Fidelity self eq one
MState.fidelity_self_eq_one
Plain-language statement
A state has perfect fidelity with itself.
Exact Lean statement
theorem fidelity_self_eq_one : fidelity ρ ρ = 1
Formal artifact
Lean source
theorem fidelity_self_eq_one : fidelity ρ ρ = 1 := by simp only [fidelity, HermitianMat.sqrt_eq_cfc_rpow_half] conv => enter [1, 1, 1, 2] rw [← HermitianMat.cfc_id ρ.M] rw [HermitianMat.cfc_conj, ← HermitianMat.cfc_comp_apply] convert ρ.tr using 2 convert ρ.M.cfc_id using 1 apply HermitianMat.cfc_congr_of_nonneg ρ.nonneg intro x hx simp only [one_div, Pi.mul_apply, id_eq, Pi.pow_apply] rw [← Real.rpow_two, Real.rpow_inv_rpow hx (by norm_num), ← sq, ← Real.rpow_two] exact Real.rpow_rpow_inv hx (by norm_num)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Distance/Fidelity.lean:35-47
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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.