Lower bound on the dimension of a quantum system given measured data

被引:104
作者
Wehner, Stephanie [1 ]
Christandl, Matthias [2 ]
Doherty, Andrew C. [3 ]
机构
[1] CALTECH, Inst Quantum Informat, Pasadena, CA 91125 USA
[2] Univ Munich, Fac Phys, Arnold Sommerfeld Ctr Theoret Phys, D-80333 Munich, Germany
[3] Univ Queensland, Sch Phys Sci, St Lucia, Qld 4072, Australia
来源
PHYSICAL REVIEW A | 2008年 / 78卷 / 06期
基金
澳大利亚研究理事会;
关键词
Bell theorem; Hilbert spaces; mathematical operators; matrix algebra; measurement theory; probability; quantum entanglement; random processes;
D O I
10.1103/PhysRevA.78.062112
中图分类号
O43 [光学];
学科分类号
070207 ; 0803 ;
摘要
We imagine an experiment on an unknown quantum mechanical system in which the system is prepared in various ways and a range of measurements are performed. For each measurement M and preparation rho the experimenter can determine, given enough time, the probability of a given outcome a: p(a parallel to M,rho). How large does the Hilbert space of the quantum system have to be in order to allow us to find density matrices and measurement operators that will reproduce the given probability distribution? In this paper, we prove a simple lower bound for the dimension of the Hilbert space. The main insight is to relate this problem to the construction of quantum random access codes, for which interesting bounds on the Hilbert space dimension already exist. We discuss several applications of our result to hidden-variable or ontological models, to Bell inequalities, and to properties of the smooth min-entropy.
引用
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页数:8
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