Daniel F. V. James

Measurement of qubits

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 that 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.

Deterministic quantum teleportation with atoms

Teleportation of a quantum state encompasses the complete transfer of information from one particle to another. The complete specification of the quantum state of a system generally requires an infinite amount of information, even for simple two-level systems (qubits). Moreover, the principles of quantum mechanics dictate that any measurement on a system immediately alters its state, while yielding at most one bit of information. The transfer of a state from one system to another (by performing measurements on the first and operations on the second) might therefore appear impossible. However, it has been shown that the entangling properties of quantum mechanics, in combination with classical communication, allow quantum-state teleportation to be performed. Teleportation using pairs of entangled photons has been demonstrated, but such techniques are probabilistic, requiring post-selection of measured photons. Here, we report deterministic quantum-state teleportation between a pair of trapped calcium ions. Following closely the original proposal, we create a highly entangled pair of ions and perform a complete Bell-state measurement involving one ion from this pair and a third source ion. State reconstruction conditioned on this measurement is then performed on the other half of the entangled pair. The measured fidelity is 75%, demonstrating unequivocally the quantum nature of the process.

Change of polarization of light beams on propagation in free space

It is shown by use of a simple model that in general the state of polarization of a light beam generated by a partially coherent source changes as the beam propagates in free space.

Nonmaximally Entangled States: Production, Characterization, and Utilization

Using a spontaneous-down-conversion photon source, we produce true nonmaximally entangled states, i.e., without the need for postselection. The degree and phase of entanglement are readily tunable, and are characterized both by a standard analysis using coincidence minima, and by quantum state tomography of the two-photon state. Using the latter, we experimentally reconstruct the reduced density matrix for the polarization. Finally, we use these states to measure the Hardy fraction, obtaining a result that is 122σ from any local-realistic result. ©1999 The American Physical Society

Correlation-induced spectral changes

This paper presents a review of research, both theoretical and experimental, concerning the influence of coherence properties of fluctuating light sources and of correlation properties of scattering media on the spectra of radiated and scattered fields. Much of this research followed a discovery made in 1986, that the spectrum of light may change on propagation, even in free space. More than 100 papers on this topic have been published to date and many of them are reviewed, or at least mentioned, in this article. After an introduction and a summary of some of the main mathematical results relating to second-order coherence theory of statistically stationary optical fields, spectral changes that may take place on superposing fields produced by two partially correlated sources are discussed. Spectral effects in fields produced by two-dimensional secondary sources and by three-dimensional primary sources are then considered. The section which follows describes spectral changes that may arise when polychromatic light is scattered on media whose physical properties vary randomly either in space and/or in time. A review is also presented of recent research, which has revealed that under certain circumstances the changes in the spectrum of light scattered on random media may imitate the Doppler effect, even though the source, the medium and the observer are all at rest with respect to one another. In the final section a brief review is given of a new emerging technique sometimes called spatial-coherence spectroscopy. It is based on the discovery that it is possible, under certain circumstances, to determine field correlations from spectral measurements.

Quantum Process Tomography of a Controlled-NOT Gate

We demonstrate complete characterization of a two-qubit entangling process--a linear optics controlled-NOT gate operating with coincident detection--by quantum process tomography. We use a maximum-likelihood estimation to convert the experimental data into a physical process matrix. The process matrix allows an accurate prediction of the operation of the gate for arbitrary input states and a calculation of gate performance measures such as the average gate fidelity, average purity, and entangling capability of our gate, which are 0.90, 0.83, and 0.73, respectively.

Maximizing the entanglement of two mixed qubits

Two-qubit states occupy a large and relatively unexplored Hilbert space. Such states can be succinctly characterized by their degree of entanglement and purity. In this article we investigate entangled mixed states and present a class of states that have the maximum amount of entanglement for a given linear entropy.

The generalized Fresnel transform and its application to optics

We consider the mathematical properties of a class of linear transforms, which we call the generalized Fresnel transforms, and which have wide applications to several areas of optics. Well-known transforms, such as the fractional Fourier transform and the Fresnel transform, can be seen to be special cases of this general transform. By use of an analogy with a quantum harmonic oscillator, we derive mathematical expressions for functions which remain invariant under such transforms, a result which has important applications in the theory of laser resonator modes.

Experimental Demonstration of a Compiled Version of Shor's Algorithm with Quantum Entanglement

Shors powerful quantum algorithm for factoring represents a major challenge in quantum computation. Here, we implement a compiled version in a photonic system. For the first time, we demonstrate the core processes, coherent control, and resultant entangled states required in a full-scale implementation. These are necessary steps on the path towards scalable quantum computing. Our results highlight that the algorithm performance is not the same as that of the underlying quantum circuit and stress the importance of developing techniques for characterizing quantum algorithms.

Ion Trap Quantum Computing with Warm Ions

We describe two schemes to manipulate the electronic qubit states of trapped ions independent of the collective vibrational state of the ions. The first scheme uses an adiabatic method, and thus is intrinsically slow. The second scheme takes the opposite approach and uses fast pulses to produce an effective direct coupling between the electronic qubits. This last scheme enables the simulation of a number of nonlinear quantum systems including systems that exhibit phase transitions, and other semiclassical bifurcations. Quantum tunnelling and entangled states occur in such systems.