Plain-language statement
The entanglement of formation of the maximally entangled state with on-site dimension 𝕕 is log(𝕕).
Exact Lean statement
theorem EoF_of_MES : EoF (pure <| Ket.MES d) = Real.log (Finset.card Finset.univ (α := d))
Formal artifact
Lean source
theorem EoF_of_MES : EoF (pure <| Ket.MES d) = Real.log (Finset.card Finset.univ (α := d)) := by simp only [EoF, convex_roof_of_pure, Finset.card_univ] simp only [KetUpToPhase.lift_mk] -- The von Neumann entropy of the maximally mixed state is log(d). have h_von_neumann : Sᵥₙ (MState.uniform : MState d) = Real.log (Fintype.card d) := by rw [MState.uniform, Sᵥₙ_ofClassical ProbDistribution.uniform, Hₛ_uniform, Finset.card_univ] simp only [NNReal.coe_mk] rw [traceRight_pure_MES] exact h_von_neumann- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entanglement.lean:280-288
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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.