\star$-isomorphism $\beta^{-1}$, recovers the original flow $\alpha$.

Let $A$ be a $\mathbb{C}$-algebra. Given a one-parameter group of $\star$-automorphisms $\alpha$ on $A$ (representing the reversible dynamics of a quantum system, where $\alpha_t$ is the automorphism at time $t \in \mathbb{R}$) and a $\star$-automorphism $\beta : A \cong A$ (representing a change of coordinates), the conjugation of $\alpha$ by $\beta$ is a new one-parameter group of $\star$-automorphisms. For any time $t$, the resulting automorphism is defined by the composition $\beta \circ \alpha_t \circ \beta^{-1}$.

Let $A$ be a $\mathbb{C}$-algebra. Let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$, where $\alpha_t$ denotes the automorphism at time $t \in \mathbb{R}$. Given a $\star$-automorphism $\beta: A \cong A$, the conjugation of the flow $\alpha$ by $\beta$ (denoted $\text{conj}_\beta \alpha$) satisfies $$(\text{conj}_\beta \alpha)_t(a) = \beta(\alpha_t(\beta^{-1}(a)))$$ for any time $t \in \mathbb{R}$ and any element $a \in A$, where $\beta^{-1}$ is the inverse $\star$-automorphism.

Let $A$ be a $\mathbb{C}$-algebra and let $\alpha$ be a one-parameter group of $\star$-automorphisms on $A$ (representing the reversible dynamics of a quantum system). Let $\text{id}_A$ denote the identity $\star$-automorphism on $A$. Then, conjugating the group $\alpha$ by $\text{id}_A$ results in the original group $\alpha$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$. Let $\alpha$ be a one-parameter group of $\star$-automorphisms of $A$ representing reversible dynamics. For any two $\star$-automorphisms $\beta, \gamma : A \cong A$, let $\alpha.\text{conj } \beta$ denote the one-parameter group defined by $(\alpha.\text{conj } \beta)_t = \beta \circ \alpha_t \circ \beta^{-1}$. Then conjugating $\alpha$ by $\beta$ and subsequently conjugating the result by $\gamma$ is equivalent to conjugating $\alpha$ by the composite $\star$-automorphism $\gamma \circ \beta$: $$(\alpha.\text{conj } \beta).\text{conj } \gamma = \alpha.\text{conj } (\gamma \circ \beta)$$ where $\gamma \circ \beta$ is the composition of $\beta$ followed by $\gamma$.

Let $A$ be a $\star$-algebra over $\mathbb{C}$. Let $\alpha$ be a one-parameter group of $\star$-automorphisms of $A$ (representing the reversible dynamics of a quantum system), and let $\beta$ be a $\star$-automorphism of $A$ with inverse $\beta^{-1}$. Conjugating the group $\alpha$ by $\beta$ and then conjugating the resulting dynamics by $\beta^{-1}$ recovers the original dynamics $\alpha$: $$(\alpha.\text{conj } \beta).\text{conj } \beta^{-1} = \alpha$$

Let $A$ be a $\star$-algebra over $\mathbb{C}$. Let $\alpha$ be a one-parameter group of $\star$-automorphisms of $A$ (representing the reversible dynamics), where $\alpha_t$ is the automorphism at time $t$. For any $\star$-automorphism $\beta$ of $A$ with inverse $\beta^{-1}$, let the conjugation of $\alpha$ by $\beta$ be the one-parameter group defined by $(\alpha.\text{conj } \beta)_t = \beta \circ \alpha_t \circ \beta^{-1}$. Then, conjugating $\alpha$ by $\beta^{-1}$ and subsequently conjugating the result by $\beta$ recovers the original dynamics: $$(\alpha.\text{conj } \beta^{-1}).\text{conj } \beta = \alpha$$