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 quantities͒ and a maximum likelihood technique which requires numerical optimization ͑but has the advantage of producing density matrices that are always non-negative definite͒. 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.

The theory of interactions between lasers and cold trapped ions as it pertains to the design of Cirac-Zoller quantum computers is discussed. The mean positions of the trapped ions, the eigenvalues and eigenmodes of the ions' oscillations, the magnitude of the Rabi frequencies for both allowed and forbidden internal transitions of the ions and the validity criterion for the required Hamiltonian are calculated. Energy level data for a variety of ion species is also presented.

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.

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 accurate prediction of the operation of the gate for arbitrary input states and 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.PACS numbers: 03.67. Lx, 03.65.Wj, 03.67.Mn, Quantum information science offers the potential for major advances such as quantum computing [1] and quantum communication [2], as well as many other quantum technologies [3]. Two-qubit entangling gates, such as the controlled-not (cnot), are fundamental elements in the archetypal quantum computer [1]. A promising proposal for achieving scalable quantum computing is that of Knill, Laflamme and Milburn (KLM), in which linear optics and a measurement-induced Kerr-like nonlinearity can be used to construct cnot gates [4]. Gates such as these can also be used to prepare the required entangled resource for optical cluster state quantum computation [5]. The nonlinearity upon which the KLM and related [6,7] cnot schemes are built can be used for other important quantum information tasks, such as quantum nondemolition measurements [8,9] and preparation of novel quantum states (for example, [10]). An essential step in realizing such advances is the complete characterization of quantum processes.A complete characterization in a particular input/output state space requires determination of the mapping from one to the other. In discrete-variable quantum information, this map can be represented as a state transfer function, expressed in terms of a process matrix χ. Experimentally, χ is obtained by performing quantum process tomography (QPT) [11,12]. QPT has been performed in a limited number of systems. A one-qubit teleportation circuit [13], and a controlled-NOT process acting on a highly mixed two-qubit state [14] have been investigated in liquid-state NMR. In optical systems, where pure qubit states are readily prepared, one-qubit processes have been investigated by both ancilla-assisted [15,16] and standard [17] QPT. Two-qubit optical QPT has been performed on a beamsplitter acting as a Bellstate filter [18].We fully characterize a two-qubit entangling gatea cnot gate acting on pure input states -by QPT, maximum-likelihood reconstruction, and analysis of the resulting process matrix. The maximum likelihood technique overcomes the problem that the naïve matrix inversion procedure in QPT, when performed on real (i.e., inherently noisy) experimental data, typically leads to an unphysical process matrix. In a previous maximumlikelihood QPT experiment [18], a reduced set of fitting constraints was used. Here we present a fully-constrained fitting technique that can be applied to any physical process. After obtaining our physical...

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 122s from any local-realistic result.PACS numbers: 03.65. Bz, 42.50.Dv Entanglement is arguably the defining characteristic of quantum mechanics, and can occur between any quantum systems, be they separate particles [1] or separate degrees of freedom of a single particle [2]. The latter can be used to realize interference-based all-optical implementations of quantum algorithms [3], while multiparticle entangled states are central in discussions of locality [4,5], and in quantum information, where they enable quantum computation [6], cryptography [7], dense coding [8], and teleportation [9]. More generally, entanglement is the underlying mechanism for measurements on, and decoherence of, quantum systems, and thus is central to understanding the quantum/classical interface.Historically, controlled production of multiparticle entangled states has proven to be nontrivial. To date, the "cleanest" and most accessible source of such entanglement arises from the process of spontaneous optical parametric down-conversion in a nonlinear crystal (for a review, see [10]). This entanglement is of a specific and limited kind: the states are maximally entangled, e.g., ͑jHV ͘ 6´jVH͒͘͞ p 1 1 j´j 2 , where H and V , respectively, represent the horizontal and vertical polarizations of two separated photons, and´1. There is no possibility of varying the intrinsic degree of entanglement, [ 11], to produce nonmaximally entangled states without compromising the purity of the state, i.e., introducing mixture [12]. Nonmaximally entangled states have been shown to reduce the required detector efficiencies for loophole-free tests of Bell inequalities [13], as well as allowing logical arguments that demonstrate the nonlocality of quantum mechanics without inequalities [14][15][16]. More generally, such states lie in a previously inaccessible range of Hilbert space, and may therefore be an important resource in quantum information applications.States with a fixed degree of entanglement,´Ӎ 4͞3, have been deterministically generated in ion traps [17], and there have been several optical experiments where nonmaximally entangled states were controllably generated via postselection, i.e., selective measurement of a product state, after the state had been produced [18,19]. The latter experiments are of considerable pedagogical interest in that they demonstrate the logic behind inequality-free locality tests. However, the underlying state is factorizable, and so is not truly entangled. In this Letter, we describe the controllable production...

This paper presents a useful compact formula for deriving an effective Hamiltonian describing the time-averaged dynamics of detuned quantum systems. The formalism also works for ensembleaveraged dynamics of stochastic systems. To illustrate the technique we give examples involving Raman processes, Bloch-Siegert shifts and Quantum Logic Gates.* To be published in Canadian Journal of Physics.

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.

Shor's 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.