Sandwiched Renyi Entropy mono trace Left
sandwichedRenyiEntropy_mono_traceLeft
Plain-language statement
Monotonicity of the sandwiched Rényi divergence under traceLeft for α > 1. Follows from sandwichedRenyiEntropy_mono_traceRight' + SWAP invariance.
Exact Lean statement
theorem sandwichedRenyiEntropy_mono_traceLeft [Nonempty dA]
(hα : 1 < α) (ρ σ : MState (dA × dB)) :
D̃_ α(ρ.traceLeft‖σ.traceLeft) ≤ D̃_ α(ρ‖σ)Formal artifact
Lean source
theorem sandwichedRenyiEntropy_mono_traceLeft [Nonempty dA] (hα : 1 < α) (ρ σ : MState (dA × dB)) : D̃_ α(ρ.traceLeft‖σ.traceLeft) ≤ D̃_ α(ρ‖σ) := by -- traceLeft = SWAP.traceRight, and SWAP preserves the SRD rw [← MState.traceRight_SWAP, ← MState.traceRight_SWAP] calc D̃_ α(ρ.SWAP.traceRight‖σ.SWAP.traceRight) ≤ D̃_ α(ρ.SWAP‖σ.SWAP) := sandwichedRenyiEntropy_mono_traceRight' hα ρ.SWAP σ.SWAP _ = D̃_ α(ρ‖σ) := sandwichedRenyiEntropy_SWAP ρ σ- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:1505-1513
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Person-level attribution pending.
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Plain-language statement
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Person-level attribution pending.