feat(QuantumInfo): Cauchy-Schwarz norm bound for Bargmann invariant#1134
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Prove `‖Δ₃‖ ≤ 1` by bridging Braket's `dot` product to the `EuclideanSpace ℂ d` inner product and applying `norm_inner_le_norm`. New results: - `Braket.norm_dot_le_one`: `‖⟨ψ₁|ψ₂⟩‖ ≤ 1` for normalized states - `norm_bargmannInvariantThree_le_one`: `‖Δ₃‖ ≤ 1` Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
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| omit [DecidableEq d] in | ||
| /-- The bra-ket product of normalized states has norm at most 1 (Cauchy-Schwarz). -/ | ||
| lemma Braket.norm_dot_le_one (ψ₁ ψ₂ : Ket d) : ‖〈ψ₁‖ψ₂〉‖ ≤ 1 := by |
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This should be in the BraKet file
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Along with all the private results above.
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Summary
Braket.norm_dot_le_one: the bra-ket product of normalized states satisfies‖⟨ψ₁|ψ₂⟩‖ ≤ 1, proved via Cauchy-Schwarznorm_bargmannInvariantThree_le_one: the Bargmann invariant satisfies‖Δ₃‖ ≤ 1, as a product of three unit-bounded factorsApproach
The proof connects
Braket.dotto theEuclideanSpace ℂ dinner product via private helpers, then applies Mathlib'snorm_inner_le_norm(Cauchy-Schwarz on unit vectors). The Bargmann bound follows by factoring‖a · b · c‖ = ‖a‖ · ‖b‖ · ‖c‖and bounding each factor.The
EuclideanSpacebridge uses three private definitions (ketToEuclidean,ketToEuclidean_norm,dot_eq_euclidean_inner) that are internal to this file and do not affect the public API.Motivation
The norm bound is needed for connecting the Bargmann invariant to the solid angle on the Bloch sphere (Pancharatnam's theorem), where
|Δ₃| = cos(Ω/4)requires|Δ₃| ≤ 1.Build
Rebased onto master (includes #1133). Zero warnings, zero errors on
lake build QuantumInfo.Finite.GeometricPhase.BargmannInvariant.🤖 Generated with Claude Code