Optimal two-qubit tomography based on local and global measurements: Maximal robustness against errors as described by condition numbers

Adam Miranowicz, Karol Bartkiewicz, Jan Peřina, Jr., Masato Koashi, Nobuyuki Imoto, and Franco Nori
Phys. Rev. A 90, 062123 – Published 18 December 2014

Abstract

We present an error analysis of various tomographic protocols based on the linear inversion for the reconstruction of an unknown two-qubit state. We solve the problem of finding a tomographic protocol which is the most robust against errors in terms of the lowest value (i.e., equal to 1) of a condition number, as required by the Gastinel–Kahan theorem. In contrast, standard tomographic protocols, including those based on mutually unbiased bases, are nonoptimal for determining all the 16 elements of an unknown two-qubit density matrix. Our method is based on the measurements of the 16 generalized Pauli operators, where twelve of them can be locally measured, and the other four require nonlocal Bell measurements. Our method corresponds to selectively measuring, one by one, all the real and imaginary elements of an unknown two-qubit density matrix. We describe two experimentally feasible setups of this protocol for the optimal reconstruction of two photons in an unknown polarization state using conventional detectors and linear-optical elements. Moreover, we define the operators for the optimal reconstruction of the states of multiqubit or multilevel (qudit) systems.

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  • Received 16 September 2014

DOI:https://doi.org/10.1103/PhysRevA.90.062123

©2014 American Physical Society

Authors & Affiliations

Adam Miranowicz1,2, Karol Bartkiewicz2,3, Jan Peřina, Jr.3, Masato Koashi4, Nobuyuki Imoto5, and Franco Nori1,6

  • 1CEMS, RIKEN, 351-0198 Wako-shi, Japan
  • 2Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland
  • 3RCPTM, Joint Laboratory of Optics of Palacký University and Institute of Physics of AS CR, Faculty of Science, Palacký University, 17. listopadu 12, 77-146 Olomouc, Czech Republic
  • 4Photon Science Center, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
  • 5Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
  • 6Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

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Issue

Vol. 90, Iss. 6 — December 2014

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