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 traceRight_pure_MES (d : Type*) [Fintype d] [DecidableEq d] [Nonempty d] :
(MState.pure (Ket.MES d)).traceRight = MState.uniformFormal artifact
Lean source
theorem traceRight_pure_MES (d : Type*) [Fintype d] [DecidableEq d] [Nonempty d] : (MState.pure (Ket.MES d)).traceRight = MState.uniform := by -- By definition of partial trace, we sum over the second system. have h_partial_trace : ∀ (i j : d), ∑ k : d, (Ket.MES d).vec (i, k) * (star (Ket.MES d).vec (j, k)) = (1 / Fintype.card d : ℝ) * (if i = j then 1 else 0) := by unfold Ket.MES intro i j simp only [one_div, Pi.star_apply, RCLike.star_def, ite_mul, zero_mul, Finset.sum_ite_eq, Finset.mem_univ, ↓reduceIte, Complex.ofReal_inv] split · subst i simp only [map_inv₀, Complex.conj_ofReal] ring_nf; norm_cast; norm_num; · grind unfold MState.pure MState.traceRight MState.uniform ext i j convert h_partial_trace i j simp_all only [Pi.star_apply, RCLike.star_def, one_div, Complex.ofReal_inv, Complex.ofReal_natCast, mul_ite, mul_one, mul_zero, HermitianMat.mat_apply, coe_ofClassical, ProbDistribution.uniform_def, Finset.card_univ] unfold HermitianMat.diagonal simp_all only [map_inv₀, map_natCast] rfl- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entanglement.lean:223-244
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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.