Abstract
We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings\^a 1D area law and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\ensuremath{\epsilon}>0$, and interaction strength at most $J$, our entropy bound is ${S}_{1D}\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$, where $X\stackrel{\mathrm{def}}{=}(J\mathrm{log}d)/\ensuremath{\epsilon}$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is $|\ensuremath{\partial}L|$, we use additional local structure in the proof and show that $S\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{|\ensuremath{\partial}L|}^{2}{\mathrm{log}}^{6}|\ensuremath{\partial}L|\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.
Highlights
One of the striking differences between quantum and classical systems is the number of parameters that are needed to describe them
A classical system of n particles can be generally described by O(n) parameters, whereas an arbitrary state of a similar quantum system would generally require 2O(n) parameters. This exponential gap is directly related to the phenomena of entanglement; quantum states do not have to be simple product states but can be an arbitrary superposition of such states. How genuine is this exponential gap? Is it an artifact of the fact that we are considering arbitrary quantum states, or is it an inherent characteristic of physical states that occur in nature, which are, a much more restricted set of states? Among the best physical systems one may consider with respect to this question are quantum many-body systems on a lattice, which are ubiquitous in condensed matter physics
Hi J, one can always pass to an equivalent system that shares the same ground space, which is made of projections and has a spectral gap /J
Summary
One of the striking differences between quantum and classical systems is the number of parameters that are needed to describe them. It has been shown by Hastings that in the ground state of gapped systems, the correlation between two local observables decays exponentially in their lattice distance.[8] It is, tempting to conclude that only the degrees of freedom near the boundary are entangled to those outside the region. 1,7,11, and 12 and references therein), it was not until a few years ago, in a seminal paper,[13] that Hastings proved that it holds for all 1D systems with a spectral gap In this case, an area law says that ground -state entropy across any cut in the 1D chain is bounded by a constant, independent. The combinatorial nature of the detectability lemma opened up the possibility of a new inherently combinatorial proof of the area law
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