Dual choi matrix
MatrixMap.dual_choi_matrix
Plain-language statement
The Choi matrix of the dual map is the transpose of the reindexed Choi matrix of the original map.
Exact Lean statement
lemma dual_choi_matrix (M : MatrixMap dIn dOut ๐) :
M.dual.choi_matrix = (M.choi_matrix.transpose).reindex (Equiv.prodComm dOut dIn) (Equiv.prodComm dOut dIn)Formal artifact
Lean source
lemma dual_choi_matrix (M : MatrixMap dIn dOut ๐) : M.dual.choi_matrix = (M.choi_matrix.transpose).reindex (Equiv.prodComm dOut dIn) (Equiv.prodComm dOut dIn) := by -- By definition of dual, we know that $(M.dual (single jโ jโ 1)) iโ iโ = (M (single iโ iโ 1)) jโ jโ$. have h_dual_def : โ (iโ : dIn) (jโ : dOut) (iโ : dIn) (jโ : dOut), (M.dual (Matrix.single jโ jโ 1)) iโ iโ = (M (Matrix.single iโ iโ 1)) jโ jโ := by intro iโ jโ iโ jโ have h_dual_def : (M.dual (Matrix.single jโ jโ 1)) iโ iโ = Matrix.trace (Matrix.single iโ iโ 1 * M.dual (Matrix.single jโ jโ 1)) := by simp [ Matrix.trace, Matrix.mul_apply ]; simp [ Matrix.single]; rw [ Finset.sum_eq_single iโ ] ยท aesop ยท intro b a a_1 simp [a_1.symm] ยท aesop rw [ h_dual_def, โ Dual.trace_eq ]; rw [ Matrix.trace ]; rw [ Finset.sum_eq_single jโ ] <;> aesop; aesop- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Dual.lean:117-133
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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.