PhyslibAlpha.AlgebraicFramework.Measurement.Postprocessing

Given a measurement $M : (\iota \to \mathbb{R}) \to E$ and a classical channel $K : (\kappa \to \mathbb{R}) \to (\iota \to \mathbb{R})$, where both are represented as positive unital linear maps in the Heisenberg convention, the postprocessing of $M$ through $K$ is the composition $M \circ K$. This resulting map is a measurement from the coarse outcome space $\kappa$ to the effect space $E$.

Given a measurement $M : (\iota \to \mathbb{R}) \to E$, a classical channel $K : (\kappa \to \mathbb{R}) \to (\iota \to \mathbb{R})$ (where both are positive unital linear maps), and a function $f : \kappa \to \mathbb{R}$, the evaluation of the postprocessed measurement at $f$ is given by $M(K(f))$.

Let $M : (\iota \to \mathbb{R}) \to E$ be a measurement, represented as a positive unital linear map from the classical system $\iota \to \mathbb{R}$ to an effect space $E$. Postprocessing $M$ through the identity channel on $\iota \to \mathbb{R}$ results in the original measurement $M$.

Let $M : (\iota \to \mathbb{R}) \to E$ be a measurement, and let $K : (\kappa \to \mathbb{R}) \to (\iota \to \mathbb{R})$ and $L : (\mu \to \mathbb{R}) \to (\kappa \to \mathbb{R})$ be classical channels (represented as positive unital linear maps). Postprocessing the measurement $M$ first through channel $K$ and then through channel $L$ is equivalent to postprocessing $M$ once through the composite channel $K \circ L$. That is: $$\text{postprocess}(\text{postprocess}(M, K), L) = \text{postprocess}(M, K \circ L)$$

Let $\iota$ and $\kappa$ be types representing finite outcome sets. This equivalence identifies a family of effects $\{e_k\}_{k \in \kappa}$ in the classical system $\mathbb{R}^\iota$ that sum to the identity (i.e., $\sum_{k \in \kappa} e_k(i) = 1$ for all $i \in \iota$) with a row-stochastic matrix $P: \iota \times \kappa \to \mathbb{R}$. Here, an effect $e_k$ is a function $\iota \to [0, 1]$, and a row-stochastic matrix $P$ satisfies $P(i, k) \geq 0$ for all $i, k$ and $\sum_{k \in \kappa} P(i, k) = 1$ for all $i \in \iota$. The correspondence is defined by the relation $P(i, k) = e_k(i)$.

For finite outcome sets $\iota$ and $\kappa$, this equivalence identifies classical channels $K: \mathbb{R}^\kappa \to \mathbb{R}^\iota$ (represented as positive unital linear maps) with row-stochastic matrices $P: \iota \times \kappa \to \mathbb{R}$. A matrix $P$ is row-stochastic if it satisfies $P(i, k) \geq 0$ for all $i \in \iota, k \in \kappa$ and the normalization condition $\sum_{k \in \kappa} P(i, k) = 1$ for every $i \in \iota$. The correspondence relates a channel $K$ to a matrix $P$ such that for any function $g: \kappa \to \mathbb{R}$, the value of the processed function at $i \in \iota$ is given by $(Kg)(i) = \sum_{k \in \kappa} P(i, k) g(k)$.

Given a map $f : \kappa \to \iota$ between outcome types, the classical pullback is the positive unital linear map $K : (\iota \to \mathbb{R}) \to (\kappa \to \mathbb{R})$ that sends a function $g : \iota \to \mathbb{R}$ to the composition $g \circ f$. This map represents a deterministic classical channel where an outcome $k \in \kappa$ is assigned the value $f(k) \in \iota$; in the Heisenberg picture, it pulls back observables (functions on $\iota$) to functions on $\kappa$.

For any set of outcomes $\iota$, the classical pullback associated with the identity function $\text{id} : \iota \to \iota$ is the identity unital positive linear map on the space of observables $(\iota \to \mathbb{R})$.

For any outcome types $\iota, \kappa$, and $\mu$, and any relabeling maps $f : \kappa \to \iota$ and $g : \mu \to \kappa$, the composition of the classical pullback along $g$ and the classical pullback along $f$ is equal to the classical pullback along the composite map $f \circ g$: $$(\text{classicalPullback } g) \circ (\text{classicalPullback } f) = \text{classicalPullback } (f \circ g)$$ where $\text{classicalPullback } f$ is the positive unital linear map that pulls back observables (functions to $\mathbb{R}$) along $f$. This demonstrates that the classical pullback is a contravariant functor from outcome types to classical channels.

Given a map between outcome types $f : \kappa \to \iota$, the classical pullback of $f$ applied to a function $g : \iota \to \mathbb{R}$ and evaluated at an outcome $k \in \kappa$ is equal to $g(f(k))$.