Plain-language statement
The intermediate operator inequality: ρ_AB ⊗ σ_C⁻¹ ≤ (ρ_A ⊗ σ_BC⁻¹).reindex(assoc⁻¹). This is derived from W_mat_sq_le_one by algebraic manipulation (conjugation and simplification).
Exact Lean statement
theorem intermediate_ineq [Nonempty dA] [Nonempty dB] [Nonempty dC]
(ρAB : HermitianMat (dA × dB) ℂ) (σBC : HermitianMat (dB × dC) ℂ)
(hρ : ρAB.mat.PosDef) (hσ : σBC.mat.PosDef) :
ρAB ⊗ₖ (σBC.traceLeft)⁻¹ ≤
(ρAB.traceRight ⊗ₖ σBC⁻¹).reindex (Equiv.prodAssoc dA dB dC).symmFormal artifact
Lean source
theorem intermediate_ineq [Nonempty dA] [Nonempty dB] [Nonempty dC] (ρAB : HermitianMat (dA × dB) ℂ) (σBC : HermitianMat (dB × dC) ℂ) (hρ : ρAB.mat.PosDef) (hσ : σBC.mat.PosDef) : ρAB ⊗ₖ (σBC.traceLeft)⁻¹ ≤ (ρAB.traceRight ⊗ₖ σBC⁻¹).reindex (Equiv.prodAssoc dA dB dC).symm := by have h_sorted : (ρAB.traceRight⁻¹.sqrt.mat ⊗ₖ σBC.sqrt.mat).reindex (Equiv.prodAssoc dA dB dC).symm (Equiv.prodAssoc dA dB dC).symm * (ρAB ⊗ₖ (σBC.traceLeft)⁻¹).mat * (ρAB.traceRight⁻¹.sqrt.mat ⊗ₖ σBC.sqrt.mat).reindex (Equiv.prodAssoc dA dB dC).symm (Equiv.prodAssoc dA dB dC).symm ≤ (1 : Matrix ((dA × dB) × dC) ((dA × dB) × dC) ℂ) := by convert W_mat_sq_le_one ρAB σBC hρ hσ using 1; convert W_mat_sq_eq_conj ρAB σBC hρ hσ |> Eq.symm using 1; convert h_sorted using 1; rw [HermitianMat.le_iff]; rw [ ← S_mat_conj_rhs_eq_one ρAB σBC hρ hσ ]; simp only [ Matrix.posSemidef_iff_dotProduct_mulVec ] constructor <;> intro h <;> simp_all · convert h_sorted using 1; convert S_mat_conj_rhs_eq_one ρAB σBC hρ hσ using 1; · have := S_mat_isUnit ρAB σBC hρ hσ; cases' this.nonempty_invertible with u hu; have h_pos_semidef : Matrix.PosSemidef ((S_mat ρAB σBC)⁻¹ * (S_mat ρAB σBC * ((ρAB.traceRight ⊗ₖ σBC⁻¹).reindex (Equiv.prodAssoc dA dB dC).symm).mat * S_mat ρAB σBC - S_mat ρAB σBC * (ρAB ⊗ₖ (σBC.traceLeft)⁻¹).mat * S_mat ρAB σBC) * (S_mat ρAB σBC)⁻¹ᴴ) := by exact Matrix.PosSemidef.mul_mul_conjTranspose_same h (S_mat ρAB σBC)⁻¹; simp_all [ Matrix.mul_assoc, Matrix.sub_mul, Matrix.mul_sub ]; simp_all [ Matrix.posSemidef_iff_dotProduct_mulVec, Matrix.IsHermitian ]; have h_conj : (S_mat ρAB σBC)ᴴ = S_mat ρAB σBC := by unfold S_mat; simp [ Matrix.conjTranspose_kronecker, Matrix.conjTranspose_submatrix ] ; simp_all [ Matrix.conjTranspose_nonsing_inv ]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/SSA.lean:761-784
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Person-level attribution pending.
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Plain-language statement
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Source project: quantumInfo
Person-level attribution pending.