Physlib.ProbabilisticTheory.Weight.Basic

Weights

i. Overview

A state assigns each positive observable a nonnegative expectation value, normalized so the certain outcome reads exactly `1`. A *weight* generalizes this by dropping the normalization and letting the values be extended nonnegative reals, possibly `+∞`. This matters especially in infinite dimensions: the trace on `B(H)` is only finite on the trace-class operators and genuinely diverges to `+∞` elsewhere. A state is then the special case that happens to be finite everywhere and normalized.

ii. Key results

- `Weight.monotone` : weights are monotone on the positive cone. - `Weight.IsFinite.isSemifinite` : a finite weight is automatically semifinite. - `Weight.IsFinite.normalize_isState` : rescaling a finite weight that's nonzero at the order unit turns it into a state.

iii. Table of contents

• A. Weights
• B. States as weights

iv. References

- G. Ludwig, *Foundations of Quantum Mechanics I*, Springer, 1983. <https://link.springer.com/book/10.1007/978-3-642-86751-4>

A. Weights

B. States as weights