Plain-language statement
The convex roof extension of g : KetUpToPhase d → ℝ≥0 applied to a pure state ψ is g (KetUpToPhase.mk ψ).
Exact Lean statement
theorem convex_roof_of_pure (ψ : Ket d) : convex_roof g (pure ψ) = g (KetUpToPhase.mk ψ)
Formal artifact
Lean source
theorem convex_roof_of_pure (ψ : Ket d) : convex_roof g (pure ψ) = g (KetUpToPhase.mk ψ) := by rw [le_antisymm_iff] constructor · apply convex_roof_le use 1; simp only [gt_iff_lt, zero_lt_one, true_and]; use trivial_pEnsemble ψ 0 constructor · exact trivial_pEnsemble_mix ψ 0 · simp only [pure_average_NNReal, Fin.isValue, ← NNReal.coe_le_coe, coe_mk] rw [trivial_pEnsemble_average ψ _ 0] rfl · apply le_convex_roof intro n hnpos e hmix apply le_of_eq simp only [pure_average_NNReal, ← NNReal.coe_inj, coe_mk] have hphase_inv : ∀ a b, Ket.PhaseEquiv.r a b → (NNReal.toReal ∘ g ∘ KetUpToPhase.mk) a = (NNReal.toReal ∘ g ∘ KetUpToPhase.mk) b := by intro a b hab simp only [Function.comp_apply] congr 1 exact congrArg g (Quotient.sound hab) rw [mix_pEnsemble_pure_average (NNReal.toReal ∘ g ∘ KetUpToPhase.mk) hphase_inv hmix] rfl- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entanglement.lean:164-185
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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.
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.
Not achieves Rate gt log dim out
CPTPMap.not_achievesRate_gt_log_dim_out
Plain-language statement
A channel cannot achieve a rate greater than log2(D), where D is the output dimension.
Source project: quantumInfo
Person-level attribution pending.