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).
Exact Lean statement
theorem id_achievesRate_log_dim : (id (dIn := d₁)).AchievesRate (Real.logb 2 (Fintype.card d₁))
Formal artifact
Lean source
theorem id_achievesRate_log_dim : (id (dIn := d₁)).AchievesRate (Real.logb 2 (Fintype.card d₁)) := by intro ε hε use 1, zero_lt_one, Fintype.card d₁, id constructor · --they are equivalent up to permutation -- TODO: Instead this proof should be `@[simp] piProd (fun x => id) = id` and `emulates_self` refine' ⟨ _, _, _ ⟩; exact CPTPMap.ofEquiv ( Fintype.equivFinOfCardEq ( by simp +decide ) ).symm; exact CPTPMap.ofEquiv ( Fintype.equivFinOfCardEq ( by simp +decide ) ); apply CPTPMap.ext; ext; simp +decide [ CPTPMap.piProd ]; unfold MatrixMap.piProd simp_all only [gt_iff_lt, PiTensorProduct.map_id, LinearMap.toMatrix_id_eq_basis_toMatrix, Module.Basis.toMatrix_self, Matrix.reindex_apply, Matrix.submatrix_one_equiv, Matrix.toLin_one] erw [ MatrixMap.submatrix_apply ] simp_all only [Equiv.symm_symm, Equiv.apply_symm_apply, Matrix.submatrix_apply] constructor · norm_num · exact εApproximates_monotone (εApproximates_self id) hε.le- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Capacity.lean:136-154
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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.
Not achieves Rate gt log dim out
CPTPMap.not_achievesRate_gt_log_dim_out
Plain-language statement
A channel cannot achieve a rate greater than log2(D), where D is the output dimension.
Source project: quantumInfo
Person-level attribution pending.