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

Quantum and approximation algorithms for maximum witnesses of Boolean matrix products

The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for n\times n Boolean matrices of the form O(n^{2.575}) has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the standard computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time \tilde{O}(n^{2+\lambda/2})=\tilde{O}(n^{2.434}), where \lambda satisfies the equation \omega(1, \lambda, 1) = 1 + 1.5 \, \lambda and \omega(1, \lambda, 1) is the exponent of the multiplication of an n \times n^{\lambda}$ matrix by an n^{\lambda} \times n matrix. Next, we consider a relaxed version of the MW problem (in the standard model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most \ell in time \tilde{O}((n/\ell)n^{\omega(1,\log_n \ell,1)}). Then, by reducing the relaxed problem to the so called k-witness problem, we provide an algorithm that reports for each non-zero entry C[i,j] of the product matrix C a witness of rank O(\lceil W_C(i,j)/k\rceil ), where W_C(i,j) is the number of witnesses for C[i,j], with high probability. The algorithm runs in \tilde{O}(n^{\omega}k^{0.4653} +n^2k) time, where \omega=\omega(1,1,1).Comment: 14 pages, 3 figure

HOW BELIEFS ABOUT HIV STATUS AFFECT RISKY BEHAVIORS: EVIDENCE FROM MALAWI

This paper examines how beliefs about own HIV status affect decisions to engage in risky sexual behavior, as measured by having extramarital sex and/or multiple sex partners. The empirical analysis is based on a panel survey of males from the 2006 and 2008 rounds of the Malawi Diffusion and Ideational Change Project (MDICP). The paper develops a behavioral model of the belief-risky behavior relationship and estimates the causal effect of beliefs on risky behavior using the Arellano and Carrasco (2003) semiparametric panel data estimator, which accommodates both unobserved heterogeneity and belief endogeneity arising from a possible dependence of current beliefs on past risky behavior. Results show that downward revisions in the belief assigned to being HIV positive increase risky behavior and upward revisions decrease it. For example, based on a linear specification, a decrease in the perceived probability of being HIV positive from 10 to 0 percentage points increases the probability of engaging in risky behavior (extramarital affairs) from 8.3 to 14.1 percentage points. We also develop and implement a modified version of the Arellano and Carrasco (2003) estimator to allow for misreporting of risky behavior and find estimates to be robust to a range of plausible misreporting levels. © 2013 The Authors. Journal of Applied Econometrics published by John Wiley & Sons, Ltd

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

Magnetization steps in Zn_(1-x)Mn_xO: Four largest exchange constants and single-ion anisotropy

Magnetization steps (MST's) from Mn pairs in several single crystals of Zn_(1-x)Mn_xO (0.0056<=x<=0.030, and in one powder (x=0.029), were observed. The largest two exchange constants, J1/kB=-18.2+/-0.5K and J1'/kB=-24.3+/-0.6K, were obtained from large peaks in the differential susceptibility, dM/dH, measured in pulsed magnetic fields, H, up to 500 kOe. These two largest J's are associated with the two inequivalent classes of nearest neighbors (NN's) in the wurtzite structure. The 29% difference between J1 and J1' is substantially larger than 13% in CdS:Mn, and 15% in CdSe:Mn. The pulsed-field data also indicate that, despite the direct contact between the samples and a superfluid-helium bath, substantial departures from thermal equilibrium occurred during the 7.4 ms pulse. The third- and fourth-largest J's were determined from the magnetization M at 20 mK, measured in dc magnetic fields H up to 90 kOe. Both field orientations H||c and H||[10-10] were studied. (The [10-10] direction is perpendicular to the c-axis, [0001].) By definition, neighbors which are not NN's are distant neighbors (DN's). The largest DN exchange constant (third-largest overall), has the value J/kB=-0.543+/-0.005K, and is associated with the DN at r=c. Because this is not the closest DN, this result implies that the J's do not decrease monotonically with the distance r. The second-largest DN exchange constant (fourth-largest overall), has the value J/kB=-0.080 K. It is associated with one of the two classes of neighbors that have a coordination number z=12, but the evidence is insufficient for a definite unique choice. The dependence of M on the direction of H gives D/kB=-0.039+/-0.008K, in fair agreement with -0.031 K from earlier EPR work.Comment: 12 pages, 10 figures. Submitted to PR

Magnetization Process of Nanoscale Iron Cluster

Low-temperature magnetization process of the nanoscale iron cluster in linearly sweeped fields is investigated by a numerical analysis of time-dependent Schr$\ddot{\rm o}$dinger equation and the quantum master equation. We introduce an effective basis method extracting important states, by which we can obtain the magnetization process effectively. We investigate the structure of the field derivative of the magnetization. We find out that the antisymmetric interaction determined from the lattice structure reproduces well the experimental results of the iron magnets and that this interaction plays an important role in the iron cluster. Deviations from the adiabatic process are also studied. In the fast sweeping case, our calculations indicate that the nonadiabatic transition dominantly occurs at the level crossing for the lowest field. In slow sweeping case, due to the influence of the thermal environment to the spin system, the field derivative of the magnetization shows an asymmetric behavior, the magnetic F$\ddot{\rm o}$hn effect, which explains the substructure of the experimental results in the pulsed field.Comment: 5 pages of text and 2 pages of 6 figures. To appear in J. Phys. Soc. Jp