Relative Ent Resource ne top
UnitalFreeStateTheory.relativeEntResource_ne_top
Plain-language statement
In a FreeStateTheory, we have free states of full rank, therefore the minimum relative entropy of any state ρ to a free state is finite.
Exact Lean statement
@[aesop (rule_sets := [finiteness]) safe] lemma relativeEntResource_ne_top (ρ : MState (H i)) : ⨅ σ ∈ IsFree, 𝐃(ρ‖σ) ≠ ⊤
Formal artifact
Lean source
@[aesop (rule_sets := [finiteness]) safe]lemma relativeEntResource_ne_top (ρ : MState (H i)) : ⨅ σ ∈ IsFree, 𝐃(ρ‖σ) ≠ ⊤ := by let ⟨w,h⟩ := free_fullRank i apply ne_top_of_le_ne_top _ (iInf_le _ w) simp only [ne_eq, iInf_eq_top, Classical.not_imp] constructor · exact h.2 · refine ne_of_apply_ne ENNReal.toEReal (qRelativeEnt_ker (ρ := ρ) (?_) ▸ EReal.coe_ne_top _) convert @bot_le _ _ (Submodule.instOrderBot) _ have := h.1.toLin_ker_eq_bot simp [LinearMap.ker_eq_bot', HermitianMat.ker] at this ⊢ intro m hm simpa only [WithLp.ofLp_eq_zero] using this m congr($hm)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/ResourceTheory/FreeState.lean:408-420
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
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.