Concentration of Measure for Quantum States with a Fixed Expectation Value
Concentration of Measure for Quantum States with a Fixed Expectation Value
复制标题
具有固定期望值的量子态测量集中度
DOI：
10.1007/s00220-011-1205-1
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发表时间：
2010
影响因子：
2.4
通讯作者：
J. Eisert
中科院分区：
文献类型：
--
作者：
Markus P. Mueller;D. Gross;J. Eisert
给定有限维量子系统上一些可观测的 H，我们研究随机状态向量 $${|\psi\rangle}$$ 的典型属性，这些状态向量对于 H 具有固定期望值 $${\langle\psi|H|\psi\rangle=E}$$。在谱上的某些条件下，我们证明这个量子态流形表现出测量集中现象：该集合上的任何连续函数几乎到处都接近其均值。我们还给出了一种解析估计相应期望值的方法，并证明了在 H 是局部可观测量之和的情况下典型的约简密度矩阵的公式。我们讨论了我们的结果作为量子信息理论的新证明工具和研究量子统计力学现象的含义。作为副产品，我们推导出一种对结果分布进行数值采样的方法，该方法概括了众所周知的高斯方法，以从球体中绘制随机状态。
Given some observable H on a finite-dimensional quantum system, we investigate the typical properties of random state vectors $${|\psi\rangle}$$ that have a fixed expectation value $${\langle\psi|H|\psi\rangle=E}$$ with respect to H. Under some conditions on the spectrum, we prove that this manifold of quantum states shows a concentration of measure phenomenon: any continuous function on this set is almost everywhere close to its mean. We also give a method to estimate the corresponding expectation values analytically, and we prove a formula for the typical reduced density matrix in the case that H is a sum of local observables. We discuss the implications of our results as new proof tools in quantum information theory and to study phenomena in quantum statistical mechanics. As a by-product, we derive a method to sample the resulting distribution numerically, which generalizes the well-known Gaussian method to draw random states from the sphere.