File: Mathlib/Algebra/Star/CHSH.lean

full_name: IsCHSHTuple
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: Prop

full_name: IsCHSHTuple.A₀B₁_commutes
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: A₀ * B₁ = B₁ * A₀

full_name: IsCHSHTuple.A₁B₁_commutes
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: A₁ * B₁ = B₁ * A₁

full_name: IsCHSHTuple.B₀_sa
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: star B₀ = B₀

full_name: IsCHSHTuple.A₀_sa
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: star A₀ = A₀

full_name: IsCHSHTuple.B₁_inv
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: B₁ ^ 2 = 1

full_name: IsCHSHTuple.A₀B₀_commutes
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: A₀ * B₀ = B₀ * A₀

full_name: IsCHSHTuple.A₁_inv
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: A₁ ^ 2 = 1

full_name: IsCHSHTuple.A₀_inv
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: A₀ ^ 2 = 1

full_name: IsCHSHTuple.A₁_sa
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: star A₁ = A₁

full_name: IsCHSHTuple.B₁_sa
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: star B₁ = B₁

full_name: IsCHSHTuple.A₁B₀_commutes
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: A₁ * B₀ = B₀ * A₁

full_name: IsCHSHTuple.B₀_inv
code: /-- A CHSH tuple in a *-monoid consists of 4 self-adjoint involutions `A₀ A₁ B₀ B₁` such that
the `Aᵢ` commute with the `Bⱼ`.

The physical interpretation is that `A₀` and `A₁` are a pair of boolean observables which
are spacelike separated from another pair `B₀` and `B₁` of boolean observables.
-/
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure IsCHSHTuple {R} [Monoid R] [StarMul R] (A₀ A₁ B₀ B₁ : R) : Prop where
  A₀_inv : A₀ ^ 2 = 1
  A₁_inv : A₁ ^ 2 = 1
  B₀_inv : B₀ ^ 2 = 1
  B₁_inv : B₁ ^ 2 = 1
  A₀_sa : star A₀ = A₀
  A₁_sa : star A₁ = A₁
  B₀_sa : star B₀ = B₀
  B₁_sa : star B₁ = B₁
  A₀B₀_commutes : A₀ * B₀ = B₀ * A₀
  A₀B₁_commutes : A₀ * B₁ = B₁ * A₀
  A₁B₀_commutes : A₁ * B₀ = B₀ * A₁
  A₁B₁_commutes : A₁ * B₁ = B₁ * A₁
goal: B₀ ^ 2 = 1

full_name: CHSH_id
code: theorem CHSH_id [CommRing R] {A₀ A₁ B₀ B₁ : R} (A₀_inv : A₀ ^ 2 = 1) (A₁_inv : A₁ ^ 2 = 1)
    (B₀_inv : B₀ ^ 2 = 1) (B₁_inv : B₁ ^ 2 = 1) :
    (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) =
      4 * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) 
goal: (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) =
  4 * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁)

full_name: CHSH_inequality_of_comm
code: /-- Given a CHSH tuple (A₀, A₁, B₀, B₁) in a *commutative* ordered `*`-algebra over ℝ,
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2`.

(We could work over ℤ[⅟2] if we wanted to!)
-/
theorem CHSH_inequality_of_comm [OrderedCommRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
    [OrderedSMul ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
    A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2 
goal: A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2

full_name: TsirelsonInequality.tsirelson_inequality_aux
code: theorem tsirelson_inequality_aux : √2 * √2 ^ 3 = √2 * (2 * (√2)⁻¹ + 4 * ((√2)⁻¹ * 2⁻¹)) 
goal: √2 * √2 ^ 3 = √2 * (2 * (√2)⁻¹ + 4 * ((√2)⁻¹ * 2⁻¹))

full_name: TsirelsonInequality.sqrt_two_inv_mul_self
code: theorem sqrt_two_inv_mul_self : (√2)⁻¹ * (√2)⁻¹ = (2⁻¹ : ℝ) 
goal: (√2)⁻¹ * (√2)⁻¹ = 2⁻¹

full_name: tsirelson_inequality
code: /-- In a noncommutative ordered `*`-algebra over ℝ,
Tsirelson's bound for a CHSH tuple (A₀, A₁, B₀, B₁) is
`A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2^(3/2) • 1`.

We prove this by providing an explicit sum-of-squares decomposition
of the difference.

(We could work over `ℤ[2^(1/2), 2^(-1/2)]` if we really wanted to!)
-/
theorem tsirelson_inequality [OrderedRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
    [OrderedSMul ℝ R] [StarModule ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
    A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ √2 ^ 3 • (1 : R) 
goal: A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ √2 ^ 3 • 1