I Inf Is Compact
OptimalHypothesisRate.iInf_IsCompact
Plain-language statement
The space of strategies T in OptimalHypothesisRate is compact.
Exact Lean statement
theorem iInf_IsCompact (ρ : MState d) (ε : Prob) : IsCompact { m | ρ.exp_val (1 - m) ≤ ε ∧ 0 ≤ m ∧ m ≤ 1 }Formal artifact
Lean source
theorem iInf_IsCompact (ρ : MState d) (ε : Prob) : IsCompact { m | ρ.exp_val (1 - m) ≤ ε ∧ 0 ≤ m ∧ m ≤ 1 } := by have hC₁ : IsCompact {m : HermitianMat d ℂ | 0 ≤ m ∧ m ≤ 1} := HermitianMat.unitInterval_IsCompact have hC₂ : IsClosed {m | ρ.exp_val (1 - m) ≤ ε} := by --This is a linear constraint and so has a closed image change IsClosed ((fun m ↦ ρ.M.innerₗ (1 - m)) ⁻¹' (Set.Iic ε)) refine IsClosed.preimage ?_ isClosed_Iic fun_prop exact hC₁.inter_left hC₂- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/ResourceTheory/HypothesisTesting.lean:61-69
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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.