Lower bound on the dimension of a quantum system given measured data
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|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.
© 2008 The American Physical Society. Received 29 August 2008; published 22 December 2008. We are indebted to the referee for helpful comments to improve the presentation of the paper. A.C.D. is supported by the Australian Research Council. S.W. is supported by NSF Grant No. PHY-04056720. S.W. thanks the University of Queensland for the generous travel support to attend QIP '07, and Oscar Dahlsten and Renato Renner for the invitation to the workshop "Information Primitives and Laws of Nature" at ETH Zurich.
Published - WEHpra08.pdf