Optimal Hypothesis Rate unique
OptimalHypothesisRate.optimalHypothesisRate_unique
Plain-language statement
On the 1D Hilbert space, the optimal hypothesis testing rate is simply 1 - ε, since there's nothing to learn. (More generally this would hold whenever ρ=σ.) -
Exact Lean statement
theorem optimalHypothesisRate_unique {d : Type*} [Fintype d] [DecidableEq d]
(ε : Prob) (ρ σ : MState d) [Unique d] : β_ ε(ρ‖{σ}) = 1 - εFormal artifact
Lean source
theorem optimalHypothesisRate_unique {d : Type*} [Fintype d] [DecidableEq d] (ε : Prob) (ρ σ : MState d) [Unique d] : β_ ε(ρ‖{σ}) = 1 - ε := by obtain rfl := Unique.eq_default ρ obtain rfl := Unique.eq_default σ rw [OptimalHypothesisRate.of_singleton] apply le_antisymm · refine iInf_le_of_le ⟨((1 - ε : Prob) : ℝ) • 1, ⟨?_, ?_, ?_⟩⟩ ?_ · simp [MState.exp_val_sub] · apply smul_nonneg ?_ zero_le_one simp · apply smul_le_of_le_one_left zero_le_one simp · simp [-Prob.coe_one_minus] · simp intro a he1 ha0 ha1 rw [MState.exp_val_sub, MState.exp_val_one, tsub_le_iff_right] at he1 rw [← tsub_le_iff_left, ← Prob.coe_one_minus] at he1 exact he1- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/ResourceTheory/HypothesisTesting.lean:536-553
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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.