Ordered POVMs and Residual Collapse

TL;DR

This paper investigates ordered realizations of discrete POVMs through a residual transform, leading to a collapse process that characterizes equivalence classes of POVMs and their structural properties.

Contribution

It introduces a residual transform-based collapse method for ordered POVMs, characterizing the range and fibers of the resulting equivalence relation.

Findings

Collapsed POVMs have mutually orthogonal non-escape coordinates.

The range of the collapse map consists of POVMs with support projections summing to identity.

Different realizations can share the same collapsed image despite different off-diagonal data.

Abstract

Ordered realizations of discrete POVMs are studied through a residual transform generated by sequential tests. One application of the transform replaces each coordinate by the effect obtained after all earlier tests have failed, and appends the remaining mass as a terminal outcome. Under natural hypotheses, iterating the transform produces a collapsed POVM whose non-escape coordinates are the parts of the original effects that survive all earlier tests. The resulting collapse map gives an equivalence relation on ordered POVM realizations. Its range and fibers are characterized. The range consists of collapsed POVMs, whose non-escape coordinates are mutually orthogonal and whose support projections strongly sum to the identity. The fiber over a collapsed POVM consists of all ordered realizations with the same residually visible compressions. In particular, different ordered realizations, including ones with different off-diagonal coupling data, can have the same collapsed image. After collapse, the non-escape coordinates are fixed under further residual iteration. The remaining dynamics takes place in the escape effect, which is fragmented by a universal scalar functional calculus.