Plain-language statement
V_rho is an isometry.
Exact Lean statement
theorem V_rho_isometry [Nonempty dB] (ρAB : HermitianMat (dA × dB) ℂ) (hρ : ρAB.mat.PosDef) :
(V_rho ρAB)ᴴ * (V_rho ρAB) = 1Formal artifact
Lean source
theorem V_rho_isometry [Nonempty dB] (ρAB : HermitianMat (dA × dB) ℂ) (hρ : ρAB.mat.PosDef) : (V_rho ρAB)ᴴ * (V_rho ρAB) = 1 := by -- Since ρA is positive definite, we can use the fact that ρA_inv_sqrt * ρA * ρA_inv_sqrt = I. have h_pos_def : (ρAB.traceRight⁻¹.sqrt : Matrix dA dA ℂ) * (ρAB.traceRight : Matrix dA dA ℂ) * (ρAB.traceRight⁻¹.sqrt : Matrix dA dA ℂ) = 1 := by convert HermitianMat.sqrt_inv_mul_self_mul_sqrt_inv_eq_one _; exact PosDef_traceRight _ hρ rw [← h_pos_def] exact V_rho_conj_mul_self_eq ρAB hρ- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/SSA.lean:193-200
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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.