PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.Lueders

For an effect $e$ in an ordered Jordan-Banach algebra $E$ (where $e$ belongs to the order interval $[0, 1]$), this function returns its unique positive square root $\sqrt{e}$ as determined by the continuous functional calculus for JB-algebras.

For an effect $e$ (an element in the order interval $[0, 1]$) in a normed Jordan algebra $E$, the Lüders map is the positive linear map from $E$ to itself defined by the quadratic representation of the positive square root of $e$, denoted by $x \mapsto U_{\sqrt{e}}(x)$. In the case of a special Jordan algebra, this operation is given by $x \mapsto \sqrt{e}x\sqrt{e}$.

Let $E$ be a normed Jordan algebra. For any effect $e \in [0, 1]$ and any element $x \in E$, the Lüders map of $e$ applied to $x$ is equal to the quadratic representation of the positive square root of $e$ applied to $x$: $$\text{luedersMap}(e)(x) = U_{\sqrt{e}}(x)$$ where $\sqrt{e}$ is the positive square root of $e$ and $U_a$ is the quadratic representation $2L_a^2 - L_{a^2}$.

Let $E$ be a normed Jordan algebra with order unit $1$. For any effect $e \in E$ (where $0 \le e \le 1$), the Lüders map $U_{\sqrt{e}}$ applied to the unit element $1$ results in the effect $e$. That is, $U_{\sqrt{e}}(1) = e$.

For an effect $e$ in a normed Jordan algebra $E$ (where $0 \le e \le 1$), the **Lüders operation** is the operation $\Phi: E \to E$ defined by the mapping $x \mapsto U_{\sqrt{e}}(x)$, where $U_{\sqrt{e}}$ is the quadratic representation of the positive square root of $e$. This operation is a positive linear map such that the image of the identity element is the effect itself ($\Phi(1) = e$), which satisfies the requirement $\Phi(1) \le 1$. It represents the intrinsic Jordan algebra analogue of a single-Kraus measurement operation.

Let $E$ be a normed Jordan algebra and $e \in E$ be an effect (meaning $0 \le e \le 1$). For any element $x \in E$, the Lüders operation of $e$ applied to $x$ is equal to the Lüders map $U_{\sqrt{e}}$ applied to $x$.

For any effect $e$ (an element satisfying $0 \le e \le 1$) in a normed Jordan algebra $E$, the outcome effect of the Lüders operation associated with $e$ is equal to $e$. Specifically, if $\Phi_e$ is the Lüders operation $x \mapsto U_{\sqrt{e}}(x)$, then the outcome effect $\Phi_e(1)$ is equal to $e$.

Let $E$ be a normed Jordan algebra and let $p \in E$ be a Jordan projection, which means it satisfies the idempotency condition $p \circ p = p$. Then the positive square root of $p$ viewed as an effect, denoted $\sqrt{p}$, is equal to $p$ itself.

Let $E$ be a normed Jordan algebra. For any Jordan projection $p \in E$ (an element satisfying $p \circ p = p$), the Lüders map of $p$ is equal to the quadratic representation $U_p$.

Let $E$ be a normed Jordan algebra. Given a state $\omega \in \mathcal{S}(\mathbb{R}, E)$ and an effect $e \in E$ (where $0 \le e \le 1$) such that the probability of the outcome $e$ is non-zero (i.e., $\omega(e) > 0$), the **Lüders conditioning** of $\omega$ on $e$ is the post-measurement state $\omega'$ defined for any $x \in E$ as: $$\omega'(x) = \frac{\omega(U_{\sqrt{e}}(x))}{\omega(e)}$$ where $U_{\sqrt{e}}$ is the quadratic representation of the positive square root of the effect $e$. This construction represents the updated state of a system after a measurement of the effect $e$ has been performed and observed.

Let $E$ be a normed Jordan algebra. Given a state $\omega \in \mathcal{S}(\mathbb{R}, E)$ and an effect $e \in E$ (where $0 \le e \le 1$) such that $\omega(e) > 0$, the Lüders conditioned state $\omega'$ applied to an element $x \in E$ is given by the formula: $$\omega'(x) = \frac{\omega(U_{\sqrt{e}}(x))}{\omega(e)}$$ where $\sqrt{e}$ is the positive square root of the effect $e$ and $U_{\sqrt{e}}$ is the quadratic representation of $\sqrt{e}$.

Let $E$ be a normed Jordan algebra. For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ and any effect $e \in E$ such that the probability of the outcome $e$ is positive ($\omega(e) > 0$), the Lüders conditioned state $\omega'$ satisfies $\omega'(1) = 1$, where $1$ is the identity element of the algebra.

Let $E$ be a real normed Jordan algebra. Let $\omega \in \mathcal{S}(\mathbb{R}, E)$ be a normal state and $e \in E$ be an effect (an element satisfying $0 \le e \le 1$) such that the probability of the outcome $e$ is strictly positive ($\omega(e) > 0$). If the Lüders operation $\Phi_e: E \to E$ defined by $x \mapsto U_{\sqrt{e}}(x)$ is normal (meaning it preserves the suprema of directed sets), then the Lüders conditioned state $\omega'$, defined for any $x \in E$ as $$\omega'(x) = \frac{\omega(U_{\sqrt{e}}(x))}{\omega(e)},$$ is also a normal state.

Let $E$ be a normed Jordan algebra, $p \in E$ be a Jordan projection (satisfying $p \circ p = p$), and $\omega$ be a state on $E$. If the probability of $p$ in state $\omega$ is positive (i.e., $\omega(p) > 0$), then the Lüders conditioned state of $\omega$ on $p$ (viewed as an effect) is equal to the state $\omega$ conditioned on the projection $p$. Specifically, for any $x \in E$, the identity $$\frac{\omega(U_{\sqrt{p}} x)}{\omega(p)} = \frac{\omega(U_p x)}{\omega(p)}$$ holds, where $U$ denotes the quadratic representation.