PhyslibAlpha.ProbabilisticTheory.Channel.Operation

Given an order unit space $E$ with an order unit denoted by $1$, an operation is defined as a positive linear map $\Phi: E \to E$ such that $\Phi(1) \le 1$. This condition ensures that the map never sends the certain event $1$ to an element greater than itself, representing a process that may lose probability but never gain it.

For an order unit space $E$, an operation $\Phi \in \text{Operation}(E)$ can be treated as a function $\Phi : E \to E$. This allows the application of an operation $\Phi$ to an element $x \in E$, denoted as $\Phi(x)$, corresponding to its underlying positive linear map.

Let $E$ be an order unit space. For any two operations $op_1$ and $op_2$ on $E$, if $op_1(x) = op_2(x)$ for all $x \in E$, then $op_1 = op_2$. An operation on $E$ is defined as a positive linear map $\Phi: E \to E$ such that $\Phi(1) \le 1$, where $1$ denotes the order unit of $E$.

Let $E$ be an order unit space. For any operation $\Phi$ on $E$ and any element $x \in E$, if $x \ge 0$, then $\Phi(x) \ge 0$. In other words, an operation never sends a non-negative element to a negative one.

Let $E$ be an order unit space with order unit $1$. For any operation $\Phi$ on $E$, the image of the certain event $1$ under $\Phi$ is less than or equal to $1$, i.e., $\Phi(1) \le 1$. This implies that an operation never sends the certain event above itself.

Let $E$ be an ordered real vector space with a unit $1$. An operation $\text{op} : E \to E$ is **normal** if it is order-continuous, meaning that for any directed set $\{x_i\}$ in $E$ that has a supremum $x$, the supremum of the set $\{\text{op}(x_i)\}$ is $\text{op}(x)$.

For an ordered real vector space $E$ with a unit element $1$, the identity operation $\text{id}_E$ is the identity map from $E$ to itself, defined by $\text{id}_E(x) = x$ for all $x \in E$. This map qualifies as an operation because it is a positive linear endomorphism and satisfies the condition $\text{id}_E(1) \le 1$ (specifically, $\text{id}_E(1) = 1$).

Let $E$ be an ordered real vector space with a unit $1$. For any element $x \in E$, the identity operation $\text{id}$ satisfies $\text{id}(x) = x$.

Let $E$ be an order unit space. The identity operation $\text{id}_E: E \to E$ is normal.

Let $E$ be an order unit space with order unit $1$. Given two operations $\phi, \psi: E \to E$ (positive linear maps satisfying $\phi(1) \le 1$ and $\psi(1) \le 1$), their sequential composition $\psi \circ \phi$ is the operation defined by $(\psi \circ \phi)(x) = \psi(\phi(x))$ for all $x \in E$. This composition is itself an operation because the positivity of $\psi$ implies it is monotone, which ensures the subunital condition $(\psi \circ \phi)(1) = \psi(\phi(1)) \le \psi(1) \le 1$ is satisfied.

Let $E$ be an order unit space. For any operations $\psi$ and $\phi$ on $E$ and any element $x \in E$, the application of the sequential composition $\psi \circ \phi$ to $x$ is equal to $\psi(\phi(x))$.

Let $E$ be an order unit space. For any operation $\phi$ on $E$ (defined as a positive linear map $\phi: E \to E$ such that $\phi(1) \le 1$), the sequential composition of the identity operation $\text{id}$ with $\phi$ is equal to $\phi$. That is, $\text{id} \circ \phi = \phi$.

Let $E$ be an order unit space. For any operation $\phi : E \to E$ (a positive linear map satisfying $\phi(1) \le 1$), the composition of $\phi$ with the identity operation $\text{id}$ is equal to $\phi$ itself, denoted as $\phi \circ \text{id} = \phi$.

Let $E$ be an order unit space. For any operations $\phi, \psi, \chi$ on $E$, the sequential composition of these operations is associative, such that $(\chi \circ \psi) \circ \phi = \chi \circ (\psi \circ \phi)$. Here, an operation is defined as a positive linear map $\Phi: E \to E$ satisfying $\Phi(1) \le 1$.

Let $E$ be an order unit space with order unit $1$. Let $\phi$ and $\psi$ be operations on $E$ (positive linear maps such that $\phi(1) \le 1$ and $\psi(1) \le 1$). If $\phi$ and $\psi$ are normal maps, then their sequential composition $\psi \circ \phi$ is also a normal operation.

Given an operation $\Phi$ on an order unit space $E$ and a state $\omega$ on $E$, provided that the probability of the outcome occurring is non-zero (i.e., $\omega(\Phi(1)) > 0$), the post-measurement state (or conditioned state) is defined as the normalized functional $\omega'$ such that for any $x \in E$: $$\omega'(x) = \frac{\omega(\Phi(x))}{\omega(\Phi(1))}$$ This construction represents the updated state of a system after an operation $\Phi$ has been performed and a specific outcome has been observed.

Let $E$ be an order unit space, $\Phi$ be an operation on $E$, and $\omega$ be a state on $E$. If the probability of the outcome occurring is strictly positive, i.e., $\omega(\Phi(1)) > 0$, then for any $x \in E$, the post-measurement state $\omega'$ (the result of conditioning $\omega$ by $\Phi$) is given by the pointwise formula: $$\omega'(x) = \frac{\omega(\Phi(x))}{\omega(\Phi(1))}$$

Let $E$ be an order unit space with order unit $1$. For any operation $\Phi$ on $E$ and any state $\omega$ on $E$ such that the probability of the outcome is strictly positive (i.e., $\omega(\Phi(1)) > 0$), the post-measurement state $\omega'$, defined by $\omega'(x) = \frac{\omega(\Phi(x))}{\omega(\Phi(1))}$, satisfies $$\omega'(1) = 1$$

Let $E$ be an order unit space. Let $\Phi: E \to E$ be an operation and $\omega$ be a state on $E$. If both the operation $\Phi$ and the state $\omega$ are normal (meaning they preserve the suprema of directed sets) and the probability of the outcome is strictly positive, i.e., $\omega(\Phi(1)) > 0$, then the conditioned state $\omega'$, defined by $$\omega'(x) = \frac{\omega(\Phi(x))}{\omega(\Phi(1))}$$ for all $x \in E$, is also normal.

For an operation $\Phi$ on an order unit space $E$, the outcome effect is defined as the image of the certain event $1$ under the map $\Phi$, denoted as $\Phi(1)$. Since $\Phi$ is a positive linear map satisfying $\Phi(1) \le 1$, the resulting value $\Phi(1)$ is an effect in $E$. This represents the probability of the operation "firing" in a given state.

For any operation $\Phi$ on an order unit space $E$ with order unit $1$, the outcome effect of $\Phi$ viewed as an element of $E$ is equal to $\Phi(1)$.