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feat(QuantumInfo): cyclic symmetry for Bargmann invariant and phase #1133

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jstoobysmith merged 1 commit into from
May 29, 2026
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feat(QuantumInfo): cyclic symmetry for Bargmann invariant and phase #1133

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24 changes: 22 additions & 2 deletions QuantumInfo/Finite/GeometricPhase/BargmannInvariant.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -24,6 +24,8 @@ geodesic triangle in projective Hilbert space.
* `bargmannPhaseThree_degenerate`: identical states give phase `= 0`
* `bargmannInvariantThree_reverse`: reversing conjugates `Δ₃`
* `bargmannPhaseThree_reverse`: reversing negates the phase (mod 2π)
* `bargmannInvariantThree_cyclic`: cyclic permutation preserves `Δ₃`
* `bargmannPhaseThree_cyclic`: cyclic permutation preserves the phase

## References
* [V. Bargmann, *Note on Wigner's theorem on symmetry operations*,
Expand All @@ -50,20 +52,23 @@ def bargmannPhaseThree (ψ₁ ψ₂ ψ₃ : Ket d) : ℝ :=

/-! ## Degenerate triangles -/

omit [DecidableEq d] in
/-- The Bargmann invariant of three identical states is 1. -/
@[simp]
lemma bargmannInvariantThree_degenerate (ψ : Ket d) :
bargmannInvariantThree ψ ψ ψ = 1 := by
unfold bargmannInvariantThree
simp [Braket.dot_self_eq_one]

omit [DecidableEq d] in
/-- The geometric phase of three identical states is 0. -/
lemma bargmannPhaseThree_degenerate (ψ : Ket d) :
bargmannPhaseThree ψ ψ ψ = 0 := by
unfold bargmannPhaseThree; simp [Complex.arg_one]

/-! ## Conjugacy -/

omit [DecidableEq d] in
/-- Reversing the cyclic order conjugates the invariant. -/
lemma bargmannInvariantThree_reverse (ψ₁ ψ₂ ψ₃ : Ket d) :
bargmannInvariantThree ψ₃ ψ₂ ψ₁ = starRingEnd ℂ (bargmannInvariantThree ψ₁ ψ₂ ψ₃) := by
Expand All @@ -73,10 +78,25 @@ lemma bargmannInvariantThree_reverse (ψ₁ ψ₂ ψ₃ : Ket d) :
← map_mul, ← map_mul]
congr 1; ring

omit [DecidableEq d] in
/-- Reversing the cyclic order negates the geometric phase (mod 2π). -/
lemma bargmannPhaseThree_reverse (ψ₁ ψ₂ ψ₃ : Ket d)
(h : bargmannInvariantThree ψ₁ ψ₂ ψ₃ ≠ 0) :
lemma bargmannPhaseThree_reverse (ψ₁ ψ₂ ψ₃ : Ket d) :
(bargmannPhaseThree ψ₃ ψ₂ ψ₁ : Real.Angle) =
-(bargmannPhaseThree ψ₁ ψ₂ ψ₃ : Real.Angle) := by
unfold bargmannPhaseThree; rw [bargmannInvariantThree_reverse]
exact Complex.arg_conj_coe_angle (bargmannInvariantThree ψ₁ ψ₂ ψ₃)

/-! ## Cyclic symmetry -/

omit [DecidableEq d] in
/-- Cyclic permutation of the three states preserves the Bargmann invariant. -/
lemma bargmannInvariantThree_cyclic (ψ₁ ψ₂ ψ₃ : Ket d) :
bargmannInvariantThree ψ₂ ψ₃ ψ₁ = bargmannInvariantThree ψ₁ ψ₂ ψ₃ := by
unfold bargmannInvariantThree; ring

omit [DecidableEq d] in
/-- Cyclic permutation preserves the geometric phase. -/
lemma bargmannPhaseThree_cyclic (ψ₁ ψ₂ ψ₃ : Ket d) :
bargmannPhaseThree ψ₂ ψ₃ ψ₁ = bargmannPhaseThree ψ₁ ψ₂ ψ₃ := by
unfold bargmannPhaseThree; rw [bargmannInvariantThree_cyclic]

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