Pure separable iff trace Left pure
MState.pure_separable_iff_traceLeft_pure
Plain-language statement
A pure state is separable iff the partial trace on the left is pure.
Exact Lean statement
theorem pure_separable_iff_traceLeft_pure (ψ : Ket (d₁ × d₂)) : IsSeparable (pure ψ) ↔
∃ ψ₁, pure ψ₁ = (pure ψ).traceLeftFormal artifact
Lean source
theorem pure_separable_iff_traceLeft_pure (ψ : Ket (d₁ × d₂)) : IsSeparable (pure ψ) ↔ ∃ ψ₁, pure ψ₁ = (pure ψ).traceLeft := by have h1 := MState.pure_separable_iff_IsProd ψ; have h2 := Ket.IsProd_iff_rank_eq_one ψ; have h3 := MState.pure_iff_rank_eq_one ( ( MState.pure ψ ).traceLeft ) simp_all have h4 : Matrix.rank ((MState.pure ψ).traceLeft.m) = Matrix.rank (Matrix.of (fun i j => ψ (i, j))) := by have h4 : (MState.pure ψ).traceLeft.m = Matrix.transpose (Matrix.conjTranspose (Matrix.of (fun i j => ψ (i, j))) * Matrix.of (fun i j => ψ (i, j))) := by ext i j simp [ MState.traceLeft, Matrix.mul_apply ] ; exact Finset.sum_congr rfl fun _ _ => mul_comm _ _; rw [ h4, Matrix.rank_transpose, Matrix.rank_conjTranspose_mul_self ]; grind- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/MState.lean:965-977
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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.