Plain-language statement
If two matrix maps satisfy the trace duality property, they are equal.
Exact Lean statement
lemma dual_unique
(M : MatrixMap dIn dOut ๐) (M' : MatrixMap dOut dIn ๐)
(h : โ A B, (M A * B).trace = (A * M' B).trace) : M.dual = M'Formal artifact
Lean source
lemma dual_unique (M : MatrixMap dIn dOut ๐) (M' : MatrixMap dOut dIn ๐) (h : โ A B, (M A * B).trace = (A * M' B).trace) : M.dual = M' := by -- By definition of dual, we know that for any A and B, the trace of (M A) * B equals the trace of A * (M.dual B). have h_dual : โ A : Matrix dIn dIn ๐, โ B : Matrix dOut dOut ๐, (M A * B).trace = (A * M.dual B).trace := by exact fun A B => Dual.trace_eq M A B; -- Since these two linear maps agree on all bases, they must be equal. have h_eq : โ A : Matrix dIn dIn ๐, โ B : Matrix dOut dOut ๐, (A * M.dual B).trace = (A * M' B).trace := by exact fun A B => h_dual A B โธ h A B; refine' LinearMap.ext fun B => _; exact Matrix.ext_iff_trace_mul_left.mpr fun x => h_eq x B- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/CPTPMap/Dual.lean:102-112
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Project documentation
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Source project: quantumInfo
Person-level attribution pending.
Convex roof of pure
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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.
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Plain-language statement
The identity channel on D dimensional space achieves a rate of log2(D).
Source project: quantumInfo
Person-level attribution pending.