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Sample-Optimal Tomography of Quantum States
Authors
Bacon D.
Belavkin VP
+8 more
David Gross GLF
Fano Robert M
Hayashi Masahito
Hayashi Masahito
Holevo A. S.
Holevo A.S.
Press ACM
Szarek Stanislaw J.
Publication date
1 June 2016
Publisher
'Institute of Electrical and Electronics Engineers (IEEE)'
Doi
Cite
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on
arXiv
Abstract
© 1963-2012 IEEE. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error \epsilon in trace distance required O(dr^{2}/\epsilon ^{2}) copies for a d -dimensional density matrix of rank r. Here, we give a theoretical measurement scheme (POVM) that requires O (dr/ \delta) \ln ~(d/\delta) copies to estimate \rho to error \delta in infidelity, and a matching lower bound up to logarithmic factors. This implies O((dr / \epsilon ^{2}) \ln ~(d/\epsilon)) copies suffice to achieve error \epsilon in trace distance. We also prove that for independent (product) measurements, \Omega (dr^{2}/\delta ^{2}) / \ln (1/\delta) copies are necessary in order to achieve error \delta in infidelity. For fixed d , our measurement can be implemented on a quantum computer in time polynomial in n
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