Is Positive
MatrixMap.IsCompletelyPositive.IsPositive
Project documentation
Definition of a CP map, but with Fintype T in the definition instead of a Fin n. -/ theorem of_Fintype {M : MatrixMap A B R} (h : IsCompletelyPositive M) (T : Type*) [Fintype T] [DecidableEq T] : (M.kron (LinearMap.id : MatrixMap T T R)).IsPositive := by obtain ⟨n, ⟨e⟩⟩ : ∃ n : ℕ, Nonempty (T ≃ Fin n) := Finite.exists_equiv_fin T convert h n using 1 h...
Exact Lean statement
theorem IsPositive {M : MatrixMap A B R}
(hM : IsCompletelyPositive M) : IsPositive MFormal artifact
Lean source
theorem IsPositive {M : MatrixMap A B R} (hM : IsCompletelyPositive M) : IsPositive M := by intro x hx let x' : Matrix (A × Fin 1) (A × Fin 1) R := x ⊗ₖ 1 let eqA : (A × Fin 1) ≃ A := (Equiv.prodCongrRight (fun _ ↦ finOneEquiv)).trans (Equiv.prodPUnit A) let eqB : (B × Fin 1) ≃ B := (Equiv.prodCongrRight (fun _ ↦ finOneEquiv)).trans (Equiv.prodPUnit B) specialize @hM 1 (x.submatrix eqA eqA) (Matrix.PosSemidef.submatrix hx _) convert Matrix.PosSemidef.submatrix hM eqB.symm; clear hM --TODO Cleanup ext i j simp only [Matrix.submatrix, Matrix.of_apply] rw [MatrixMap.kron_def] suffices h : M x = ∑ a₁, ∑ a₂, x a₁ a₂ • M (Matrix.single a₁ a₂ 1) by simp [h, Matrix.sum_apply, Matrix.single, eqA, eqB] ac_rfl simp only [← M.map_smul, ← map_sum] congr ext k l simp [Matrix.sum_apply, Matrix.single] rw [Finset.sum_eq_single k] · simp · simp +contextual · simp +contextual- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Unbundled.lean:277-301
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Related declarations
Conj Transpose isometry mul isometry le one
conjTranspose_isometry_mul_isometry_le_one
Project documentation
The operator norm of the conjugate transpose is equal to the operator norm. -/ theorem Matrix.opNorm_conjTranspose_eq_opNorm {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] (A : Matrix m n 𝕜) : Matrix.opNorm Aᴴ = Matrix.opNorm A := by unfold Matrix.opNorm rw [← ContinuousLinearMap.adjoint.norm_map (toEuclideanLin A).toContinuousLine...
Source project: quantumInfo
Person-level attribution pending.
Convex roof of pure
convex_roof_of_pure
Plain-language statement
The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).
Source project: quantumInfo
Person-level attribution pending.
Id achieves Rate log dim
CPTPMap.id_achievesRate_log_dim
Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.