feat(QuantumInfo): Fisher-Rao metric and Cramer-Rao bound on finite simplex

[naype888-cloud]

Summary

• Define the Fisher–Rao inner product on the open probability simplex
• Prove Cauchy–Schwarz for the Fisher–Rao inner product (sorry-free)
• Derive the classical Cramér–Rao lower bound on estimator variance (sorry-free)
• Single new file QuantumInfo/ForMathlib/FisherRao.lean plus its import line

t-quantum-info

Definitions added

Name	Meaning
FisherRao.OpenSimplex	Point on the open probability simplex Δ°(α)
FisherRao.OpenSimplex.fisherRaoInner	gₚ(u, v) = Σᵢ uᵢvᵢ/pᵢ
FisherRao.fisherInfo	Classical Fisher information I(θ) = Σᵢ (∂pᵢ/∂θ)²/pᵢ
FisherRao.expect	Weighted expectation E_p[f]
FisherRao.variance	Weighted variance Var_p(f)

Theorems

Name	Statement
fisherRao_cauchy_schwarz	gₚ(u,v)² ≤ gₚ(u,u) · gₚ(v,v)
fisherRaoSq_eq_zero_iff	Positive definiteness of the Fisher–Rao form
fisherInfo_eq_fisherRaoSq	I(θ) = ‖dp/dθ‖²_FR
cramerRao	Var(T) ≥ 1/I(θ) for unbiased estimators

Context

Neither Mathlib nor PhysLean currently have Fisher information, the Fisher–Rao metric, or the Cramér–Rao bound formalized. This PR fills that gap using only Mathlib primitives (Finset.sum, div, field_simp).

The Cramér–Rao proof reduces to Cauchy–Schwarz on the Fisher–Rao inner product — the same obstruction kernel underlying the Robertson–Schrödinger uncertainty relation already in PhysLean.

Test plan

• CI passes (Build, Lint, imports check)
• No sorry, no new axioms beyond standard

AI-assisted-by: Anthropic Claude

[github-actions]

Thank you for this pull-request (PR). If this is your first PR, welcome to the community!

Below is what will happen next. Please read carefully if you are not familiar with the process. You may open other PRs while this one is being reviewed, and can stack PRs on top of each other, so don't let these steps slow you down.

1. Some automated checks will be run on your PR. You can see the results of these checks at the buttom of your PR page. If any of these checks fail, you will need to fix the issues before your PR can be merged. You can learn more about these here, including how to run them locally, which is sometimes quicker than relying on the GitHub Actions. If you have never had a PR merged before, you may have to wait for a reviewer to manually start these checks (this is for security).

2. A reviewer will look at your PR and may ask you to make changes. This may happen a couple of days after you submit your PR, so you may need to be patient. But it should not be longer than that - if it is please bring it to the attention of the community on the Zulip. The level of review will depend on where your PR is submitted. If it is submitted to ./Physlib or ./QuantumInfo, the review will be more thorough than if it is submitted to ./PhyslibAlpha. You can find out more about what the review process is looking for in our review guidelines. If a reviewer adds an awaiting-author label to your PR, address the review comments, then please remove that label by adding a comment with -awaiting-author. This helps us keep track of reviews.

3. The reviewer will either approve your PR, or request more changes (in which case we return to step 2). Once your PR is approved, it will be merged by a maintainer, this should happen shortly after approval, though you may get more comments at this stage.

Tip: The easiest way to get have a fast review is to submit a PR that is small and self-contained, and has clear documentation explaining why things are the way they are in your chages.

If you have any problems or questions, please reach out to the community on the Zulip.

[naype888-cloud]

This PR formalizes the Fisher–Rao metric and the Cramér–Rao bound for finite sample spaces — neither exists in Mathlib or PhysLean today.

Why here: The Cramér–Rao proof reduces to Cauchy–Schwarz on a weighted inner product, which is the same obstruction kernel behind Robertson–Schrödinger (already in PhysLean via Cipollina and Diem). Having Fisher–Rao in ForMathlib/ makes it available as infrastructure for quantum information geometry — in particular, connecting the algebraic uncertainty framework to the Fubini-Study/Fisher-Rao bridge that @Zayn described in the Integrating QuantumInfo thread.

What it gives you:

• fisherRaoInner — the metric on the open simplex, with positive definiteness proved
• fisherRao_cauchy_schwarz — sorry-free
• cramerRao — sorry-free, derived from the above
• fisherInfo_eq_fisherRaoSq — Fisher information = FR squared norm of the score vector
  No new axioms, no sorry, Mathlib-only imports. Happy to adjust naming or placement if reviewers prefer a different location.

atte:Eduardo Nava Hernandez (Lalo)

[naype888-cloud]

im herre to write the personal coment, by mysself, i do this to help @zyan to complete his own comit, about bloch sphere i think. so i need it to, becouse the uncerntanly dimensional elemental principle can be fited here on Hilbert 4+

[naype888-cloud]

Closing in favor of #1657, which ports this content to PhyslibAlpha.AlgebraicFramework.InformationGeometry.FisherRao (the intended home for the estimation-geometry layer, per maintainer feedback in the 'Fisher information and Fubini-Study' thread). This QuantumInfo version is superseded; please review #1657 instead.