关于零熵密度猜想的尖锐性
On the sharpness of the zero-entropy-density conjecture
摘要：
零熵密度猜想指出，定义为S（N）/N在无穷远处的极限的熵密度，对于自旋链上的所有自旋不变纯态都为零。或者等价地说，S（N），这种状态的冯诺伊曼熵限制在N个连续的自旋，是次线性的。本文证明了这一猜想不能被锐化，即，不变性状态引起任意快速次线性熵增长。证明是建设性的，是基于一类来自准自由状态的CAR代数的状态。纯准自由态的熵增长是否可以是任意快次线性的问题首先由Fannes等人[J.Math.Phys.44，6005（2003）]提出。除了主要定理，它还表明，所有纯平移不变的非平凡准自由态的熵渐近至少是对数。
英文摘要：
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.
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