Plain-language statement
The kernel condition σ.M.ker ≤ ρ.M.ker is preserved under partial trace. This follows because supp(ρ) ⊆ supp(σ) implies supp(Tr_B ρ) ⊆ supp(Tr_B σ): if v ∈ supp(Tr_B ρ), then ⟨v, (Tr_B ρ) v⟩ > 0, so for some basis vector e_b we have v ⊗ e_b ∈ supp(ρ) ⊆ supp(σ), hence ⟨v, (Tr_B σ) v⟩ ≥ ⟨v ⊗ e_b, σ (v ⊗ e_b)⟩ > 0.
Exact Lean statement
theorem ker_le_traceRight {ρ σ : MState (dA × dB)}
(hker : σ.M.ker ≤ ρ.M.ker) :
σ.traceRight.M.ker ≤ ρ.traceRight.M.kerFormal artifact
Lean source
theorem ker_le_traceRight {ρ σ : MState (dA × dB)} (hker : σ.M.ker ≤ ρ.M.ker) : σ.traceRight.M.ker ≤ ρ.traceRight.M.ker := by simp only [MState.traceRight_M] intro v hv rw [HermitianMat.mem_ker_iff_mulVec_zero] at hv ⊢ have hv' : σ.M.mat.traceRight.mulVec v.ofLp = 0 := by rwa [HermitianMat.traceRight_mat] at hv have h_inner_zero : star v.ofLp ⬝ᵥ σ.M.mat.traceRight.mulVec v.ofLp = 0 := by rw [hv']; simp [dotProduct] rw [inner_traceRight_eq_sum_inner_vecTensorBasis] at h_inner_zero have hσ_psd := HermitianMat.zero_le_iff.mp σ.nonneg have h_each_zero : ∀ b : dB, star (vecTensorBasis v.ofLp b) ⬝ᵥ σ.M.mat.mulVec (vecTensorBasis v.ofLp b) = 0 := by have h_nonneg : ∀ b, (0 : ℂ) ≤ star (vecTensorBasis v.ofLp b) ⬝ᵥ σ.M.mat.mulVec (vecTensorBasis v.ofLp b) := fun b => hσ_psd.dotProduct_mulVec_nonneg _ intro b exact Finset.sum_eq_zero_iff_of_nonneg (fun b _ => h_nonneg b) |>.mp h_inner_zero b (Finset.mem_univ _) have h_σ_zero : ∀ b : dB, σ.M.mat.mulVec (vecTensorBasis v.ofLp b) = 0 := fun b => (hσ_psd.dotProduct_mulVec_zero_iff _).mp (h_each_zero b) have h_ρ_zero : ∀ b : dB, ρ.M.mat.mulVec (vecTensorBasis v.ofLp b) = 0 := by intro b have hmem_σ : (WithLp.toLp 2 (vecTensorBasis v.ofLp b) : EuclideanSpace ℂ _) ∈ σ.M.ker := by rw [HermitianMat.mem_ker_iff_mulVec_zero]; exact h_σ_zero b have hmem_ρ := hker hmem_σ rwa [HermitianMat.mem_ker_iff_mulVec_zero] at hmem_ρ exact traceRight_mulVec_zero_of_vecTensorBasis_zero ρ.M.mat v.ofLp h_ρ_zero- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:1375-1402
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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.
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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.