Sandwiched Renyi Entropy mono trace Right
sandwichedRenyiEntropy_mono_traceRight
Plain-language statement
The sandwiched Rényi divergence is monotone under partial trace for α > 1. This follows from monotonicity of the trace functional together with the fact that D̃_α = log(Q̃_α) / (α - 1) and both log and 1/(α-1) are order-preserving for α > 1.
Exact Lean statement
theorem sandwichedRenyiEntropy_mono_traceRight [Nonempty dB]
(hα : 1 < α) (ρ σ : MState (dA × dB))
(hker : σ.M.ker ≤ ρ.M.ker) :
D̃_ α(ρ.traceRight‖σ.traceRight) ≤ D̃_ α(ρ‖σ)Formal artifact
Lean source
theorem sandwichedRenyiEntropy_mono_traceRight [Nonempty dB] (hα : 1 < α) (ρ σ : MState (dA × dB)) (hker : σ.M.ker ≤ ρ.M.ker) : D̃_ α(ρ.traceRight‖σ.traceRight) ≤ D̃_ α(ρ‖σ) := by have hα₀ : 0 < α := by linarith have hα₁ : α ≠ 1 := hα.ne' have hker_tr := ker_le_traceRight hker -- Rewrite both sides as log(Q̃) / (α - 1) rw [sandwichedRelRentropy_eq_log_traceFunctional hα₀ hα₁ hker, sandwichedRelRentropy_eq_log_traceFunctional hα₀ hα₁ hker_tr] apply ENNReal.ofReal_le_ofReal apply div_le_div_of_nonneg_right _ (by linarith : 0 < α - 1).le exact Real.log_le_log (sandwichedTraceFunctional_pos ρ.traceRight σ.traceRight hker_tr) (sandwichedTraceFunctional_mono_traceRight hα ρ σ hker)- Project
- quantumInfo
- License
- MIT
- Commit
- 56e83a9288a3
- Source
- QuantumInfo/Finite/Entropy/DPI.lean:1407-1420
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Source project: quantumInfo
Person-level attribution pending.