Plain-language statement
Two maps are equal if they agree on all positive inputs with trace one
Exact Lean statement
theorem funext_pos_trace [Fintype dIn]
(h : ∀ M : HermitianMat dIn ℂ, 0 ≤ M → M.trace = 1 → CΛ₁.map M = CΛ₂.map M) :
CΛ₁ = CΛ₂Formal artifact
Lean source
theorem funext_pos_trace [Fintype dIn] (h : ∀ M : HermitianMat dIn ℂ, 0 ≤ M → M.trace = 1 → CΛ₁.map M = CΛ₂.map M) : CΛ₁ = CΛ₂ := by apply funext_pos intro M hM' rcases hM'.eq_or_lt with rfl | hM · simp have h_tr : 0 < M.trace := M.trace_pos hM have := h (M.trace⁻¹ • M) ?_ ?_ · simp only [HermitianMat.mat_smul, LinearMap.map_smul_of_tower] at this convert congr(M.trace • $this) · rw [smul_smul] field_simp simp · rw [smul_smul] field_simp simp · apply smul_nonneg (by positivity) hM' · simp [field]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Bundled.lean:114-132
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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.