On the Measurement of Qubits - HP · PDF...

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On the Measurement of Qubits Daniel F.V. James 1 , Paul G. Kwiat 2 , William J. Munro, Andrew G. White 3 Trusted E-Services Laboratory HP Laboratories Bristol HPL-2001-160 July 2 nd , 2001* quantum tomography, state reconstruction, maximum likelihood methods, tomographic reconstruction 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. * Internal Accession Date Only Approved for External Publication 1 Theory Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. 2 Dept. of Physics, University of Illinois, Urbana-Champaign, Illinois, USA. 3 Special Research Centre for Quantum Computer Technology, University of Queensland, Brisbane, Australia Copyright Hewlett-Packard Company 2001

Transcript of On the Measurement of Qubits - HP · PDF...

On the Measurement of Qubits Daniel F.V. James1, Paul G. Kwiat2, William J. Munro, Andrew G. White3 Trusted E-Services Laboratory HP Laboratories Bristol HPL-2001-160 July 2nd , 2001* quantum tomography, state reconstruction, maximum likelihood methods, tomographic reconstruction

We describe in detail the theory underpinning the measurement of density matrices of a pair of quantumtwo-level systems ("qubits"). Our particular emphasisis 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 ofproducing 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.

* Internal Accession Date Only Approved for External Publication 1 Theory Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. 2 Dept. of Physics, University of Illinois, Urbana-Champaign, Illinois, USA. 3 Special Research Centre for Quantum Computer Technology, University of Queensland, Brisbane, Australia Copyright Hewlett-Packard Company 2001

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