Sᵥₙ strong subadditivity
Sᵥₙ_strong_subadditivity
Plain-language statement
Strong subadditivity on a tripartite system
Exact Lean statement
theorem Sᵥₙ_strong_subadditivity (ρ₁₂₃ : MState (d₁ × d₂ × d₃)) :
let ρ₁₂Formal artifact
Lean source
theorem Sᵥₙ_strong_subadditivity (ρ₁₂₃ : MState (d₁ × d₂ × d₃)) : let ρ₁₂ := ρ₁₂₃.assoc'.traceRight; let ρ₂₃ := ρ₁₂₃.traceLeft; let ρ₂ := ρ₁₂₃.traceLeft.traceRight; Sᵥₙ ρ₁₂₃ + Sᵥₙ ρ₂ ≤ Sᵥₙ ρ₁₂ + Sᵥₙ ρ₂₃ := by have _ := ρ₁₂₃.nonempty |> nonempty_prod.mp |>.right |> nonempty_prod.mp |>.right -- Apply weak monotonicity to ρBCR, then substitute purification identities have h_wm := Sᵥₙ_wm (ρBCR ρ₁₂₃) rw [S_BC_of_BCR_eq, S_CR_of_BCR_eq, S_B_of_BCR_eq, S_R_of_BCR_eq] at h_wm linarith- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/SSA.lean:1179-1188
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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.