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.

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

A priori probability that a qubit-qutrit pair is separable

We extend to arbitrarily coupled pairs of qubits (two-state quantum systems) and qutrits (three-state quantum systems) our earlier study (quant-ph/0207181), which was concerned with the simplest instance of entangled quantum systems, pairs of qubits. As in that analysis -- again on the basis of numerical (quasi-Monte Carlo) integration results, but now in a still higher-dimensional space (35-d vs. 15-d) -- we examine a conjecture that the Bures/SD (statistical distinguishability) probability that arbitrarily paired qubits and qutrits are separable (unentangled) has a simple exact value, u/(v Pi^3)= >.00124706, where u = 2^20 3^3 5 7 and v = 19 23 29 31 37 41 43 (the product of consecutive primes). This is considerably less than the conjectured value of the Bures/SD probability, 8/(11 Pi^2) = 0736881, in the qubit-qubit case. Both of these conjectures, in turn, rely upon ones to the effect that the SD volumes of separable states assume certain remarkable forms, involving "primorial" numbers. We also estimate the SD area of the boundary of separable qubit-qutrit states, and provide preliminary calculations of the Bures/SD probability of separability in the general qubit-qubit-qubit and qutrit-qutrit cases.Comment: 9 pages, 3 figures, 2 tables, LaTeX, we utilize recent exact computations of Sommers and Zyczkowski (quant-ph/0304041) of "the Bures volume of mixed quantum states" to refine our conjecture

Spectroradiometric calibration of the thematic mapper and multispectral scanner system

The results of an analysis that relates thematic mapper (TM) saturation level to ground reflectance, calendar date, latitude, and atmospheric condition is provided. A revised version of the preprint included with the last quarterly report is also provided for publication in the IEEE Transactions on Geoscience and Remote Sensing

Spectroradiometric calibration of the Thematic Mapper and Multispectral Scanner system

The effects of the atmosphere on propagating radiation must be known in order to calibrate an in orbit sensor using ground based measurements. A set of model atmosphere parameters, applicable to the White Sands (New Mexico) area is defined with particular attention given to those parameters which are required as input to the Herman Code. The radial size distribution, refractive index, vertical distribution, and visibility of aerosols are discussed as well as the molecular absorbers in the visible and near IR wavelength which produce strong absorption lines. Solar irradiance is also considered

Spectroradiometric calibration of the Thematic Mapper and Multispectral Scanner system

The results of analyses of Thematic Mapper (TM) images acquired on July 8 and October 28, 1984, and of a check of the calibration of the 1.22-m integrating sphere at Santa Barbara Research Center (SBRC) are described. The results obtained from the in-flight calibration attempts disagree with the pre-flight calibrations for bands 2 and 4. Considerable effort was expended in an attempt to explain the disagreement. The difficult point to explain is that the difference between the radiances predicted by the radiative transfer code (the code radiances) and the radiances predicted by the preflight calibration (the pre-flight radiances) fluctuate with spectral band. Because the spectral quantities measured at White Sands show little change with spectral band, these fluctuations are not anticipated. Analyses of other targets at White Sands such as clouds, cloud shadows, and water surfaces tend to support the pre-flight and internal calibrator calibrations. The source of the disagreement has not been identified. It could be due to: (1) a computational error in the data reduction; (2) an incorrect assumption in the input to the radiative transfer code; or (3) incorrect operation of the field equipment

Spectroradiometric calibration of the thematic mapper and multispectral scanner system

The results obtained for the absolute calibration of TM bands 2, 3, and 4 are presented. The results are based on TM image data collected simultaneously with ground and atmospheric data at White Sands, New Mexico. Also discussed are the results of a moments analysis to determine the equivalent bandpasses, effective central wavelengths and normalized responses of the TM and MSS spectral bands; the calibration of the BaSO, plate used at White Sands; and future plans

Spectroradiometric calibration of the Thematic Mapper and Multispectral Scanner system

The newly built Caste spectropolarimeters gave satisfactory performance during tests in the solar radiometer and helicopter modes. A bandwidth normalization technique based on analysis of the moments of the spectral responsivity curves was used to analyze the spectral bands of the MSS and TM subsystems of LANDSAT 4 and 5 satellites. Results include the effective wavelength, the bandpass, the wavelength limits, and the normalized responsivity for each spectral channel. Temperature coefficients for TM PF channel 6 were also derived. The moments normalization method used yields sensor parameters whose derivation is independent of source characteristics (i.e., incident solar spectral irradiance, atmospheric transmittance, or ground reflectance). The errors expected using these parameters are lower than those expected using other normalization methods

Spectroradiometric calibration of the Thematic Mapper and Multispectral Scanner system

The reduction of the data measured on July 8, 1984 at White Sands, New Mexico is summarized. The radiance incident at the entrance pupil of the LANDSAT 5 sensors have been computed for bands 1 to 4. When these are compared to the digital counts of the TM image, the ground based calibration for this sensor will be given. The image was received from Goddard SFC and is presently being analyzed

Spectroradiometric calibration of the Thematic Mapper and multispectral scanner system

The eleventh quarterly report on Spectroradiometric Calibration of the Thematic Mapper (Contract NAS5-27832) discusses calibrations made at White Sands on 24 May 1985. An attempt is made to standardize test results. Critical values used in the final steps of the data reduction and the comparison of the results of the pre-flight and internal calibration (IC) data are summarized