Entanglement, Entropy and Negentropy - Metabolism of Life

Abstract

The third law of thermodynamics places absolute zero at the bottom of the temperature scale, but this floor is not where entanglement fails: it is where entanglement is maximal. We argue that what temperature destroys is the purity of the state rather than quantum correlation itself, and we test this view in four numerical experiments on the critical transverse-field Ising chain (N = 4–8 spins). (i) In equilibrium, the half-chain logarithmic negativity decreases monotonically with temperature and vanishes at 𝑇𝑑 ≈ 1.28 𝐽, next to the maximum of the heat capacity (1.23 J), while the López-Ruiz–Mancini–Calbet statistical complexity peaks at T ≈ 0.58 J. (ii) Two thermal baths at the chain ends raise the entanglement by up to 26% relative to the Gibbs state of equal energy, but shift sudden death only to ≈ 1.33 J. (iii) Independent baths on every spin double the entanglement at fixed energy and extend it to 𝑇𝑒𝑓𝑓 ≈ 1.64 𝐽; the gain correlates with the entropy deficit relative to equilibrium (r = 0.62), which we identify with Brillouin's negentropy at fixed energy, 𝐷(𝜌‖𝛾) = 𝛽[𝐹(𝜌) − 𝐹𝑒𝑞]. A convexity argument yields an upper bound showing that high-energy states can be strongly entangled. (iv) A periodic drive combined with a cold bath sustains entanglement up to 𝑇𝑒𝑓𝑓 ≈ 5.7 𝐽, about 4.5 times the equilibrium sudden-death temperature, with an optimal intermediate drive amplitude. Coherent work coupled to a cold entropy sink thus manufactures purity far from the cold, something heat alone cannot do. The results provide a quantitative minimal-model reading of Schrödinger's idea that living systems feed on negative entropy.