Département de Physique, Université de Sherbrooke, Québec, J1K 2R1, Canada.
Phys Rev Lett. 2011 Feb 25;106(8):080403. doi: 10.1103/PhysRevLett.106.080403. Epub 2011 Feb 24.
We present a lower bound for the free energy of a quantum many-body system at finite temperature. This lower bound is expressed as a convex optimization problem with linear constraints, and is derived using strong subadditivity of von Neumann entropy and a relaxation of the consistency condition of local density operators. The dual to this minimization problem leads to a set of quantum belief propagation equations, thus providing a firm theoretical foundation to that approach. The minimization problem is numerically tractable, and we find good agreement with quantum Monte Carlo calculations for spin-1/2 Heisenberg antiferromagnet in two dimensions. This lower bound complements other variational upper bounds. We discuss applications to Hamiltonian complexity theory and give a generalization of the structure theorem of [P. Hayden et al., Commun. Math. Phys. 246, 359 (2004).] to trees in an appendix.
我们给出了有限温度下量子多体系统自由能的下界。这个下界可以表示为一个具有线性约束的凸优化问题，是利用 von Neumann 熵的强次可加性和局部密度算子一致性条件的松弛推导出来的。这个最小化问题的对偶问题导致了一组量子置信传播方程，从而为该方法提供了坚实的理论基础。这个最小化问题在数值上是可处理的，我们发现它与二维自旋-1/2 海森堡反铁磁体的量子蒙特卡罗计算有很好的一致性。这个下界补充了其他变分上界。我们讨论了它在哈密顿复杂性理论中的应用，并在附录中推广了 [P. Hayden 等人，Commun. Math. Phys. 246, 359 (2004).] 的树结构定理。
