How many orthonormal bases are needed to distinguish all pure quantum states?

We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d=2 the answer is three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4 the answer is either three or four. Curiously, the exact number in d=4 seems to be an open problem

Survivability of copper projectiles during hypervelocity impacts in porous ice: A laboratory investigation of the survivability of projectiles impacting comets or other bodies

AbstractDuring hypervelocity impact (>a few kms−1) the resulting cratering and/or disruption of the target body often outweighs interest on the outcome of the projectile material, with the majority of projectiles assumed to be vaporised. However, on Earth, fragments, often metallic, have been recovered from impact sites, meaning that metallic projectile fragments may survive a hypervelocity impact and still exist within the wall, floor and/or ejecta of the impact crater post-impact. The discovery of the remnant impactor composition within the craters of asteroids, planets and comets could provide further information regarding the impact history of a body. Accordingly, we study in the laboratory the survivability of 1 and 2mm diameter copper projectiles fired onto ice at speeds between 1.00 and 7.05kms−1. The projectile was recovered intact at speeds up to 1.50kms−1, with no ductile deformation, but some surface pitting was observed. At 2.39kms−1, the projectile showed increasing ductile deformation and broke into two parts. Above velocities of 2.60kms−1 increasing numbers of projectile fragments were identified post impact, with the mean size of the fragments decreasing with increasing impact velocity. The decrease in size also corresponds with an increase in the number of projectile fragments recovered, as with increasing shock pressure the projectile material is more intensely disrupted, producing smaller and more numerous fragments. The damage to the projectile is divided into four classes with increasing speed and shock pressure: (1) minimal damage, (2) ductile deformation, start of break up, (3) increasing fragmentation, and (4) complete fragmentation. The implications of such behaviour is considered for specific examples of impacts of metallic impactors onto Solar System bodies, including LCROSS impacting the Moon, iron meteorites onto Mars and NASA’s “Deep Impact” mission where a spacecraft impacted a comet

Hypervelocity impact fragmentation of basalt and shale projectiles

Results are presented for the fragmentation of projectiles in laboratory experiments. 1.5 mm cubes and spheres of basalt and shale were impacted onto water at normal incidence and speeds from 0.39 to 6.13 km s−1; corresponding to peak shock pressures 0.7–32 GPa. Projectile fragments were collected and measured (over 100,000 fragments in some impacts, at sizes down to 10 µm). Power laws were fitted to the cumulative fragment size distributions and the evolution of the exponent vs. impact speed and peak shock pressure found. The gradient of each of these power laws increased with increasing impact speed/peak shock pressure. The percentage of the projectiles recovered in the impacts was found and used to estimate projectile remnant survival in different solar system impact scenarios at the mean impact speed appropriate to that scenario. For Pluto, the Moon and in the asteroid belt approximately 55%, 40% and 15%, respectively, of an impactor could survive and be recovered at an impact site. Finally, the catastrophic disruption energy densities of basalt and shale were measured and found to be 24 × 104 J kg−1 and 9 × 104 J kg−1, respectively, a factor of ∼2.5 difference. These corresponded to peak shock pressures of 1 to 1.5 GPa (basalt), and 0.8 GPa (shale). This is for near normal-incidence impacts where tensile strength is dominant. For shallow angle impacts we suggest shear effects dominate, resulting in lower critical energy densities and peak shock pressures. We also determine a method to ascertain information about fragment sizes in solar system impact events using a known size of impactor. The results are used to predict projectile fragments sizes for the Veneneia and Rheasilvia crater forming impacts on Vesta, and similar impacts on Ceres

Forward pi^0 Production and Associated Transverse Energy Flow in Deep-Inelastic Scattering at HERA

Deep-inelastic positron-proton interactions at low values of Bjorken-x down to x \approx 4.10^-5 which give rise to high transverse momentum pi^0 mesons are studied with the H1 experiment at HERA. The inclusive cross section for pi^0 mesons produced at small angles with respect to the proton remnant (the forward region) is presented as a function of the transverse momentum and energy of the pi^0 and of the four-momentum transfer Q^2 and Bjorken-x. Measurements are also presented of the transverse energy flow in events containing a forward pi^0 meson. Hadronic final state calculations based on QCD models implementing different parton evolution schemes are confronted with the data.Comment: 27 pages, 8 figures and 3 table

Search for Doubly-Charged Higgs Boson Production at HERA

A search for the single production of doubly-charged Higgs bosons H^{\pm \pm} in ep collisions is presented. The signal is searched for via the Higgs decays into a high mass pair of same charge leptons, one of them being an electron. The analysis uses up to 118 pb^{-1} of ep data collected by the H1 experiment at HERA. No evidence for doubly-charged Higgs production is observed and mass dependent upper limits are derived on the Yukawa couplings h_{el} of the Higgs boson to an electron-lepton pair. Assuming that the doubly-charged Higgs only decays into an electron and a muon via a coupling of electromagnetic strength h_{e \mu} = \sqrt{4 \pi \alpha_{em}} = 0.3, a lower limit of 141 GeV on the H^{\pm\pm} mass is obtained at the 95% confidence level. For a doubly-charged Higgs decaying only into an electron and a tau and a coupling h_{e\tau} = 0.3, masses below 112 GeV are ruled out.Comment: 15 pages, 3 figures, 1 tabl

Low Q^2 Jet Production at HERA and Virtual Photon Structure

The transition between photoproduction and deep-inelastic scattering is investigated in jet production at the HERA ep collider, using data collected by the H1 experiment. Measurements of the differential inclusive jet cross-sections dsigep/dEt* and dsigmep/deta*, where Et* and eta* are the transverse energy and the pseudorapidity of the jets in the virtual photon-proton centre of mass frame, are presented for 0 < Q2 < 49 GeV2 and 0.3 < y < 0.6. The interpretation of the results in terms of the structure of the virtual photon is discussed. The data are best described by QCD calculations which include a partonic structure of the virtual photon that evolves with Q2.Comment: 20 pages, 5 Figure

Hadron Production in Diffractive Deep-Inelastic Scattering

Characteristics of hadron production in diffractive deep-inelastic positron-proton scattering are studied using data collected in 1994 by the H1 experiment at HERA. The following distributions are measured in the centre-of-mass frame of the photon dissociation system: the hadronic energy flow, the Feynman-x (x_F) variable for charged particles, the squared transverse momentum of charged particles (p_T^{*2}), and the mean p_T^{*2} as a function of x_F. These distributions are compared with results in the gamma^* p centre-of-mass frame from inclusive deep-inelastic scattering in the fixed-target experiment EMC, and also with the predictions of several Monte Carlo calculations. The data are consistent with a picture in which the partonic structure of the diffractive exchange is dominated at low Q^2 by hard gluons.Comment: 16 pages, 6 figures, submitted to Phys. Lett.

Measurement of D* Meson Cross Sections at HERA and Determination of the Gluon Density in the Proton using NLO QCD

With the H1 detector at the ep collider HERA, D* meson production cross sections have been measured in deep inelastic scattering with four-momentum transfers Q^2>2 GeV2 and in photoproduction at energies around W(gamma p)~ 88 GeV and 194 GeV. Next-to-Leading Order QCD calculations are found to describe the differential cross sections within theoretical and experimental uncertainties. Using these calculations, the NLO gluon momentum distribution in the proton, x_g g(x_g), has been extracted in the momentum fraction range 7.5x10^{-4}< x_g <4x10^{-2} at average scales mu^2 =25 to 50 GeV2. The gluon momentum fraction x_g has been obtained from the measured kinematics of the scattered electron and the D* meson in the final state. The results compare well with the gluon distribution obtained from the analysis of scaling violations of the proton structure function F_2.Comment: 27 pages, 9 figures, 2 tables, submitted to Nucl. Phys.

Measurement of Leading Proton and Neutron Production in Deep Inelastic Scattering at HERA

Deep--inelastic scattering events with a leading baryon have been detected by the H1 experiment at HERA using a forward proton spectrometer and a forward neutron calorimeter. Semi--inclusive cross sections have been measured in the kinematic region 2 <= Q^2 <= 50 GeV^2, 6.10^-5 <= x <= 6.10^-3 and baryon p_T <= MeV, for events with a final state proton with energy 580 <= E' <= 740 GeV, or a neutron with energy E' >= 160 GeV. The measurements are used to test production models and factorization hypotheses. A Regge model of leading baryon production which consists of pion, pomeron and secondary reggeon exchanges gives an acceptable description of both semi-inclusive cross sections in the region 0.7 <= E'/E_p <= 0.9, where E_p is the proton beam energy. The leading neutron data are used to estimate for the first time the structure function of the pion at small Bjorken--x.Comment: 30 pages, 9 figures, 2 tables, submitted to Eur. Phys.

Search for direct production of charginos and neutralinos in events with three leptons and missing transverse momentum in √s = 7 TeV pp collisions with the ATLAS detector

A search for the direct production of charginos and neutralinos in final states with three electrons or muons and missing transverse momentum is presented. The analysis is based on 4.7 fb−1 of proton–proton collision data delivered by the Large Hadron Collider and recorded with the ATLAS detector. Observations are consistent with Standard Model expectations in three signal regions that are either depleted or enriched in Z-boson decays. Upper limits at 95% confidence level are set in R-parity conserving phenomenological minimal supersymmetric models and in simplified models, significantly extending previous results