Christopher J. Foster

Alternative fidelity measure between quantum states

We propose an alternative fidelity measure (namely, a measure of the degree of similarity) between quantum states and benchmark it against a number of properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple function of the linear entropy and the Hilbert-Schmidt inner product between the given states and is thus, in comparison, not as computationally demanding. It also features several remarkable properties such as being jointly concave and satisfying all of Jozsas axioms. The trade-off, however, is that it is supermultiplicative and does not behave monotonically under quantum operations. In addition, metrics for the space of density matrices are identified and the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is established.

Remote sensing the magnetosphere using ground-based observations of ULF waves

Ground-based magnetometers record the signature of ultra-low frequency (ULF) wave energy incident from the magnetosphere. A portion of the signal comes from resonant ULF wave structures that form between the northern and southern ionospheres, known as field line resonances (FLRs) in the near-Earth space plasma. It is well known that the resonant frequency depends on the length, the strength, and the distribution of plasma mass along the geomagnetic field from one ionosphere to the other. Given an accurate description of the geomagnetic field in space, the ULF resonant frequencies may be used to remote sense the plasma mass density in the magnetosphere. Identifying the continuum or more directly driven resonant frequencies is the key to the remote sensing process. Techniques involving amplitude, phase, and single and multi-instrument data are discussed. The procedures developed for identifying FLRs at various locations on the Earths surface have traditionally been accomplished using magnetometers but may also be applied to other ground-based instrumentation such as high-frequency radars, Doppler sounders and meridian scanning photometers. Data analysis methods are reviewed and the development of new analysis methods is encouraged by providing code to compute the Wavelet transform. Remote sensed estimates of plasma mass density are compared with spacecraft data, leading to a discussion of accuracy and further refinements.

Bell Inequalities for Continuous-Variable Correlations

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at least second moments. The derivation stems from a simple variance inequality by setting local commutators to zero. We show that above a constant detector efficiency threshold, the continuous-variable Bell violation can survive even in the macroscopic limit of large n. This method can be used to derive other well-known Bell inequalities, shedding new light on the importance of non-commutativity for violations of local realism.

Analog of the Clauser–Horne–Shimony–Holt inequality for steering

The Clauser–Horne–Shimony–Holt (CHSH) inequality and its permutations are necessary and sufficient criteria for Bell nonlocality in the simplest Bell-nonlocality scenario: two parties, two measurements per party and two outcomes per measurement. Here we derive an inequality for Einstein–Podolsky–Rosen (EPR)-steering that is an analog of the CHSH, in that it is necessary and sufficient in this same scenario. However, since in the case of steering the device at Bob’s site must be specified (as opposed to the Bell case, in which it is a black box), the scenario we consider is that where Alice performs two (black-box) dichotomic measurements, and Bob performs two mutually unbiased qubit measurements. We show that this inequality is strictly weaker than the CHSH, as expected, and use it to decide whether a recent experiment [Phys. Rev. Lett.110, 130401 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.130401] involving a single-photon split between two parties has demonstrated EPR-steering.

Vortex pairing in two-dimensional Bose gases

Recent experiments on ultracold Bose gases in two dimensions have provided evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase via analysis of the interference between two independent systems. In this work we study the two-dimensional quantum degenerate Bose gas at finite temperature using the projected Gross-Pitaevskii equation classical field method. Although this describes the highly occupied modes of the gas below a momentum cutoff, we have developed a method to incorporate the higher momentum states in our model. We concentrate on finite-sized homogeneous systems in order to simplify the analysis of the vortex pairing. We determine the dependence of the condensate fraction on temperature and compare this to the calculated superfluid fraction. By measuring the first order correlation function we determine the boundary of the Bose-Einstein condensate and BKT phases, and find it is consistent with the superfluid fraction decreasing to zero. We reveal the characteristic unbinding of vortex pairs above the BKT transition via a coarse-graining procedure. Finally, we model the procedure used in experiments to infer system correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its level of agreement with directly calculated in situ correlation functions.

Superfluidity and anomalous correlations in a two-dimensional Bose gas

We show that the onset of superfluidity in a finite homogeneous 2D Bose gas is associated with the appearance of an anomalous thermal density induced by the presence of a finite-size condensate. We investigate the temporal frequency signatures of the Bose field, and show that thermally excited vortices in the field are weakly coupled to the longitudinal excitations. We apply the concept of locking of the superfluid velocity field by the condensate to estimate the superfluid density in a metastable circulating flow of a superfluid Bose gas in a 2D toroidal geometry.