On the measurement of qubits

By D. James, Paul G. Kwiat, William J. Munro, A. White. 2001

1,921 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ebb006597dabc5613632564c391fcf43c33a2720

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Citations in the corpus

Continuous-variable optical quantum-state tomography (2009, methodology): “A textbook example of the above is the work of James et al. 2001 .”
Quantum random number generators (2016, methodology): “The device includes a testing phase in which it performs full tomography of the input state (James et al., 2001) from a set of measurements in order to determine the 2×2 matrix that describes the photonic two level system for a single photon or the effective two-dimensional Hilbert space of…”
Simplifying quantum logic using higher-dimensional Hilbert spaces (2009, methodology): “Our state and process tomography uses maximum likelihood estimation to reconstruct physical states and Monte Carlo simulation for error analysi”
Demonstration of two-qubit algorithms with a superconducting quantum processor (2009, methodology): “The 15 measured values are then input to a maximum likelihood estimator 34 of ρ (see Supplementary Information).”
Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement (2010, background): “The first is a direct determination of H ( X |B)+ H ( Z |B) (shown in the figures using blue datapoints) which requires us to experimentally find the reduced density matrix of Bob’s photon, conditioned on Alice’s measurement choice and outcom”
Quantum computing with trapped ions (2008, methodology): “This problem is avoided by employing a maximum likelihood estimation of the density matrix (Hradil, 1997; James et al., 2001) that makes use of the estimated expectation values for searching in the set of meaningful density matrices the ’most likely one’.”
Experimental one-way quantum computing (2005, methodology): “Instead of a direct linear combination of measurement results which can lead to unphysical density matrices due to experimental noise, we employ a maximum-likelihood reconstruction technique 30-32 .”
Photonic State Tomography (2005, background): “In the photon case, for example, errors in counting statistics and waveplate settings were analyzed in some detail in reference (James et al., 2001), giving errors in both density matrices and commonly derived quantities, such as the tangle and the linear entropy.”
Heralded generation of entangled photon pairs (2010, methodology): “The most likely physical density matrix for our two-qubit state was extracted using a maximum-likelihood reconstructio”