Pos Def mix of ne one
MState.PosDef_mix_of_ne_one
Plain-language statement
If the second state is positive definite and the mixture is nondegenerate, their mixture is also positive definite.
Exact Lean statement
theorem PosDef_mix_of_ne_one {d : Type*} [Fintype d] [DecidableEq d] {σ₁ σ₂ : MState d}
(hσ₂ : σ₂.m.PosDef) (p : Prob) (hp : p ≠ 1) : (p [σ₁ ↔ σ₂]).m.PosDefFormal artifact
Lean source
theorem PosDef_mix_of_ne_one {d : Type*} [Fintype d] [DecidableEq d] {σ₁ σ₂ : MState d} (hσ₂ : σ₂.m.PosDef) (p : Prob) (hp : p ≠ 1) : (p [σ₁ ↔ σ₂]).m.PosDef := by have : 0 < 1 - p := by --TODO this is ridiculous, move to Prob contrapose! hp have : (1 : ℝ) - (p : ℝ) = (0 : ℝ) := by have := le_antisymm hp (1 - p).zero_le rw [Subtype.ext_iff] at this simpa using this ext change (p : ℝ) = 1 linarith exact (hσ₂.smul this).posSemidef_add (σ₁.psd.rsmul p.zero_le)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/MState.lean:1337-1349
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.