Sᵥₙ pure tripartite triangle
Sᵥₙ_pure_tripartite_triangle
Plain-language statement
Triangle inequality for pure tripartite states: S(A) ≤ S(B) + S(C).
Exact Lean statement
theorem Sᵥₙ_pure_tripartite_triangle (ψ : Ket ((d₁ × d₂) × d₃)) :
Sᵥₙ (MState.pure ψ).traceRight.traceRight ≤
Sᵥₙ (MState.pure ψ).traceRight.traceLeft + Sᵥₙ (MState.pure ψ).traceLeftFormal artifact
Lean source
theorem Sᵥₙ_pure_tripartite_triangle (ψ : Ket ((d₁ × d₂) × d₃)) : Sᵥₙ (MState.pure ψ).traceRight.traceRight ≤ Sᵥₙ (MState.pure ψ).traceRight.traceLeft + Sᵥₙ (MState.pure ψ).traceLeft := by have h_subadd := Sᵥₙ_subadditivity (MState.pure ψ).assoc.traceLeft obtain ⟨ψ', hψ'⟩ : ∃ ψ', (MState.pure ψ).assoc = _ := MState.relabel_pure_exists ψ _ grind [Sᵥₙ_of_partial_eq, MState.traceLeft_left_assoc, MState.traceLeft_right_assoc, MState.traceRight_assoc]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/SSA.lean:1204-1211
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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.