The Groverian Measure of Entanglement for Mixed States

The Groverian entanglement measure introduced earlier for pure quantum states [O. Biham, M.A. Nielsen and T. Osborne, Phys. Rev. A 65, 062312 (2002)] is generalized to the case of mixed states, in a way that maintains its operational interpretation. The Groverian measure of a mixed state of n qubits is obtained by a purification procedure into a pure state of 2n qubits, followed by an optimization process based on Uhlmann's theorem, before the resulting state is fed into Grover's search algorithm. The Groverian measure, expressed in terms of the maximal success probability of the algorithm, provides an operational measure of entanglement of both pure and mixed quantum states of multiple qubits. These results may provide further insight into the role of entanglement in making quantum algorithms powerful.Comment: 6 pages, 2 figure

Algebraic analysis of quantum search with pure and mixed states

An algebraic analysis of Grover's quantum search algorithm is presented for the case in which the initial state is an arbitrary pure quantum state of n qubits. This approach reveals the geometrical structure of the quantum search process, which turns out to be confined to a four-dimensional subspace of the Hilbert space. This work unifies and generalizes earlier results on the time evolution of the amplitudes during the quantum search, the optimal number of iterations and the success probability. Furthermore, it enables a direct generalization to the case in which the initial state is a mixed state, providing an exact formula for the success probability.Comment: 13 page

Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure

The Groverian entanglement measure, G(psi), is applied to characterize a variety of pure quantum states |psi> of multiple qubits. The Groverian measure is calculated analytically for certain states of high symmetry, while for arbitrary states it is evaluated using a numerical procedure. In particular, it is calculated for the class of Greenberger-Horne-Zeilinger states, the W states as well as for random pure states of n qubits. The entanglement generated by Grover's algorithm is evaluated by calculating G(psi) for the intermediate states that are obtained after t Grover iterations, for various initial states and for different sets of the marked states.Comment: 28 pages, 5 figure

Formation of Multipartite Entanglement Using Random Quantum Gates

The formation of multipartite quantum entanglement by repeated operation of one and two qubit gates is examined. The resulting entanglement is evaluated using two measures: the average bipartite entanglement and the Groverian measure. A comparison is made between two geometries of the quantum register: a one dimensional chain in which two-qubit gates apply only locally between nearest neighbors and a non-local geometry in which such gates may apply between any pair of qubits. More specifically, we use a combination of random single qubit rotations and a fixed two-qubit gate such as the controlled-phase gate. It is found that in the non-local geometry the entanglement is generated at a higher rate. In both geometries, the Groverian measure converges to its asymptotic value more slowly than the average bipartite entanglement. These results are expected to have implications on different proposed geometries of future quantum computers with local and non-local interactions between the qubits.Comment: 7 pages, 5 figure

Electromigration-Induced Propagation of Nonlinear Surface Waves

Due to the effects of surface electromigration, waves can propagate over the free surface of a current-carrying metallic or semiconducting film of thickness h_0. In this paper, waves of finite amplitude, and slow modulations of these waves, are studied. Periodic wave trains of finite amplitude are found, as well as their dispersion relation. If the film material is isotropic, a wave train with wavelength lambda is unstable if lambda/h_0 < 3.9027..., and is otherwise marginally stable. The equation of motion for slow modulations of a finite amplitude, periodic wave train is shown to be the nonlinear Schrodinger equation. As a result, envelope solitons can travel over the film's surface.Comment: 13 pages, 2 figures. To appear in Phys. Rev.

Necessity of Superposition of Macroscopically Distinct States for Quantum Computational Speedup

For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer performing Shor's factoring algorithm [A. Ukena and A. Shimizu, Phys. Rev. A69 (2004) 022301], it needs to be generalized for it to be applicable to a large class of algorithms and/or other models such as measurement-based quantum computers. To treat such general cases, we first generalize the indices for the superposition of macroscopically distinct states. We then generalize the conjecture, using the generalized indices, in such a way that it is unambiguously applicable to general models if a quantum algorithm achieves exponential speedup. On the basis of this generalized conjecture, we further extend the conjecture to Grover's quantum search algorithm, whose speedup is large but quadratic. It is shown that this extended conjecture is also correct. Since Grover's algorithm is a representative algorithm for unstructured problems, the present result further supports the conjecture.Comment: 18 pages, 5 figures. Fixed typos throughout the manuscript. This version has been publishe

Electromigration-Induced Flow of Islands and Voids on the Cu(001) Surface

Electromigration-induced flow of islands and voids on the Cu(001) surface is studied at the atomic scale. The basic drift mechanisms are identified using a complete set of energy barriers for adatom hopping on the Cu(001) surface, combined with kinetic Monte Carlo simulations. The energy barriers are calculated by the embedded atom method, and parameterized using a simple model. The dependence of the flow on the temperature, the size of the clusters, and the strength of the applied field is obtained. For both islands and voids it is found that edge diffusion is the dominant mass-transport mechanism. The rate limiting steps are identified. For both islands and voids they involve detachment of atoms from corners into the adjacent edge. The energy barriers for these moves are found to be in good agreement with the activation energy for island/void drift obtained from Arrhenius analysis of the simulation results. The relevance of the results to other FCC(001) metal surfaces and their experimental implications are discussed.Comment: 9 pages, 13 ps figure