Mixed convex roof of pure
mixed_convex_roof_of_pure
Plain-language statement
The mixed convex roof extension of f : MState d → ℝ≥0 applied to a pure state ψ is f (pure ψ).
Exact Lean statement
theorem mixed_convex_roof_of_pure (ψ : Ket d) : mixed_convex_roof f (pure ψ) = f (pure ψ)
Formal artifact
Lean source
theorem mixed_convex_roof_of_pure (ψ : Ket d) : mixed_convex_roof f (pure ψ) = f (pure ψ) := by rw [le_antisymm_iff] constructor · apply mixed_convex_roof_le use 1; simp only [gt_iff_lt, zero_lt_one, true_and]; use trivial_mEnsemble (pure ψ) 0 constructor · exact trivial_mEnsemble_mix (pure ψ) 0 · simp only [average_NNReal, Fin.isValue, ← NNReal.coe_le_coe, coe_mk] rw [trivial_mEnsemble_average _ (pure ψ) 0] rfl · apply le_mixed_convex_roof intro n hnpos e hmix replace hpure := mix_mEnsemble_pure_iff_pure.mp hmix apply le_of_eq simp only [average_NNReal, ← NNReal.coe_inj, coe_mk] rw [mix_mEnsemble_pure_average (toReal ∘ f) hmix, Function.comp_apply]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entanglement.lean:189-204
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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.