PhyslibAlpha.AlgebraicFramework.JordanOrderUnit.FiniteRank

Given a real Jordan algebra $E$ equipped with trace-determinant data $\tau$, the set of states is defined as the collection of elements $\rho \in E$ that are sums of squares (i.e., satisfy `IsSumSq ρ`) and have a trace equal to 1, such that $\text{tr}(\rho) = 1$.

Given a real Jordan algebra $E$ equipped with trace-determinant data $\tau$, the set of pure states is defined as the subset of density observables $\rho \in \tau.\text{states}$ that are idempotent, satisfying the condition $\rho^2 = \rho$.

Given a real Jordan algebra $E$ with trace-determinant data $\tau$, and a state $\rho$ (an element of the state space $\tau.\text{states}$), the expectation is the linear map from $E$ to $\mathbb{R}$ defined by mapping an element $a \in E$ to the trace of the Jordan product of $\rho$ and $a$, denoted as $\text{tr}(\rho \cdot a)$.

Let $E$ be a real Jordan algebra equipped with trace-determinant data $\tau$. For any state $\rho \in \tau.\text{states}$ and any element $a \in E$, the expectation value of $a$ with respect to the state $\rho$, denoted by $\tau.\text{expectation}(\rho)(a)$, is equal to the trace of the Jordan product of $\rho$ and $a$, denoted $\text{tr}(\rho \cdot a)$.

Let $E$ be a formally real Jordan algebra. For any element $x \in E$, if $x$ belongs to the sum-of-squares cone, then for any real number $r \in \mathbb{R}$, the scalar multiple $(r^2) \cdot x$ also belongs to the sum-of-squares cone.

Let $E$ be a formally real Jordan algebra equipped with trace-determinant data $\tau$. Let $\tau.\text{states} = \{ \rho \in E \mid \text{IsSumSq } \rho, \text{tr}(\rho) = 1 \}$ denote the set of states (also called finite-rank density observables). For any states $\rho, \sigma \in \tau.\text{states}$ and any real numbers $r, s \in \mathbb{R}$ such that $r^2 + s^2 = 1$, the square-weighted mixture $(r^2) \cdot \rho + (s^2) \cdot \sigma$ also belongs to $\tau.\text{states}$.

Let $E$ be a formally real Jordan algebra with trace-determinant data $\tau$. Let $\tau.\text{states}$ denote the set of states $\rho \in E$ such that $\rho$ is a sum of squares and $\text{tr}(\rho) = 1$. For any states $\rho, \sigma \in \tau.\text{states}$ and real numbers $r, s \in \mathbb{R}$ satisfying $r^2 + s^2 = 1$, the expectation of an element $a \in E$ with respect to the square-weighted mixture $(r^2) \rho + (s^2) \sigma$ is given by the weighted sum of the individual expectations: $$\text{tr}(((r^2) \rho + (s^2) \sigma) \cdot a) = r^2 \text{tr}(\rho \cdot a) + s^2 \text{tr}(\sigma \cdot a)$$ where $\text{tr}(\rho \cdot a)$ denotes the expectation of $a$ given the state $\rho$.