Project documentation
Sub-lemma for Step 1b: the conj of H_hat by σ^{−γ} simplifies to (ρ.M.conj (σ^γ).mat)^{α−1}. This uses σ^{−γ} · σ^γ = identity (on support) to cancel the outer σ^γ factors.
Exact Lean statement
theorem H_hat_conj_sigma (hα : 1 < α) (ρ σ : MState d) :
let γFormal artifact
Lean source
theorem H_hat_conj_sigma (hα : 1 < α) (ρ σ : MState d) : let γ := (1 - α) / (2 * α) (H_hat α ρ σ).conj (σ.M ^ (-γ)).mat = (ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1) := by intro γ have hγ : γ ≠ 0 := by simp only [γ]; rw [div_ne_zero_iff]; exact ⟨by linarith, by linarith⟩ have hα1 : α - 1 ≠ 0 := by linarith show (((ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1)).conj (σ.M ^ γ).mat).conj (σ.M ^ (-γ)).mat = (ρ.M.conj (σ.M ^ γ).mat) ^ (α - 1) rw [HermitianMat.conj_conj] rw [HermitianMat.rpow_neg_mul_rpow_eq_supportProj σ.nonneg hγ] exact conj_supportProj_eq_of_ker_le σ.M _ (ker_sigma_le_ker_conj_rpow ρ σ hγ hα1)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:411-422
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Person-level attribution pending.
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Person-level attribution pending.
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Source project: quantumInfo
Person-level attribution pending.