Renyi Entropy Measures for Quantum Entanglement Characterization

Abstract

The Renyi entropy family generalizes the von Neumann entanglement entropy through a single order parameter
alpha, but the practical question of which alpha values are most informative and most numerically tractable for
characterizing entanglement in interacting quantum many-body systems is not settled by the formalism alone. We
computed Renyi entanglement entropies S_alpha for alpha in {0.5, 1, 2, 3, 4, infinity} across bipartitions of a
one-dimensional transverse-field Ising spin chain with 24 sites, simulated using matrix product states with bond
dimension up to 200, spanning 60 values of the transverse field h across the paramagnetic-to-ferromagnetic
quantum phase transition at h_c = 1.0. Near criticality, S_1 (the von Neumann limit) scaled with subsystem size L
as (c/3) ln L with extracted central charge c = 0.501 (95% CI: 0.487-0.516), consistent with the expected Ising
universality class value of 0.5 to within 2%. Across all 60 field values, S_alpha decreased monotonically with alpha
at every point, from a mean S_0.5 of 1.84 to a mean S_infinity (min-entropy) of 0.97, a reduction of 47%, with the
rate of this decrease sharpest near the critical point, where the ratio S_0.5/S_4 peaked at 2.31 compared to 1.42
averaged over the gapped phases away from criticality. The second Renyi entropy S_2, computable in principle
from a physically implementable swap-test measurement protocol, tracked S_1 closely across the full field range,
with a mean absolute deviation of 0.041 and a maximum deviation of 0.098 near the critical point, whereas
S_infinity showed systematically larger deviations from S_1 (mean absolute deviation 0.31), indicating that
low-order Renyi entropies retain most of the discriminating power of the full von Neumann entropy for locating the
phase transition while remaining far more experimentally accessible. Finite-bond-dimension convergence tests
showed that S_1 and S_2 converged to within 0.5% of their extrapolated chi = infinity values already at bond
dimension 120, whereas S_infinity required chi >= 180 to reach comparable convergence, reflecting its greater
sensitivity to the tail of the entanglement spectrum. These results indicate that S_2 offers a favorable trade-off
between experimental accessibility, numerical convergence cost, and sensitivity to critical entanglement scaling,
supporting its continued use as a practical proxy for full entanglement characterization in both numerical and
experimental quantum many-body studies.