Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems
Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems
复制标题
量子自旋晶格系统上具有有限相关长度的量子态的浓度界限
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Anurag Anshu
中科院分区：
文献类型：
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作者：
Anurag Anshu
我们考虑的问题，确定的量子态的能量分布，满足指数衰减的相关和产品的状态，相对于一个自旋晶格上的量子局域哈密顿量。对于D维格点上的量子态，它具有相关长度σ，并且相对于给定的局域哈密顿量具有平均能量e（具有n个局域项，每个局域项的范数至多为1），我们证明了这个态与能量f的本征空间的重叠至多为exp（−（（e-f）2 σ）1 D + 1 /n1 D + 1 D σ）。这个界限在任何时候都成立，只要e − f> 2 D n σ。因此，在一维晶格上，能量分布的尾部随能量呈指数衰减。对于乘积态，我们改进了上述结果，得到了能量的高斯衰减，即使对于没有晶格结构的量子自旋系统也是如此。给定一个自旋集合上的乘积态，它相对于一个局部哈密顿量具有平均能量e（有n个局部项，每个局部项最多与m个其他局部项重叠），我们证明了这个态与能量f的本征空间的重叠最多为exp（−（e-f）2 / nm 2）。这个界限在任何时候都成立。
We consider the problem of determining the energy distribution of quantum states that satisfy exponential decay of correlation and product states, with respect to a quantum local Hamiltonian on a spin lattice. For a quantum state on a D-dimensional lattice that has correlation length σ and has average energy e with respect to a given local Hamiltonian (with n local terms, each of which has norm at most 1), we show that the overlap of this state with eigenspace of energy f is at most exp ( − ( ( e − f ) 2 σ ) 1 D + 1 / n 1 D + 1 D σ ) . This bound holds whenever ∣ e − f ∣ > 2 D n σ . Thus, on a one-dimensional lattice, the tail of the energy distribution decays exponentially with the energy. For product states, we improve above result to obtain a Gaussian decay in energy, even for quantum spin systems without an underlying lattice structure. Given a product state on a collection of spins which has average energy e with respect to a local Hamiltonian (with n local terms and each local term overlapping with at most m other local terms), we show that the overlap of this state with eigenspace of energy f is at most exp ( − ( e − f ) 2 / nm 2 ) . This bound holds whenever ∣ e − f ∣ > m n .
DOI：
10.1145/2488608.2488719
发表时间：
2013-06
期刊：
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影响因子：
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作者：
F. Brandão;A. Harrow
通讯作者：
F. Brandão;A. Harrow
影响因子：
19.6
作者：
Brandão F
通讯作者：
Brandão F