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On the Measurement of Qubits

James, Daniel F.V.; Kwiat, Paul G.; Munro, William J.; White, Andrew G.

HPL-2001-160

Keyword(s): quantum tomography; state reconstruction; maximum likelihood methods; tomographic reconstruction

Abstract: We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems ("qubits"). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices which are always non- negative definite). In addition a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results. Notes: Daniel F.V. James, Theory Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. Paul G. Kwiat, Dept. of Physics, University of Illinois, Urbana-Champaign, Illinois, USA. Andrew G. White, Special Research Centre for Quantum Computer Technology, University of Queensland, Brisbane, AUSTRALIA

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