feat(QuantumInfo): angle-parameterized qubit ket for Pancharatnam connection

QuantumInfo/States/Pure/Pancharatnam.lean

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+/-
+Copyright (c) 2026 Anand Nambakam. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Anand Nambakam
+-/
+module
+
+public import QuantumInfo.States.Pure.BlochSphere
+
+/-!
+# Pancharatnam Connection
+
+The angle-parameterized qubit ket `qubitKet α θ` represents the quantum
+state `cos(α/2)|0⟩ + e^{iθ}sin(α/2)|1⟩`, mapping to the Bloch sphere
+at polar angle `α` and azimuthal angle `θ`.
+
+## Important definitions
+ * `qubitKet`: qubit state from Bloch sphere angles
+
+## Important results
+ * `dot_qubitKet`: inner product in half-angle form
+
+## References
+ * [S. Pancharatnam, *Generalized theory of interference, and its
+   applications*, Proc. Indian Acad. Sci. A 44, 247–262 (1956)][pancharatnam1956]
+-/
+
+open Braket Complex
+
+noncomputable section
+
+/-- The raw vector for `qubitKet`. Internal helper. -/
+private def qubitKetVec (α θ : ℝ) : Fin 2 → ℂ :=
+  ![↑(Real.cos (α / 2)), exp (↑θ * I) * ↑(Real.sin (α / 2))]
+
+/-- A qubit state parameterized by Bloch sphere polar angle `α` and
+    azimuthal angle `θ`: `|ψ⟩ = cos(α/2)|0⟩ + e^{iθ}sin(α/2)|1⟩`. -/
+def qubitKet (α θ : ℝ) : Ket (Fin 2) where
+  vec := qubitKetVec α θ
+  normalized' := by
+    simp only [qubitKetVec, Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one,
+      Complex.norm_real, norm_mul, norm_exp_ofReal_mul_I, one_mul, Real.norm_eq_abs, sq_abs]
+    linarith [Real.sin_sq_add_cos_sq (α / 2)]
+
+@[simp]
+lemma qubitKet_apply_zero (α θ : ℝ) :
+    (qubitKet α θ) (0 : Fin 2) = ↑(Real.cos (α / 2)) := rfl
+
+@[simp]
+lemma qubitKet_apply_one (α θ : ℝ) :
+    (qubitKet α θ) (1 : Fin 2) = exp (↑θ * I) * ↑(Real.sin (α / 2)) := rfl
+
+/-- The inner product of two angle-parameterized qubit states:
+    `⟨ψ₁|ψ₂⟩ = cos(α₁/2)cos(α₂/2) + e^{i(θ₂-θ₁)}sin(α₁/2)sin(α₂/2)`. -/
+lemma dot_qubitKet (α₁ θ₁ α₂ θ₂ : ℝ) :
+    〈qubitKet α₁ θ₁‖qubitKet α₂ θ₂〉 =
+    ↑(Real.cos (α₁ / 2) * Real.cos (α₂ / 2)) +
+    exp (↑(θ₂ - θ₁) * I) * ↑(Real.sin (α₁ / 2) * Real.sin (α₂ / 2)) := by
+  unfold Braket.dot
+  simp only [Fin.sum_univ_two, Bra.eq_conj, qubitKet_apply_zero, qubitKet_apply_one]
+  simp only [map_mul, conj_ofReal, ofReal_mul]
+  rw [show starRingEnd ℂ (exp (↑θ₁ * I)) = exp (-(↑θ₁ * I)) from by
+    rw [← Complex.exp_conj]; congr 1; simp [conj_ofReal, conj_I, mul_neg]]
+  congr 1
+  have : exp (-(↑θ₁ * I)) * ↑(Real.sin (α₁ / 2)) *
+      (exp (↑θ₂ * I) * ↑(Real.sin (α₂ / 2))) =
+      exp (-(↑θ₁ * I) + ↑θ₂ * I) *
+      (↑(Real.sin (α₁ / 2)) * ↑(Real.sin (α₂ / 2))) := by
+    rw [exp_add]; ring
+  rw [this]
+  congr 1
+  congr 1
+  push_cast
+  ring