Plain-language statement
The dual of TracePreserving map is not trace-preserving, it's unital, that is, M*(I) = I.
Exact Lean statement
theorem dual_Unital (h : M.IsTracePreserving) : M.dual.Unital
Formal artifact
Lean source
theorem dual_Unital (h : M.IsTracePreserving) : M.dual.Unital := by -- By definition of dual, we know that for any matrix A, Tr(M(A) * I) = Tr(A * M*(I)). have h_dual_trace : โ A : Matrix dIn dIn ๐, (M A * 1).trace = (A * M.dual 1).trace := by exact fun A => Dual.trace_eq M A 1; ext i j specialize h_dual_trace ( Matrix.of ( fun k l => if k = j then if l = i then 1 else 0 else 0 ) ) simp_all [ Matrix.trace, Matrix.mul_apply ] ; specialize h ( Matrix.of ( fun k l => if k = j then if l = i then 1 else 0 else 0 ) ) simp_all [ Matrix.trace ] simp [ Matrix.one_apply, eq_comm ]- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Dual.lean:86-95
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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 ๐) : Matrix.opNorm Aแดด = Matrix.opNorm A := by unfold Matrix.opNorm rw [โ 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 โ โโฅ0 applied to a pure state ฯ is g (KetUpToPhase.mk ฯ).
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.