ar X iv : m at h - ph / 0 50 70 22 v 3 1 9 D ec 2 00 5 On the sharpness of the zero - entropy - density conjec - ture
نویسندگان
چکیده
The zero-entropy-density conjecture states that the entropy density defined as s ≔ lim N→∞ S N /N 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 en-tropy asymptotics of all pure shift-invariant nontrivial quasifree states is at least logarithmic.
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