Advances in delimiting the Hilbert-Schmidt separability probability of real two-qubit systems

We seek to derive the probability--expressed in terms of the Hilbert-Schmidt (Euclidean or flat) metric--that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres-Horodecki test on the partial transposes (PT's) of the associated 4 x 4 density matrices). But the full implementation of the test--requiring that the determinant of the PT be nonnegative for separability to hold--appears to be, at least presently, computationally intractable. So, we have previously implemented--using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)--the weaker implied test of nonnegativity of the six 2 x 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of 1024/{135 pi^2} =0.76854$. Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 x 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of 22/35 = 0.628571. Then, we conclude that a still further improved upper bound of 1129/2100 = 0.537619 can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 x 3 principal minors. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESF's that comes close to reproducing our numerical estimate of the true separability function.Comment: 16 pages, 9 figures, a few inadvertent misstatements made near the end are correcte

Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence

The generic real (b=1) and complex (b=2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert-Schmidt and Bures metrics are obtainable as special cases of formulas of Zyczkowski and Sommers. We claim that if one could determine certain metric-independent 3-dimensional "eigenvalue-parameterized separability functions" (EPSFs), then these formulas could be readily modified so as to yield the Hilbert-Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities). Motivated by analogous earlier analyses of "diagonal-entry-parameterized separability functions", we further explore the possibility that such 3-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable--which we hypothesize to be the maximal concurrence (0 < C <1) over spectral orbits. Extensive numerical results we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at C=1/2, as well as a number of additional matching discontinuities.Comment: 12 pages, 7 figures, new abstract, revised for J. Phys.

d1005+68: A New Faint Dwarf Galaxy in the M81 Group

We present the discovery of d1005+68, a new faint dwarf galaxy in the M81 Group, using observations taken with the Subaru Hyper Suprime-Cam. d1005+68's color-magnitude diagram is consistent with a distance of $3.98_{-0.43}^{+0.39}$ Mpc, establishing group membership. We derive an absolute $V$-band magnitude, from stellar isochrone fitting, of $M_{V} = -7.94_{-0.50}^{+0.38}$, with a half-light radius of $r_{h} = 188_{-41}^{+39}$ pc. These place d1005+68 within the radius-luminosity locus of Local Group and M81 satellites and among the faintest confirmed satellites outside the Local Group. Assuming an age of 12 Gyr, d1005+68's red giant branch is best fit by an isochrone of [Fe/H] $= -1.90 \pm 0.24$. It has a projected separation from nearby M81 satellite BK5N of only 5 kpc. As this is well within BK5N's virial radius, we speculate that d1005+68 may be a satellite of BK5N. If confirmed, this would make d1005+68 one of the first detected satellites-of-a-satellite.Comment: 7 pages, 4 figures, 1 table, additional affiliations include

Andromeda XXIX: A New Dwarf Spheroidal Galaxy 200 kpc from Andromeda

We report the discovery of a new dwarf galaxy, Andromeda XXIX (And XXIX), using data from the recently released Sloan Digital Sky Survey Data Release 8, and confirmed by Gemini North telescope Multi-Object Spectrograph imaging data. And XXIX appears to be a dwarf spheroidal galaxy, separated on the sky by a little more than 15° from M31, with a distance inferred from the tip of the red giant branch of 730 ± 75 kpc, corresponding to a three-dimensional separation from M31 of 207 +20 – 2 kpc (close to M31's virial radius). Its absolute magnitude, as determined by comparison to the red giant branch luminosity function of the Draco dwarf spheroidal, is M V = –8.3 ± 0.4. And XXIX's stellar populations appear very similar to Draco's; consequently, we estimate a metallicity for And XXIX of [Fe/H] ~–1.8. The half-light radius of And XXIX is 360 ± 60 pc and its ellipticity is 0.35 ± 0.06, typical of dwarf satellites of the Milky Way and M31 at this absolute magnitude range.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90744/1/2041-8205_742_1_L15.pd

Andromeda XXVIII: A Dwarf Galaxy More Than 350 kpc from Andromeda

We report the discovery of a new dwarf galaxy, Andromeda XXVIII, using data from the recently-released SDSS DR8. The galaxy is a likely satellite of Andromeda, and, at a separation of $365^{+17}_{-1}$ kpc, would be one of the most distant of Andromeda's satellites. Its heliocentric distance is $650^{+150}_{-80}$ kpc, and analysis of its structure and luminosity show that it has an absolute magnitude of $M_V = -8.5^{+0.4}_{-1.0}$ and half-light radius of $r_h = 210^{+60}_{-50}$ pc, similar to many other faint Local Group dwarfs. With presently-available imaging we are unable to determine if there is ongoing or recent star formation, which prevents us from classifying it as a dwarf spheroidal or dwarf irregular.Comment: Accepted to ApJ Letter

Price Discovery and the Accuracy of Consolidated Data Feeds in the U.S. Equity Markets

Both the scientific community and the popular press have paid much attention to the speed of the Securities Information Processor, the data feed consolidating all trades and quotes across the US stock market. Rather than the speed of the Securities Information Processor, or SIP, we focus here on its accuracy. Relying on Trade and Quote data, we provide various measures of SIP latency relative to high-speed data feeds between exchanges, known as direct feeds. We use first differences to highlight not only the divergence between the direct feeds and the SIP, but also the fundamental inaccuracy of the SIP. We find that as many as 60 percent or more of trades are reported out of sequence for stocks with high trade volume, therefore skewing simple measures such as returns. While not yet definitive, this analysis supports our preliminary conclusion that the underlying infrastructure of the SIP is currently unable to keep pace with the trading activity in today's stock market.Comment: 18 pages, 20 figures, 2 table

Theory for Magnetic Anisotropy of Field-Induced Insulator-to-Metal Transition in Cubic Kondo Insulator YbB_{12}

Magnetization and energy gap of Kondo insulator YbB_{12} are calculated theoretically based on the previously proposed tight-binding model composed of Yb 5d$\epsilon$ and 4f $\Gamma_8$ orbitals. It is found that magnetization curves are almost isotropic, naturally expected from the cubic symmetry, but that the gap-closing field has an anisotropy: the gap closes faster for the field in (100) direction than in (110) and (111) directions, in accord with the experiments. This is qualitatively understood by considering the maximal eigenvalues of the total angular momentum operators projected on each direction of the magnetic field. But the numerical calculation based on the band model yields better agreement with the experiment.Comment: 4 pages, 4 figures, to appear in J. Phys. Soc. Jp