PhyslibAlpha.ProbabilisticTheory.Examples.GBit

The generalized bit (gbit) is an order-unit space defined on the product $\mathbb{R} \times \mathbb{R}^3$. It is constructed as a norm cone where $\mathbb{R}^3$ is equipped with the supremum norm $\|\cdot\|_\infty$. An element $(r, v) \in \mathbb{R} \times \mathbb{R}^3$ is non-negative if and only if it satisfies the condition $\|v\|_\infty \leq r$.

The state space of the generalized bit (gbit), denoted as $\mathcal{S}[\mathbb{R}, \text{GBit}]$, is equivalent to the octahedron in $\mathbb{R}^3$ defined by the closed $\ell^1$ unit ball $\{ a \in \mathbb{R}^3 \mid \sum_{i=0}^2 |a_i| \le 1 \}$.

There is a linear equivalence between the space of continuous linear functionals on $\mathbb{R}^3$, denoted $((\text{Fin } 3 \to \mathbb{R}) \to_L \mathbb{R})$, and the vector space $\mathbb{R}^3$ itself. This isomorphism maps a functional $f$ to a vector $a \in \mathbb{R}^3$ whose components are the values of $f$ on the standard basis, i.e., $a_i = f(e_i)$ for $i \in \{0, 1, 2\}$.

Let $\text{dualEquiv}^{-1} : \mathbb{R}^3 \to ((\mathbb{R}^3) \to_L \mathbb{R})$ be the inverse of the linear isomorphism between the dual space of $\mathbb{R}^3$ and $\mathbb{R}^3$. The image of the $\ell_1$ unit ball $\{a \in \mathbb{R}^3 \mid \sum_{i=0}^2 |a_i| \leq 1\}$ under $\text{dualEquiv}^{-1}$ is equal to the unit ball of the dual space $\{f \mid \|f\| \leq 1\}$.

Let $j = (i, b)$ be an index where $i \in \{0, 1, 2\}$ and $b \in \{\text{true}, \text{false}\}$. The function `GBit.purePoint` defines a state $\omega_j$ on the generalized bit $GBit \cong \mathbb{R} \times \mathbb{R}^3$. For an element $(r, v) \in \mathbb{R} \times \mathbb{R}^3$, the state is given by the formula: $$\omega_j(r, v) = r + \sigma v_i$$ where $v_i$ is the $i$-th coordinate of $v$, and $\sigma \in \{1, -1\}$ is the sign determined by the boolean $b$. These six states correspond to the vertices of the octahedron that forms the state space of the gbit, where the underlying functional is $\pm 1$ times a coordinate projection.

Let $\text{GBit}$ be the generalized bit, an order-unit space defined on $\mathbb{R} \times \mathbb{R}^3$ equipped with the supremum norm. For any index $j = (i, b)$ where $i \in \{0, 1, 2\}$ and $b \in \{\text{true}, \text{false}\}$, the state $\omega_j$ defined by $\omega_j(r, v) = r + \sigma v_i$ (where $\sigma = 1$ if $b$ is true and $\sigma = -1$ if $b$ is false) is a pure state. That is, $\omega_j$ is an extreme point of the state space $\mathcal{S}(\text{GBit})$.

The function `GBit.purePoint`, which maps each index $j \in \{0, 1, 2\} \times \{\text{true}, \text{false}\}$ to a state $\omega_j$ on the generalized bit $GBit$, is injective. That is, for any indices $j_1, j_2$, if $\omega_{j_1} = \omega_{j_2}$, then $j_1 = j_2$.

The dimension (finite rank) of the generalized bit $\text{GBit}$ as a vector space over the real numbers $\mathbb{R}$ is $4$.

The algebraic state space of the generalized bit $\text{GBit}$ is not a simplex. This implies that the $\text{GBit}$ is a non-classical system, as its state space has 6 pure states (vertices) in a 4-dimensional vector space, which exceeds the number of vertices allowed for a simplex in that dimension.