PhyslibAlpha.QuantumMechanics.OperatorAlgebra.Observables.Projection

Let $A$ be an operator algebra. A projection (also referred to as a projection observable or a sharp effect) in $A$ is an element $p \in A$ that is self-adjoint, satisfying $p^* = p$, and idempotent, satisfying $p^2 = p$.

Let $A$ be an operator algebra. Given a projection $p \in A$ (which is an element satisfying $p^* = p$ and $p^2 = p$), this mapping treats $p$ as an observable by utilizing its self-adjoint property. In quantum mechanics, this represents the fact that every projection (or sharp effect) is a specific type of observable.

Let $A$ be an operator algebra. Given a projection $p \in A$ (an element satisfying $p^* = p$ and $p^2 = p$), this mapping treats $p$ as an effect. An effect is an observable $a \in A$ that satisfies $0 \le a \le \mathbb{1}$. Since every projection is self-adjoint and satisfies the bounds $0 \le p \le \mathbb{1}$, it naturally defines an effect in $A$.

Let $A$ be an operator algebra. For any projection $p \in A$ (an element that is self-adjoint, $p^* = p$, and idempotent, $p^2 = p$), the real spectrum of $p$, denoted $\text{spectrum}_{\mathbb{R}}(p)$, is a subset of $\{0, 1\}$.

Given a projection $p$ in an operator algebra $A$, its complement is the projection defined by the element $1 - p$.

Let $A$ be an operator algebra. For any projection $p \in A$ (a self-adjoint and idempotent element), let its complement be defined as $1 - p$. Taking the complement of the complement of $p$ returns the original projection $p$.