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

Wehner, Stephanie, Christandl, Matthias and Doherty, Andrew C. (2008) Lower bound on the dimension of a quantum system given measured data. Physical Review A, 78 6: 062112-1-062112-8. doi:10.1103/PhysRevA.78.062112

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Author Wehner, Stephanie
Christandl, Matthias
Doherty, Andrew C.
Title Lower bound on the dimension of a quantum system given measured data
Journal name Physical Review A   Check publisher's open access policy
ISSN 1050-2947
Publication date 2008-12-22
Sub-type Article (original research)
DOI 10.1103/PhysRevA.78.062112
Open Access Status File (Publisher version)
Volume 78
Issue 6
Start page 062112-1
End page 062112-8
Total pages 8
Place of publication College Park, MD, United States
Publisher American Physical Society
Language eng
Abstract 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 the experimenter can determine, given enough time, the probability of a given outcome a: p(a|M,). 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.
Keyword Bell theorem
Hilbert spaces
Mathematical operators
Matrix algebra
Measurement theory
Quantum entanglement
Random processes
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: 2009 Higher Education Research Data Collection
School of Mathematics and Physics
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Citation counts: TR Web of Science Citation Count  Cited 53 times in Thomson Reuters Web of Science Article | Citations
Scopus Citation Count Cited 56 times in Scopus Article | Citations
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Created: Sat, 18 Apr 2009, 01:42:57 EST by Jo Hughes on behalf of School of Mathematics & Physics