PhyslibAlpha.AlgebraicFramework.Representation.Covariance.Finite

For a group $G$ acting on a finite set $\iota$ of outcome labels, this definition is the linear map from the space of functions $\iota \to \mathbb{R}$ to itself induced by an element $\sigma \in G$. It maps each function $f$ to the function $i \mapsto f(\sigma^{-1} \cdot i)$, representing the standard action of a group on the space of functions over a $G$-set by pulling back along the inverse action.

For a group $G$ acting on a finite set $\iota$ of outcome labels, let $f: \iota \to \mathbb{R}$ be a function and $\sigma \in G$. The linear map induced by $\sigma$, denoted as $\text{inducedLinearMap}(\sigma)$, maps $f$ to a new function whose value at an index $i \in \iota$ is given by $f(\sigma^{-1} \cdot i)$.

For an element $\sigma$ of a group $G$ acting on a finite set $\iota$ of outcome labels, this definition is the unital positive linear map (channel) from the space of functions $\iota \to \mathbb{R}$ to itself induced by $\sigma$. It maps each function $f$ to the function $i \mapsto f(\sigma^{-1} \cdot i)$. The map is positive because permuting coordinates preserves non-negativity, and it is unital because it preserves the constant function $\mathbf{1}$.

Let $G$ be a group acting on a finite set $\iota$. For any element $\sigma \in G$, function $f : \iota \to \mathbb{R}$, and index $i \in \iota$, the value of the induced channel $(\text{inducedChannel } \sigma)$ applied to $f$ and evaluated at $i$ is given by $f(\sigma^{-1} \cdot i)$.

Let $G$ be a group acting on a finite set $\iota$ of outcome labels. For any element $\sigma \in G$, the induced channel $M_\sigma: (\iota \to \mathbb{R}) \to (\iota \to \mathbb{R})$, defined as the unital positive linear map such that $(M_\sigma f)(i) = f(\sigma^{-1} \cdot i)$, is an order-automorphism. This means that $M_\sigma$ is a bijection and preserves the pointwise ordering of functions (i.e., $f \le g$ if and only if $M_\sigma f \le M_\sigma g$).

For an element $\sigma$ of a group $G$ acting on a finite set $\iota$ of outcome labels, the induced symmetry is the order-automorphism of the space of functions $\iota \to \mathbb{R}$. It bundles the induced channel $M_\sigma$—which maps a function $f$ to the function $i \mapsto f(\sigma^{-1} \cdot i)$—with the proof that this map is an order-automorphism (i.e., a bijective map that preserves the pointwise ordering of functions).

Let $G$ be a group acting on a finite set $\iota$ of outcome labels. For any element $\sigma \in G$, the underlying unital positive linear map (channel) of the symmetry $\text{inducedSymmetry}(\sigma)$ on the space of functions $\iota \to \mathbb{R}$ is equal to the induced channel $\text{inducedChannel}(\sigma)$.

The induced action is a group homomorphism $G \to \text{Symmetry}(\iota \to \mathbb{R})$ derived from the action of a group $G$ on a finite set of outcome labels $\iota$. For each element $\sigma \in G$, the homomorphism provides a symmetry (specifically, an order-preserving automorphism) of the space of real-valued functions $\iota \to \mathbb{R}$. This symmetry transforms a function $f$ into a new function $i \mapsto f(\sigma^{-1} \cdot i)$, which ensures that the mapping behaves as a left group action.

For any element $\sigma$ of a group $G$ acting on a finite set $\iota$ of outcome labels, the underlying map of the symmetry $\text{inducedAction}(\sigma)$ on the space of functions $\iota \to \mathbb{R}$ is equal to the induced channel $\text{inducedChannel}(\sigma)$.

Let $G$ be a group acting on finite sets of outcome labels $\iota$ and $\kappa$. For any such set $X$, let the induced action of $G$ on the space of functions $X \to \mathbb{R}$ be defined by $(\sigma \cdot f)(x) = f(\sigma^{-1} \cdot x)$ for $\sigma \in G$. Let $E$ be a physical system (an ordered $\mathbb{R}$-module) and $\rho : G \to \text{Symmetry}(E)$ be a group homomorphism representing the symmetry on $E$. Consider a finite-outcome measurement $M : (\iota \to \mathbb{R}) \to E$ and a classical post-processing channel $K : (\kappa \to \mathbb{R}) \to (\iota \to \mathbb{R})$. If $M$ is covariant with respect to the induced action on $\iota \to \mathbb{R}$ and $\rho$, and $K$ is equivariant (covariant with respect to the induced actions on $\kappa \to \mathbb{R}$ and $\iota \to \mathbb{R}$), then the post-processed measurement $M \circ K$ is covariant with respect to the induced action on $\kappa \to \mathbb{R}$ and $\rho$.

Let $G$ be a group, and let $\iota$ and $\kappa$ be types (representing outcome label sets) equipped with $G$-actions. Suppose $f: \kappa \to \iota$ is a $G$-equivariant map, such that for all $g \in G$ and $k \in \kappa$, the condition $f(g \cdot k) = g \cdot f(k)$ holds. Then the classical pullback channel $(\text{classicalPullback } f) : (\iota \to \mathbb{R}) \to (\kappa \to \mathbb{R})$, defined by precomposing a function with $f$, is covariant with respect to the induced group actions on the function spaces. The induced action of $g \in G$ on a function $p$ is given by $(g \cdot p)(x) = p(g^{-1} \cdot x)$.

Let $G$ be a group acting on finite sets of outcome labels $\iota$ and $\kappa$. Let $E$ be a physical system represented by an ordered real vector space with a unit $1 \in E$, and let $\rho: G \to \text{Symmetry}(E)$ be a group homomorphism representing the symmetry action of $G$ on the system. Suppose $J : (\iota \times \kappa \to \mathbb{R}) \to E$ is a joint measurement (a unital positive linear map). If $J$ is covariant with respect to the diagonal action of $G$ on $\iota \times \kappa$ and the symmetry $\rho$ on $E$ (i.e., $J(\sigma \cdot f) = \rho(\sigma)(J(f))$ for all $\sigma \in G$, where the induced action on functions is $(\sigma \cdot f)(x) = f(\sigma^{-1} \cdot x)$), then the marginal measurement on $\iota$ is also covariant. The marginal measurement on $\iota$ is defined as the post-processing of $J$ by the classical channel mapping $f \in (\iota \to \mathbb{R})$ to $f \circ \text{pr}_1 \in (\iota \times \kappa \to \mathbb{R})$, where $\text{pr}_1: \iota \times \kappa \to \iota$ is the projection onto the first factor. This marginal is covariant with respect to the action of $G$ on $\iota$ and the symmetry $\rho$.