Sₙ = (1/(1-n)) log Tr(ρⁿ) — generalizing von Neumann entropy
The replica trick computes Tr(ρⁿ) by analytically continuing n copies of a system.
Sₙ = (1/(1-n)) log Tr(ρⁿ). As n→1, Sₙ → von Neumann entropy S = -Tr(ρ log ρ).
S₂ = -log Tr(ρ²) = -log purity. S∞ = -log λ_max (min-entropy).
For a pure state all Sₙ = 0. For maximally mixed state on d dimensions, Sₙ = log(d).
Left: density matrix eigenvalue spectrum. Right: Rényi entropy vs replica index n.