Plain-language statement
For Antitone functions, the InfRegularized is the supremum of values.
Exact Lean statement
theorem anti_inf (h : Antitone fn) :
InfRegularized fn hl hu = sInf (Set.range fn)Formal artifact
Lean source
theorem anti_inf (h : Antitone fn) : InfRegularized fn hl hu = sInf (Set.range fn) := by unfold InfRegularized; simp +decide [ Filter.liminf_eq ] ; rw [ @csSup_eq_of_forall_le_of_forall_lt_exists_gt ]; · exact ⟨ _, ⟨ 0, fun n _ => hl n ⟩ ⟩; · rintro a ⟨ n, hn ⟩; exact le_csInf ⟨ _, Set.mem_range_self n ⟩ fun x hx => by rcases hx with ⟨ m, rfl ⟩ ; exact hn _ ( le_max_left _ _ ) |> le_trans <| h <| le_max_right _ _; · exact fun w hw => ⟨ _, ⟨ 0, fun n _ => csInf_le ⟨ _lb, Set.forall_mem_range.2 hl ⟩ ⟨ n, rfl ⟩ ⟩, hw ⟩- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Regularized.lean:47-54
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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.