PhyslibAlpha.ProbabilisticTheory.Channel.Symmetry

A channel $\phi: E \to E$ is an **order-automorphism** if there exists a channel $\psi: E \to E$ that acts as its two-sided inverse, such that $\psi \circ \phi = \text{id}_E$ and $\phi \circ \psi = \text{id}_E$. This is equivalent to the condition that $\phi$ is a bijective linear map where both $\phi$ and its inverse $\phi^{-1}$ are positive and unital.

The identity channel $\text{id}_E: E \to E$ on a system $E$ is an order-automorphism.

Given a channel $\phi: E \to E$ and a proof $h$ that $\phi$ is an order-automorphism, this function returns the channel $\phi^{-1}: E \to E$ that acts as the two-sided inverse of $\phi$ such that $\phi^{-1} \circ \phi = \text{id}_E$ and $\phi \circ \phi^{-1} = \text{id}_E$.

Let $\phi$ be a channel on a space $E$. If $\phi$ is an order-automorphism and $\phi^{-1}$ denotes its inverse channel, then the composition of $\phi^{-1}$ and $\phi$ is the identity channel on $E$: $$\phi^{-1} \circ \phi = \text{id}_E$$

Let $\phi$ be a channel on a space $E$. If $\phi$ is an order-automorphism and $\phi^{-1}$ denotes its inverse channel, then the composition of $\phi$ and $\phi^{-1}$ is the identity channel on $E$: $$\phi \circ \phi^{-1} = \text{id}_E$$

Let $\phi$ be a channel on a space $E$. If $\phi$ is an order-automorphism and $\phi^{-1}$ denotes its inverse channel, then for any $x \in E$, applying $\phi$ and then $\phi^{-1}$ returns the original element $x$: $$\phi^{-1}(\phi(x)) = x$$

Let $\phi$ be an order-automorphism on a space $E$, and let $\phi^{-1}$ denote its inverse channel. For any $x \in E$, applying the inverse channel followed by the channel $\phi$ yields the original element: $$\phi(\phi^{-1}(x)) = x$$

Let $\phi$ be a channel. If $\phi$ is an order-automorphism, then its inverse channel $\phi^{-1}$ is also an order-automorphism.

Let $E$ be an order unit space. If $\phi: E \to E$ and $\psi: E \to E$ are channels that are both order-automorphisms, then their composition $\phi \circ \psi$ is also an order-automorphism. An order-automorphism is a channel that possesses a two-sided inverse which is also a channel.

Let $\phi$ be a channel. If $\phi$ is an order-automorphism, then $\phi$ is a bijective function.

For an order unit space $E$, a **symmetry** is defined as a channel $\phi : E \to E$ that is an order-automorphism. A channel $\phi$ is an order-automorphism if there exists a channel $\psi : E \to E$ such that $\psi \circ \phi = \text{id}_E$ and $\phi \circ \psi = \text{id}_E$. The set of all such symmetries forms a group under composition.

Let $E$ be an order unit space. For any two symmetries $\phi$ and $\psi$ of $E$, if $\phi(x) = \psi(x)$ for all $x \in E$, then $\phi = \psi$.

The identity element $1$ of the symmetry group $\text{Symmetry}(E)$ for an order unit space $E$ is defined as the identity channel $\text{id}_E$ on $E$.

For an order unit space $E$, the multiplication operation on the symmetry group $\text{Symmetry}(E)$ is defined such that for any two symmetries $\phi, \psi \in \text{Symmetry}(E)$, their product $\phi \cdot \psi$ is given by the composition of the underlying channels $\phi \circ \psi$.

For an order unit space $E$, this instance defines the inverse operation for the symmetry group $\text{Symmetry}(E)$. For any symmetry $\phi \in \text{Symmetry}(E)$, its inverse $\phi^{-1}$ is the unique order-automorphism that satisfies $\phi^{-1} \circ \phi = \text{id}_E$ and $\phi \circ \phi^{-1} = \text{id}_E$.

Let $E$ be an order unit space. The underlying map of the identity element $1$ in the symmetry group $\text{Symmetry}(E)$ is the identity channel $\text{id}_E$ on $E$.

Let $E$ be an order unit space. For any two symmetries $\phi, \psi \in \text{Symmetry}(E)$, the underlying map of the product $\phi \cdot \psi$ is equal to the composition of their underlying maps, denoted as $\phi \circ \psi$.

Let $E$ be an order unit space. For any symmetry $\phi \in \text{Symmetry}(E)$, the underlying map of the group inverse $\phi^{-1}$ is equal to the inverse channel of $\phi$ (the two-sided inverse channel $\phi^{-1}$ that exists by virtue of $\phi$ being an order-automorphism).

For an order unit space $E$, the type $\text{Symmetry}(E)$ forms a group. The group operation is the composition of channels $\phi \circ \psi$, the identity element is the identity channel $\text{id}_E$, and the inverse of a symmetry $\phi$ is the inverse channel $\phi^{-1}$.

For an order unit space $E$, the group of symmetries $\text{Symmetry}(E)$ is canonically isomorphic to the group of units $(\text{Channel}(E, E))^\times$ of the monoid of channels (unital positive linear maps from $E$ to itself). This isomorphism identifies a symmetry $\phi$, defined as an order-automorphism, with the invertible element in the channel monoid whose value is $\phi$ and whose inverse is the inverse channel $\phi^{-1}$.

For an order unit space $E$, the group of symmetries $\text{Symmetry}(E)$ acts on the state space $\mathcal{S}(E)$ (denoted as `𝓢[ℝ, E]`). Given a symmetry $\phi \in \text{Symmetry}(E)$ and a state $\omega \in \mathcal{S}(E)$, the action is defined by $\phi \cdot \omega = \omega \circ \phi^{-1}$. This transformation corresponds to the Schrödinger-picture pushforward of the state $\omega$ along the channel $\phi^{-1}$, where the composition $\omega \circ \phi^{-1}$ remains a valid state (unital and positive) because both $\omega$ and $\phi^{-1}$ are unital positive linear maps.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space (the set of unital positive linear maps from $E$ to $\mathbb{R}$). For any symmetry $\phi \in \text{Symmetry}(E)$ and any state $\omega \in \mathcal{S}(E)$, the action of the symmetry on the state, denoted $\phi \cdot \omega$, is defined by the composition $\phi \cdot \omega = \omega \circ \phi^{-1}$, where $\phi^{-1}$ is the inverse symmetry of $\phi$.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space (denoted as `𝓢[ℝ, E]`). Given a group $G$ and a group homomorphism $\rho : G \to \text{Symmetry}(E)$, where $\text{Symmetry}(E)$ is the group of order-automorphisms (symmetries) of $E$, this definition specifies the induced action of $G$ on the state space $\mathcal{S}(E)$. For any group element $g \in G$ and state $\omega \in \mathcal{S}(E)$, the action results in a new state defined by $\omega \circ \rho(g)^{-1}$.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space (the set of unital positive linear maps from $E$ to $\mathbb{R}$). Let $G$ be a group and $\rho : G \to \text{Symmetry}(E)$ be a group homomorphism, where $\text{Symmetry}(E)$ is the group of order-automorphisms of $E$. For any state $\omega \in \mathcal{S}(E)$, the induced action of the group identity element $1 \in G$ on the state $\omega$ returns the original state, i.e., $\text{stateSMul}(\rho, 1, \omega) = \omega$.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space. Given a group $G$ and a group homomorphism $\rho : G \to \text{Symmetry}(E)$, where $\text{Symmetry}(E)$ is the group of order-automorphisms of $E$, let $g \cdot \omega$ denote the induced action of an element $g \in G$ on a state $\omega \in \mathcal{S}(E)$ (formally defined as `stateSMul`). For any $g, h \in G$ and $\omega \in \mathcal{S}(E)$, the action of the product $g \cdot h$ on the state $\omega$ satisfies $$(g \cdot h) \cdot \omega = g \cdot (h \cdot \omega).$$

For an order unit space $E$, an element $G$ of the one-parameter automorphism group (representing the reversible dynamics of the system) is coerced into a function that maps a real number $t \in \mathbb{R}$ to a quantum channel $G(t) : E \to E$. In this context, each $G(t)$ is a unital completely positive (UCP) map.

For an order unit space $E$ and a one-parameter automorphism group $\alpha$ representing the reversible dynamics of the system, the channel corresponding to the time parameter $t = 0$ is the identity channel $\text{id}_E$.

For a one-parameter automorphism group $\alpha$ on an order unit space $E$, and for any real numbers $s$ and $t$, the transformation $\alpha(s + t)$ is equal to the composition of the transformations $\alpha(s)$ and $\alpha(t)$, such that $\alpha(s + t) = \alpha(s) \circ \alpha(t)$.

Let $E$ be a physical system and $\alpha$ be a one-parameter group of symmetries on $E$. For any $t \in \mathbb{R}$, the transformation $\alpha(t)$ is an order-automorphism. This means that $\alpha(t)$ is a channel with a channel inverse $\alpha(-t)$ such that $\alpha(-t) \circ \alpha(t) = \text{id}_E$ and $\alpha(t) \circ \alpha(-t) = \text{id}_E$.

For a one-parameter automorphism group $\alpha$ on an order unit space $E$, this definition constructs the group homomorphism from the additive group of real numbers $(\mathbb{R}, +)$, represented as $\text{Multiplicative } \mathbb{R}$, to the group of symmetries $\text{Symmetry}(E)$. The homomorphism maps each $t \in \mathbb{R}$ to the corresponding symmetry $\alpha(t) \in \text{Symmetry}(E)$.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space (denoted as `𝓢[ℝ, E]`). For a one-parameter automorphism group $\alpha$ on $E$, a time $t \in \mathbb{R}$, and an initial state $\omega \in \mathcal{S}(E)$, this definition describes the evolved state at time $t$. The evolution is defined by the action of the symmetry $\alpha(t)$ on the state space, specifically mapping $\omega$ to $\omega \circ \alpha(t)^{-1}$. This evolution "moves" the state forward by time $t$ and satisfies the group property, meaning $\alpha.\text{stateEvolution}(s+t, \omega) = \alpha.\text{stateEvolution}(s, \alpha.\text{stateEvolution}(t, \omega))$.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space. For a one-parameter automorphism group $\alpha$ on $E$ and any initial state $\omega \in \mathcal{S}(E)$, the state evolution at time $t=0$ is equal to the initial state, i.e., $\alpha.\text{stateEvolution}(0, \omega) = \omega$.

Let $E$ be an order unit space and $\mathcal{S}(E)$ be its state space (denoted as `𝓢[ℝ, E]`). For a one-parameter automorphism group $\alpha$ on $E$, any times $s, t \in \mathbb{R}$, and an initial state $\omega \in \mathcal{S}(E)$, the state evolution satisfies the group property: $$\alpha.\text{stateEvolution}(s + t, \omega) = \alpha.\text{stateEvolution}(s, \alpha.\text{stateEvolution}(t, \omega))$$ This means that evolving a state for time $s+t$ is equivalent to evolving the state for time $t$ and then evolving the resulting state for an additional time $s$.