For two classical Brownian particles an analog of continuous-variable quantum entanglement is presented: The common probability distribution of the two coordinates and the corresponding coarse-grained velocities cannot always be prepared via mixing of any factorized distributions referring to the two particles separately. This is possible for particles which have interacted in the past, but do not interact at present. Three factors are crucial for the effect: (1) separation of time scales of coordinate and momentum which motivates the definition of coarse-grained velocities; (2) the resulting uncertainty relations between the coordinate of the Brownian particle and the change of its coarse-grained velocity; (3) the fact that the coarse-grained velocity, though pertaining to a single Brownian particle, is defined on a common context of two particles. The Brownian entanglement is a consequence of a coarse-grained description and disappears for a finer resolution of the Brownian motion. Analogies with the quantum situation are discussed, as well as possibilities of experimental realization of the effect in examples of macroscopic Brownian motion.

We present a model, motivated by the criterion of reality put forward by Einstein, Podolsky, and Rosen and supplemented by classical communication, which correctly reproduces the quantum-mechanical predictions for measurements of all products of Pauli operators on an $n$-qubit GHZ state (or “cat state”). The $n-2\text{bits}$ employed by our model are shown to be optimal for the allowed set of measurements, demonstrating that the required communication overhead scales linearly with $n$. We formulate a connection between the generation of the local values utilized by our model and the stabilizer formalism, which leads us to conjecture that a generalization of this method will shed light on the content of the Gottesman-Knill theorem.

We describe an approach for characterizing the process performed by a quantum gate using quantum process tomography, by first modeling the gate in an extended Hilbert space, which includes nonqubit degrees of freedom. To prevent unphysical processes from being predicted, present quantum process tomography procedures incorporate mathematical constraints, which make no assumptions as to the actual physical nature of the system being described. By contrast, the procedure presented here assumes a particular class of physical processes, and enforces physicality by fitting the data to this model. This allows quantum process tomography to be performed using a smaller experimental data set, and produces parameters with a direct physical interpretation. The approach is demonstrated by example of mode matching in an all-optical controlled-NOT gate. The techniques described are general and could be applied to other optical circuits or quantum computing architectures.

In order to place an upper bound on the tolerable amount of photon loss at the detectors, we calculate the error threshold for the linear optics quantum computing proposal by Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)] under an error model where photon detectors have efficiency $<100\%$ but all other components—such as single photon sources, beam splitters, and phase shifters—are perfect and introduce no errors. We make use of the fact that the error model induced by the lossy hardware is that of an erasure channel, i.e., the error locations are always known. Using a method based on a Markov chain description of the error correction procedure, our calculations show that, with the 7 qubit Calderbank–Shor–Steane quantum code, the gate error threshold for fault tolerant quantum computation is bounded below by a value between 1.78% and 11.5% depending on the construction of the entangling gates.

We investigate the unambiguous comparison of quantum states in a scenario that is more general than the one that was originally suggested by Barnett et al. First, we find the optimal solution for the comparison of two states taken from a set of two pure states with arbitrary a priori probabilities. We show that the optimal coherent measurement is always superior to the optimal incoherent measurement. Second, we develop a strategy for the comparison of two states from a set of $N$ pure states, and find an optimal solution for some parameter range when $N=3$. In both cases we use the reduction method for the corresponding problem of mixed-state discrimination, as introduced by Raynal et al., which reduces the problem to the discrimination of two pure states only for $N=2$. Finally, we provide a necessary and sufficient condition for unambiguous comparison of mixed states to be possible.

The band structure of a Bose-Einstein condensate is studied for lattice traps of sinusoidal, Jacobi elliptic, and Kronig-Penney form, all in the context of the nonlinear Schrödinger equation. It is demonstrated that the physical properties of the system are independent of the type of lattice. The Kronig-Penney potential, which admits a full exact solution in closed analytical form, is then used to understand the swallowtails, or loops, that form in the band structure. The appearance of swallowtails is explained by adiabatically tuning a second lattice with half the period. Such a two-color lattice, which can be easily realized in experiments, has intriguing physical properties. For instance, swallowtails appear even for weak nonlinearity, which is the experimental regime. We determine the stability properties of this system and relate them to current experiments.

We discuss the necessity of using nonstandard boson operators for diagonalizing quadratic bosonic forms which are not positive definite and its convenience for describing the temporal evolution of the system. Such operators correspond to non-Hermitian coordinates and momenta and are associated with complex frequencies. As application, we examine a bosonic version of a BCS-like pairing Hamiltonian, which, in contrast with the fermionic case, is stable just for limited values of the gap parameter and requires the use of the present extended treatment for a general diagonal representation. The dynamical stability of such forms and the occurrence of nondiagonalizable cases are also discussed.

We construct the Hamiltonian description of the Chern-Simons theory with $Z_n$ gauge group on a triangular lattice. We show that the $Z_2$ model can be mapped onto free Majorana fermions and compute the excitation spectrum. In the bulk the spectrum turns out to be gapless but acquires a gap if a magnetic term is added to the Hamiltonian. On a lattice edge one gets additional nongauge invariant (matter) gapless degrees of freedom whose number grows linearly with the edge length. Therefore, a small hole in the lattice plays the role of a charged particle characterized by a nontrivial projective representation of the gauge group, while a long edge provides a decoherence mechanism for the fluxes. We discuss briefly the implications for the implementations of protected qubits.

In this paper we give sharp two-sided estimates of the volume of the set of separable states on $N$ qubits. In particular, the magnitude of the “effective radius” of that set in the sense of volume is determined up to a factor which is a (small) power of $N$, and thus precisely on the scale of powers of its dimension. We also identify an ellipsoid that appears to optimally approximate the set of separable states. Additionally, one of the appendixes contains sharp estimates (by known methods) for the expected values of norms of the Gaussian unitary ensemble random matrices. We employ standard tools of classical convexity, high-dimensional probability, and geometry of Banach spaces.

Measurements of fully differential cross sections for electron impact ionization of atoms have been performed for over 30 years. However, only within the last ten years has agreement between experiment and theory been achieved for ionization of hydrogen and helium. For the heavier inert gases, reasonably good agreement between experiment and theory has only been achieved for high incident energies while serious discrepancies are common for intermediate and low incident energies. It is believed that a major source of the problem stems from an improper/inadequate treatment of exchange distortion (ED) and the effects of post-collision interactions (PCIs). In this paper, two different methods for including ED are examined—one based upon the $R$ matrix (close-coupling) approach and one originating from the single-configuration Hartree-Fock approach. In general, these two methods predict significant, but different, ED effects. The importance of PCI is studied by including the final-state Coulomb interaction directly in the final-state wave function. This procedure guarantees that PCI effects will be included to all orders of perturbation theory. For intermediate energies, PCI is an important effect and leads to an overall improvement in the agreement between experiment and theory.

Quantum signatures of a free electron in interaction with a continuous-wave radiation field are investigated by looking for negativities in the Wigner function of the system. The free-electron wave function in the radiation field is calculated fully analytically by solving the appropriate Schrödinger equation in the Krammers-Henneberger frame. It is found that pronounced quantum signatures show up already for a laser peak field of magnitude $E_0=1\text{–}2\mathrm{a.u.}$ and a frequency $\omega =1\mathrm{a.u.}$ However, the nonclassical behavior gets lost if the interaction with the radiation field is taken in the dipole approximation.

A quantum theory of two-photon-state generation via four-wave mixing in optical fibers is studied, with emphasis on the case where the pump is a classical, narrow (picosecond-duration) pulse. One of the experiments performed in our lab is discussed and analyzed. Numerical predictions from the theory are shown to be in good agreement with the experimental results.

The thermal entanglement in a two-qubit Heisenberg $XXZ$ spin chain is investigated under an inhomogeneous magnetic field $b$. We show that the ground-state entanglement is independent of the interaction of $z$-component $J_z$. The thermal entanglement at the fixed temperature can be enhanced when $J_z$ increases. We strictly show that for any temperature $T$ and $J_z$, the entanglement is symmetric with respect to zero inhomogeneous magnetic field, and the critical inhomogeneous magnetic field $b_c$ is independent of $J_z$. The critical magnetic field $B_c$ increases with the increasing $\mid b\mid$ but the maximum entanglement value that the system can arrive at becomes smaller.

The first quantum cryptography protocol, proposed by Bennett and Brassard in 1984 (BB84), has been widely studied in recent years. This protocol uses four states (more precisely, two complementary bases) for the encoding of the classical bit. Recently, it has been noticed that by using the same four states, but a different encoding of information, one can define a protocol which is more robust in practical implementations, specifically when attenuated laser pulses are used instead of single-photon sources [V. Scarani et al., Phys. Rev. Lett. 92, 057901 (2004), referred to as the SARG04 protocol]. We present a detailed study of SARG04 in two different regimes. In the first part, we consider an implementation with a single-photon source: we derive bounds on the error rate $Q$ for security against all possible attacks by the eavesdropper. The lower and the upper bound obtained for SARG04 ($Q\lesssim 10.95\%$ and $Q\gtrsim 14.9\%$, respectively) are close to those obtained for BB84 ($Q\lesssim 12.4\%$ and $Q\gtrsim 14.6\%$, respectively). In the second part, we consider a realistic source consisting of an attenuated laser and improve on previous analysis by allowing Alice to optimize the mean number of photons as a function of the distance. The SARG04 protocol is found to perform better than BB84, both in secret-key rate and in maximal achievable distance, for a wide class of Eve’s attacks.

We analyze the security of so-called quantum-tomographic cryptography with the source producing entangled photons via an experimental scheme proposed by Fattal et al. [Phys. Rev. Lett. 92, 37903 (2004)]. We determine the range of the experimental parameters for which the protocol is secure against the most general incoherent attacks.

We present a theoretical study of the atomic fragment distributions formed in dissociative recombination of $\mathrm{He}{\mathrm{H}}^{+}$ at low energies $(<2\mathrm{eV})$ using wave packets propagating on the potential-energy curves of HeH, coupled by nonadiabatic and rotational interactions. We show that at zero electron collision energy, dissociation into $\mathrm{H}(n=2)+\mathrm{He}$ will completely dominate over dissociation into the ground-state fragments. When the $n=3$ channel become energetically open at $E=0.334\mathrm{eV}$ a rapid switchover to this channel will occur. This phenomenon has previously been observed experimentally, but here we present a theoretical model showing these jumps in the branching ratios. We present an explanation for the observed sudden change in branching ratios and we also discuss how the branching ratios depend on the nonadiabatic couplings used in the model.

The fourth-rank hypermagnetizability tensor of a series of planar conjugated molecules—i.e., aromatic naphthalene, nonaromatic borazine, antiaromatic flattened cyclo-octatetraene, pentalene, indacene, and the 60-carbon fullerene,—has been evaluated at the coupled Hartree-Fock level of accuracy, within the conventional common-origin approach, via extended Gaussian basis sets. The theoretical predictions indicate that antiaromatic molecules are characterized by out-of-plane hypermagnetizability components much bigger than benzene’s. The fullerene cage has a hypermagnetizability exceeding that of planar aromatics by three orders of magnitude. However, the experimental determination of the hypermagnetizabilities constitutes a big challenge. Chemically substituted carbon clusters seem good candidates for detection of cubic magnetic response. Understanding of the calculated hypermagnetizabilities is eased by plots of the differential electron density induced by the applied field. It is found that a strong magnetic field perpendicular to the plane of antiaromatic molecules causes a distortion of the electron charge density, which tends to break C-C double bonds. This charge stretching has a dynamical origin and may be qualitatively explained as a feedback effect due to the Lorentz force acting on the electron current density.

We examine the properties of an atom laser produced by outcoupling from a Bose-Einstein condensate with squeezed light. We model the multimode dynamics of the output field and show that a significant amount of squeezing can be transferred from an optical mode to a propagating atom laser beam. We use this to demonstrate that two-mode squeezing can be used to produce twin atom laser beams with continuous variable entanglement in amplitude and phase.

We present a scheme that produces a conditionally prepared state that can be used for a Bell test based on homodyne detection. The state is near optimal for Bell-inequality violations based on quadrature-phase homodyne measurements that use correlated photon-number states. The scheme utilizes a Gaussian entanglement distillation protocol and uses only beam splitters and photodetection to conditionally prepare a non-Gaussian state from a source of two-mode squeezed states with low squeezing parameter.

We propose an experimentally feasible scheme to generate Greenberger-Horne-Zeilinger type of maximal entanglement among three atomic ensembles, and show one of the applications, controlled secure direct communication. The scheme involves laser manipulation of atomic ensembles, quarter- and half-wave plates, beam splitters, polarizing beam splitters, and single-photon detectors, which are within the reach of current experimental technology.