Plain-language statement
Entanglement of Formation of bipartite systems. It is the convex roof extension of the von Neumann entropy of one of the subsystems (here chosen to be the left one, but see Entropy.Sᵥₙ_of_partial_eq). The function Sᵥₙ ∘ traceRight ∘ pure is phase-invariant because pure maps phase-equivalent kets to the same mixed state, so it descends to `KetUpToPha...
Exact Lean statement
theorem Sᵥₙ_ofClassical {d : Type*} [Fintype d] [DecidableEq d] (dist : ProbDistribution d) :
Sᵥₙ (MState.ofClassical dist) = Hₛ distFormal artifact
Lean source
theorem Sᵥₙ_ofClassical {d : Type*} [Fintype d] [DecidableEq d] (dist : ProbDistribution d) : Sᵥₙ (MState.ofClassical dist) = Hₛ dist := by -- Let's unfold the definition of `Sᵥₙ` using `Sᵥₙ_eq_trace_cfc`. have h_def : Sᵥₙ (MState.ofClassical dist) = (HermitianMat.cfc (MState.ofClassical dist).M Real.negMulLog).trace := by exact Sᵥₙ_eq_trace_cfc (ofClassical dist); convert h_def using 1; -- By definition of $MState.ofClassical$, we know that $(MState.ofClassical dist).M$ is a diagonal matrix with entries $dist i$. have h_diag : (MState.ofClassical dist).M = HermitianMat.diagonal ℂ (fun x => dist x) := by exact rfl; rw [ h_diag, HermitianMat.cfc_diagonal, HermitianMat.trace_diagonal ] ; aesop- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entanglement.lean:268-277
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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.