PhyslibAlpha.AlgebraicFramework.OrderUnit.State.Discrimination

The success probability of discriminating between two states $\omega_0$ and $\omega_1$ with respective prior probabilities $p_0$ and $p_1$ using a binary test $e$ is defined as $$p_0 \omega_0(e) + p_1 \omega_1(1 - e)$$ where $e$ is an effect (an element such that $0 \le e \le 1$) and $1 - e$ is its complement. This value represents the probability of correctly identifying the state by guessing $0$ when the effect $e$ occurs and guessing $1$ when its complement occurs.

Let $E$ be an ordered additive commutative group with a unit element $1$. For two states $\omega_0, \omega_1 : E \to \mathbb{R}$ (represented as unital positive linear maps) and prior probabilities $p_0, p_1 \in \mathbb{R}$, the success probability of a binary test $e \in E$ (where $0 \le e \le 1$) is given by: $$\text{successProb}(\omega_0, \omega_1, p_0, p_1, e) = p_1 + (p_0 \omega_0(e) - p_1 \omega_1(e))$$ This formula expresses the success probability as the prior probability of guessing state $1$ ($p_1$) plus the discrimination advantage $p_0 \omega_0(e) - p_1 \omega_1(e)$ provided by the test $e$.

Given two states $\omega_0, \omega_1$ (represented as unital positive linear maps from an ordered additive commutative group $E$ with a unit $1$ to $\mathbb{R}$) and their respective prior probabilities $p_0, p_1 \in \mathbb{R}$, the **optimal success probability** is defined as the supremum of the success probabilities over all possible effects $e \in E$ ($0 \le e \le 1$): $$\sup_{0 \le e \le 1} (p_0 \omega_0(e) + p_1 \omega_1(1 - e))$$ This value represents the maximum possible probability of correctly identifying whether the system is in state $\omega_0$ or $\omega_1$ based on a single binary test.

Let $E$ be an ordered additive commutative group with a unit element $1$. For any unital positive linear maps $\omega_0, \omega_1: E \to \mathbb{R}$, non-negative real numbers $p_0, p_1 \ge 0$, and any effect $e \in E$ (defined such that $0 \le e \le 1$), the discrimination advantage satisfies the inequality: $$p_0 \omega_0(e) - p_1 \omega_1(e) \le p_0$$

Let $E$ be a type equipped with an additive commutative group structure, a partial order, and a unit element $1$. Let $\omega_0$ and $\omega_1$ be states (unital positive linear maps from $E$ to $\mathbb{R}$), and let $p_0$ and $p_1$ be non-negative real prior probabilities. Then the set of values $\{p_0 \omega_0(e) - p_1 \omega_1(e) \mid 0 \le e \le 1\}$, representing the "advantage" or distinguishing power of an effect $e$, is bounded above in $\mathbb{R}$.

Let $E$ be an ordered additive commutative group with a unit element $1$. Let $\omega_0, \omega_1: E \to \mathbb{R}$ be states (unital positive linear maps) and $p_0, p_1 \ge 0$ be their respective prior probabilities. The optimal success probability for distinguishing $\omega_0$ and $\omega_1$ is given by $$\text{optimalSuccessProb}(\omega_0, \omega_1, p_0, p_1) = p_1 + \sup_{0 \le e \le 1} (p_0 \omega_0(e) - p_1 \omega_1(e))$$ where the supremum is taken over all effects $e \in E$ (elements satisfying $0 \le e \le 1$). This formula, known as the abstract Helstrom bound, expresses the maximum probability of correctly identifying a state as $p_1$ plus the maximum "advantage" provided by a single binary test.

Let $E$ be an ordered additive commutative group with a unit element $1$. Given two states $\omega_0, \omega_1 : E \to \mathbb{R}$ (unital positive linear maps) and real weights $p_0, p_1$, the `weightedStateBaseNorm` is the base norm of the signed functional $p_0 \omega_0 - p_1 \omega_1$. Operationally, it is defined as $$\|p_0 \omega_0 - p_1 \omega_1\| = 2 \left( \sup_{0 \le e \le 1} (p_0 \omega_0(e) - p_1 \omega_1(e)) \right) - (p_0 - p_1)$$ where the supremum is taken over all effects $e \in E$ (elements satisfying $0 \le e \le 1$). This quantity is normalized such that if $p_0 + p_1 = 1$, the optimal success probability for distinguishing the states is given by the abstract Helstrom formula $\frac{1 + \|p_0 \omega_0 - p_1 \omega_1\|}{2}$.

Let $E$ be an ordered additive commutative group with a unit element $1$. Let $\omega_0, \omega_1: E \to \mathbb{R}$ be two states, defined as unital positive linear maps. For any non-negative weights $p_0, p_1 \ge 0$, the operational base norm of the weighted difference satisfies: $$\|p_0 \omega_0 - p_1 \omega_1\| \le p_0 + p_1$$ where the base norm is defined as $\|p_0 \omega_0 - p_1 \omega_1\| = 2 \left( \sup_{0 \le e \le 1} (p_0 \omega_0(e) - p_1 \omega_1(e)) \right) - (p_0 - p_1)$, with the supremum taken over all effects $e \in E$.

Let $E$ be an ordered additive commutative group with a unit element $1$. Let $\omega_0$ and $\omega_1$ be two states on $E$ (represented as unital positive linear maps to $\mathbb{R}$). For any non-negative real weights $p_0, p_1 \ge 0$, the absolute difference between the weights is bounded by the operational base norm of the weighted difference of the states: $$|p_0 - p_1| \le \|p_0 \omega_0 - p_1 \omega_1\|$$ where the operational base norm is defined as $$\|p_0 \omega_0 - p_1 \omega_1\| = 2 \left( \sup_{0 \le e \le 1} (p_0 \omega_0(e) - p_1 \omega_1(e)) \right) - (p_0 - p_1)$$ and the supremum is taken over all effects $e \in E$ satisfying $0 \le e \le 1$.

Let $E$ be an ordered additive commutative group with a unit element $1$. Let $\omega_0, \omega_1$ be two states on $E$, represented as unital positive linear maps to $\mathbb{R}$. For any non-negative prior probabilities $p_0, p_1 \ge 0$ such that $p_0 + p_1 = 1$, the optimal success probability for discriminating between the two states is given by the abstract Helstrom formula: $$\text{optimalSuccessProb}(\omega_0, \omega_1, p_0, p_1) = \frac{1 + \|p_0 \omega_0 - p_1 \omega_1\|}{2}$$ where $\|p_0 \omega_0 - p_1 \omega_1\|$ is the operational base norm of the weighted difference between the states.

Let $\varphi : E \to F$ be a unital positive linear map between spaces with ordered structures and unit elements. Let $\omega_0, \omega_1$ be states on $F$ and $p_0, p_1$ be non-negative real weights. The maximal weighted advantage obtained by choosing an effect $e \in E$ (where $0 \le e \le 1$) for the pullback states $\omega_0 \circ \varphi$ and $\omega_1 \circ \varphi$ is less than or equal to the maximal weighted advantage obtained by choosing an effect in $F$: $$\sup_{e \in [0, 1]_E} \left( p_0 (\omega_0 \circ \varphi)(e) - p_1 (\omega_1 \circ \varphi)(e) \right) \le \sup_{e \in [0, 1]_F} \left( p_0 \omega_0(e) - p_1 \omega_1(e) \right)$$ where the pullback state $\omega \circ \varphi$ maps $x \in E$ to $\omega(\varphi(x))$.

Let $\varphi : E \to F$ be a unital positive linear map between spaces with an ordered structure and a unit element. Let $\omega_0, \omega_1$ be states on $F$ (i.e., unital positive linear maps $F \to \mathbb{R}$) and $p_0, p_1 \ge 0$ be non-negative real weights. Let $\omega_0 \circ \varphi$ and $\omega_1 \circ \varphi$ denote the pullback states on $E$. Then the base norm of the weighted difference of the pullback states is less than or equal to the base norm of the weighted difference of the original states: $$\|p_0 (\omega_0 \circ \varphi) - p_1 (\omega_1 \circ \varphi)\| \le \|p_0 \omega_0 - p_1 \omega_1\|$$ where the weighted base norm is defined as $\|p_0 \omega_0 - p_1 \omega_1\| = 2 \left( \sup_{0 \le e \le 1} (p_0 \omega_0(e) - p_1 \omega_1(e)) \right) - (p_0 - p_1)$.

Let $E$ and $F$ be ordered additive commutative groups with unit elements. Let $\varphi : E \to F$ be a unital positive linear map (representing a channel in the Heisenberg picture). For any two states $\omega_0, \omega_1$ on $F$ and non-negative prior probabilities $p_0, p_1 \ge 0$, the optimal success probability for discriminating the states after the action of the channel (the pullback states $\omega_0 \circ \varphi$ and $\omega_1 \circ \varphi$) is less than or equal to the optimal success probability for discriminating the original states: $$\text{optimalSuccessProb}(\omega_0 \circ \varphi, \omega_1 \circ \varphi, p_0, p_1) \le \text{optimalSuccessProb}(\omega_0, \omega_1, p_0, p_1)$$ where the optimal success probability is defined as the supremum over all effects $e$ ($0 \le e \le 1$): $$\text{optimalSuccessProb}(\omega, \omega', p, p') = \sup_{0 \le e \le 1} (p \omega(e) + p' \omega'(1 - e)).$$

The operational distance between two states $\omega, \phi \in \mathcal{S}(\mathbb{R}, E)$ is defined as the supremum of the absolute difference between the probabilities they assign to the same effect, expressed as $\sup_{e} |\omega(e) - \phi(e)|$, where the supremum is taken over all effects $e \in E$ such that $0 \le e \le 1$.

Let $E$ be an ordered additive commutative group with a unit element $1$. For any two states $\omega, \phi \in \mathcal{S}[\mathbb{R}, E]$ (unital positive linear maps from $E$ to $\mathbb{R}$) and any effect $e \in E$ (where $0 \le e \le 1$), the absolute difference between the values assigned to the effect by the two states satisfies $$|\omega(e) - \phi(e)| \le 1.$$

Let $E$ be a type equipped with an additive commutative group structure, a partial order, and a unit element $1$. Let $\omega$ and $\varphi$ be unital positive linear maps from $E$ to $\mathbb{R}$. An effect $e$ is defined as an element of $E$ such that $0 \le e \le 1$. Then the set of values $\{ |\omega(e) - \varphi(e)| : 0 \le e \le 1 \}$ is bounded above.

Let $E$ be an ordered additive commutative group with a unit element. For any two states $\omega, \phi \in \mathcal{S}(\mathbb{R}, E)$ (unital positive linear maps from $E$ to $\mathbb{R}$), the operational distance between them is non-negative: $$0 \le \text{operationalDistance}(\omega, \phi).$$ The operational distance is defined as the supremum of the absolute difference between the probabilities the states assign to the same effect $e$, expressed as $\sup_{e} |\omega(e) - \phi(e)|$, where the supremum is taken over all effects $e \in E$ such that $0 \le e \le 1$.

Let $E$ be an ordered additive commutative group with a unit element $1$. For any two states $\omega, \phi \in \mathcal{S}(\mathbb{R}, E)$ (defined as unital positive linear maps from $E$ to $\mathbb{R}$), the operational distance between them is at most $1$: $$\text{operationalDistance}(\omega, \phi) \le 1.$$ The operational distance is defined as the supremum of the absolute difference between the values assigned by the states to the same effect $e$, expressed as $\sup_{e} |\omega(e) - \phi(e)|$, where $e$ ranges over all effects in $E$ such that $0 \le e \le 1$.

For any state $\omega \in \mathcal{S}(\mathbb{R}, E)$ (a unital positive linear map from $E$ to $\mathbb{R}$), the operational distance from the state to itself is zero: $$\text{operationalDistance}(\omega, \omega) = 0.$$

For any two states $\omega, \phi \in \mathcal{S}(\mathbb{R}, E)$, the operational distance between $\omega$ and $\phi$ is equal to the operational distance between $\phi$ and $\omega$: $$\text{operationalDistance}(\omega, \phi) = \text{operationalDistance}(\phi, \omega)$$ where the operational distance is defined as the supremum of the absolute difference between the probabilities assigned to an effect $e$ by the two states, $\sup_{e} |\omega(e) - \phi(e)|$.

For any two states $\omega, \phi \in \mathcal{S}(\mathbb{R}, E)$, the operational distance between them is zero if and only if the states are identical ($\omega = \phi$). Here, the operational distance is defined as $d_{\text{op}}(\omega, \phi) = \sup_{0 \le e \le 1} |\omega(e) - \phi(e)|$, where the supremum is taken over all effects $e$.

For any states $\omega, \phi, \psi \in \mathcal{S}(\mathbb{R}, E)$, the operational distance satisfies the triangle inequality: $$\text{operationalDistance}(\omega, \psi) \le \text{operationalDistance}(\omega, \phi) + \text{operationalDistance}(\phi, \psi)$$ where the operational distance between two states is defined as the supremum of the absolute difference between the probabilities they assign to the same effect $e$, i.e., $\sup_{e} |\omega(e) - \phi(e)|$.

Let $E$ and $F$ be spaces and $\varphi : E \to F$ be a unital positive linear map (representing a quantum channel in the Heisenberg picture). For any two states $\omega, \psi$ on $F$, the operational distance between their pullback states $\omega \circ \varphi$ and $\psi \circ \varphi$ on $E$ is less than or equal to the operational distance between $\omega$ and $\psi$: $$d_{\text{op}}(\omega \circ \varphi, \psi \circ \varphi) \le d_{\text{op}}(\omega, \psi)$$ where the operational distance is defined as $d_{\text{op}}(\omega, \psi) = \sup_{0 \le e \le 1} |\omega(e) - \psi(e)|$, with the supremum taken over all effects $e$.