PhyslibAlpha.ProbabilisticTheory.LocalNet.State

Let $\sigma$ be a state of a local net $N$. For any two regions $X$ and $Y$ such that $X \le Y$, and for any observable $a$ in the region $X$, the expectation value of $a$ is consistent when viewed in the larger region $Y$. That is, if $N.incl$ is the inclusion map of observables from $X$ to $Y$, then the expectation value assigned by $\sigma$ in region $Y$ to the included observable $N.incl(a)$ equals the expectation value assigned by $\sigma$ in region $X$ to $a$: $$\sigma.\omega_Y(N.incl(a)) = \sigma.\omega_X(a)$$ where $\sigma.\omega_X$ and $\sigma.\omega_Y$ are the expectation value functionals (states) associated with the respective regions.

Given a net of observables $N$ and two states $\sigma, \tau$ of this net (represented as `NetState N`), along with a probability $t \in [0, 1]$, this function constructs a new state of the net. This new state is the convex combination $t\sigma + (1-t)\tau$. For any region $X$ and any local observable $a$ in that region, the expectation value assigned by the mixed state is calculated as $t \sigma_X(a) + (1 - t) \tau_X(a)$.

Let $N$ be a net of observables. For any two states $\sigma$ and $\tau$ of the net, any probability $t \in [0, 1]$, any region $X$, and any local observable $a \in A(X)$, the expectation value of $a$ in the mixed state $t\sigma + (1-t)\tau$ is the convex combination of the individual expectation values: $$(t\sigma + (1-t)\tau)_X(a) = t \cdot \sigma_X(a) + (1 - t) \cdot \tau_X(a)$$ where $\sigma_X(a)$ and $\tau_X(a)$ denote the expectation values assigned to the observable $a$ by states $\sigma$ and $\tau$ in region $X$, respectively.

A state $\sigma$ of a local net is invariant under a symmetry action $\alpha$ if, for every group element $g \in G$ and every region $X$, the expectation values satisfy $$\omega_{gX} \circ \tau_{g,X} = \omega_X$$ where $\omega_X$ denotes the expectation value functional (state) on the observables in region $X$, and $\tau_{g,X}$ is the symmetry transformation mapping observables in region $X$ to those in the transformed region $g \cdot X$. In other words, the expectation value of a measurement is unchanged when both the measurement and the state are moved by the symmetry.

Given a net state $\sigma$, a symmetry group action $\alpha$ of $G$ on a net of observables, a group element $g \in G$, and a region $X$, the function `pull` returns the local state on the observables $A(X)$ obtained by pulling back the state $\omega_{g \cdot X}$ (the state assigned by $\sigma$ to the region $g \cdot X$) along the symmetry transformation $\tau_g(X) : A(X) \to A(g \cdot X)$.

A state $\sigma$ of a local net $N$ (which assigns a local state $\omega_X$ to each region $X$) is invariant under a symmetry action $\alpha$ of a group $G$ if and only if, for every group element $g \in G$ and every region $X$, the pullback of the local state $\omega_{g \cdot X}$ along the symmetry transformation $\tau_g(X)$ is equal to the local state $\omega_X$. That is, $$\text{IsInvariant}(\alpha, \sigma) \iff \forall g \in G, \forall X, \text{pull}(\alpha, g, X, \sigma) = \omega_X$$

Let $\sigma$ be a state of a local net $N$, which assigns a local state (expectation value functional) $\omega_X$ to every region $X$. For any two regions $X$ and $Y$ such that $X = Y$, let $\text{eqIso}(e)$ be the canonical transport isomorphism between the observables of the two regions induced by the equality $e: X = Y$. Then, pulling back the local state $\omega_Y$ along this isomorphism yields the local state $\omega_X$, i.e., $(\text{eqIso}(e))^* \omega_Y = \omega_X$.

For any region $X$ and any state $\sigma$ of a local net $N$, the pullback of the local state by the identity element $1$ of the symmetry group $G$ at region $X$ is equal to the local state $\omega_X$ assigned by $\sigma$ to $X$. That is, $$\text{pull}(\alpha, 1, X, \sigma) = \omega_X$$ where $\alpha$ is the symmetry action of $G$ on the net.

Let $G$ be a symmetry group acting on a local net of observables $N$ such that for each $g \in G$ and region $X$, there is a symmetry transformation $\tau_g(X) : A(X) \to A(g \cdot X)$. For a state $\sigma$ of the net, let $\text{pull}(g, X, \sigma)$ denote the pullback of the local state at region $g \cdot X$ along the morphism $\tau_g(X)$. For any $g, h \in G$ and region $X$, it holds that $$\text{pull}(gh, X, \sigma) = (\tau_h(X))^* (\text{pull}(g, h \cdot X, \sigma))$$ where $(\tau_h(X))^*$ denotes the pullback operation on the local state space.

Given a net action $\alpha$ of a group $G$ on a net of observables $N$, and an element $g \in G$, the action of $g$ on a net state $\sigma$ (denoted $g \cdot \sigma$) is a new net state. For any region $X$ and any local observable $a \in A(X)$, the expectation value assigned by the transformed state is: $$(g \cdot \sigma)_X(a) = \sigma_{g^{-1} \cdot X}(\tau_{g^{-1}}(X)(a))$$ where $\tau_{g^{-1}}(X) : A(X) \to A(g^{-1} \cdot X)$ is the symmetry transformation corresponding to $g^{-1}$ at the region $X$.

Let $\alpha$ be a net action of a group $G$ on a net of observables $N$. For any group element $g \in G$, net state $\sigma$, and region $X$, the local state (expectation value functional) assigned by the transformed state $g \cdot \sigma$ to the region $X$ is equal to the pullback of the state $\sigma$ from the region $g^{-1} \cdot X$ via the symmetry transformation $\tau_{g^{-1}}(X)$. This is expressed as: $$(g \cdot \sigma)_X = \text{pull}(g^{-1}, X, \sigma)$$

Let $\alpha$ be a net action of a group $G$ on a net of observables $N$. For any group element $g \in G$, net state $\sigma$, region $X$, and local observable $a$ belonging to the algebra $A(X)$ of region $X$, the expectation value of $a$ in the transformed state $g \cdot \sigma$ is given by: $$(g \cdot \sigma)_X(a) = \sigma_{g^{-1} \cdot X}(\tau_{g^{-1}}(X)(a))$$ where $\tau_{g^{-1}}(X) : A(X) \to A(g^{-1} \cdot X)$ is the symmetry transformation corresponding to $g^{-1}$ acting on the region $X$.

Given a net action $\alpha$ of a group $G$ on a net of observables $N$, for any state $\sigma$ of the net, the action of the identity element $1 \in G$ on the state $\sigma$ leaves the state unchanged, i.e., $1 \cdot \sigma = \sigma$.

Let $G$ be a group acting on a net of observables $N$ via a net action $\alpha$. For any group elements $g, h \in G$ and any net state $\sigma \in \text{NetState}(N)$, the action of the product $g \cdot h$ on the state $\sigma$ is equivalent to the action of $g$ applied to the state resulting from the action of $h$ on $\sigma$: $$\text{act}(g \cdot h, \sigma) = \text{act}(g, \text{act}(h, \sigma))$$ This identity confirms that the group action on the space of net states is compatible with the group's multiplication law.

Given a net action $\alpha$ of a group $G$ on a net of observables $N$, the space of net states $\text{NetState}(N)$ is equipped with a group action (specifically a `MulAction`) of $G$. For any group element $g \in G$ and net state $\sigma$, the action $g \cdot \sigma$ is defined by the transformation $\text{act}(\alpha, g, \sigma)$. This action satisfies the identity law $1 \cdot \sigma = \sigma$ and the associativity law $(gh) \cdot \sigma = g \cdot (h \cdot \sigma)$.

Let $\sigma$ be a state of a local net of observables $N$ and $\alpha$ be a symmetry action of a group $G$ on $N$. The state $\sigma$ is invariant under $\alpha$ if and only if for every group element $g \in G$, the transformed state $g \cdot \sigma$ (the action of $g$ on the state $\sigma$) is equal to $\sigma$.

Let $\sigma$ be a state of a local net $N$ and let $\alpha$ be an action of a group $G$ on $N$. If $\sigma$ is invariant under $\alpha$, then for every group element $g \in G$, the transformed state under the symmetry action satisfies $g \cdot \sigma = \sigma$.

Let $\rho$ be a net representation that maps the local observables $A(X)$ of a net $N$ into a global system $B$. Given a global state $\omega$ on $B$ (represented as a unital positive linear map), this definition induces a net state on $N$. For each region $X$, the corresponding local state is defined as the composition $\omega \circ \pi_X$, where $\pi_X: A(X) \to B$ is the representation map for the region $X$.

Let $\rho$ be a net representation that maps the local observables $A(X)$ of a net into a global system $B$. For any global state $\omega$ on $B$, any region $X$, and any local observable $a \in A(X)$, the expectation value of $a$ in the induced net state $(\rho.\text{stateOf}(\omega))$ at region $X$ is given by the application of the global state to the represented observable: $$(\rho.\text{stateOf}(\omega))(X)(a) = \omega(\pi_X(a))$$ where $\pi_X: A(X) \to B$ is the representation map for the region $X$.

Let $\rho$ be a covariant net representation that maps a local net $N$ into a global system $B$, where a group $G$ acts on $B$ via symmetries $\beta_g$. If a global state $\omega$ on $B$ is invariant under the group action, such that $\omega \circ \beta_g = \omega$ for all $g \in G$, then the induced net state $\rho.\text{stateOf}(\omega)$ is invariant under the symmetry action $\alpha$ on the net. In other words, an invariant target state induces an invariant compatible family of local states.