On the sharpness of the zero-entropy-density conjecture

TL;DR

This paper demonstrates that the zero-entropy-density conjecture for translation-invariant pure states on spin chains cannot be strengthened, showing such states can have arbitrarily fast sublinear entropy growth, with implications for quantum entropy behavior.

Contribution

It proves that the zero-entropy-density conjecture cannot be sharpened, providing a constructive proof that states can exhibit arbitrarily fast sublinear entropy growth.

Findings

States can have arbitrarily fast sublinear entropy growth.

Pure shift-invariant quasifree states have at least logarithmic entropy growth.

The conjecture's limit on entropy density is not restrictive for pure states.

Abstract

The zero-entropy-density conjecture states that the entropy density, defined as the limit of S(N)/N at infinity, vanishes for all translation-invariant pure states on the spin chain. Or equivalently, S(N), the von Neumann entropy of such a state restricted to N consecutive spins, is sublinear. In this paper it is proved that this conjecture cannot be sharpened, i.e., translation-invariant states give rise to arbitrary fast sublinear entropy growth. The proof is constructive, and is based on a class of states derived from quasifree states on a CAR algebra. The question whether the entropy growth of pure quasifree states can be arbitrary fast sublinear was first raised by Fannes et al. [J. Math. Phys. 44, 6005 (2003)]. In addition to the main theorem it is also shown that the entropy asymptotics of all pure shift-invariant nontrivial quasifree states is at least logarithmic.