9,939 research outputs found
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
High-Temperature Expansions of Bures and Fisher Information Priors
For certain infinite and finite-dimensional thermal systems, we obtain ---
incorporating quantum-theoretic considerations into Bayesian thermostatistical
investigations of Lavenda --- high-temperature expansions of priors over
inverse temperature beta induced by volume elements ("quantum Jeffreys' priors)
of Bures metrics. Similarly to Lavenda's results based on volume elements
(Jeffreys' priors) of (classical) Fisher information metrics, we find that in
the limit beta -> 0, the quantum-theoretic priors either conform to Jeffreys'
rule for variables over [0,infinity], by being proportional to 1/beta, or to
the Bayes-Laplace principle of insufficient reason, by being constant. Whether
a system adheres to one rule or to the other appears to depend upon its number
of degrees of freedom.Comment: Six pages, LaTeX. The title has been shortened (and then further
modified), at the suggestion of a colleague. Other minor change
Bures distance between two displaced thermal states
The Bures distance between two displaced thermal states and the corresponding
geometric quantities (statistical metric, volume element, scalar curvature) are
computed. Under nonunitary (dissipative) dynamics, the statistical distance
shows the same general features previously reported in the literature by
Braunstein and Milburn for two--state systems. The scalar curvature turns out
to have new interesting properties when compared to the curvature associated
with squeezed thermal states.Comment: 3 pages, RevTeX, no figure
Volume of the quantum mechanical state space
The volume of the quantum mechanical state space over -dimensional real,
complex and quaternionic Hilbert-spaces with respect to the canonical Euclidean
measure is computed, and explicit formulas are presented for the expected value
of the determinant in the general setting too. The case when the state space is
endowed with a monotone metric or a pull-back metric is considered too, we give
formulas to compute the volume of the state space with respect to the given
Riemannian metric. We present the volume of the space of qubits with respect to
various monotone metrics. It turns out that the volume of the space of qubits
can be infinite too. We characterize those monotone metrics which generates
infinite volume.Comment: 17 page
Structure and Response in the World Trade Network
We examine how the structure of the world trade network has been shaped by
globalization and recessions over the last 40 years. We show that by treating
the world trade network as an evolving system, theory predicts the trade
network is more sensitive to evolutionary shocks and recovers more slowly from
them now than it did 40 years ago, due to structural changes in the world trade
network induced by globalization. We also show that recession-induced change to
the world trade network leads to an \emph{increased} hierarchical structure of
the global trade network for a few years after the recession.Comment: 4 pages, 4 figures, to appear in Phys. Rev. Let
Traveling sealer for contoured table Patent
Sealing apparatus for joining two pieces of frangible material
Electric field driven donor-based charge qubits in semiconductors
We investigate theoretically donor-based charge qubit operation driven by
external electric fields. The basic physics of the problem is presented by
considering a single electron bound to a shallow-donor pair in GaAs: This
system is closely related to the homopolar molecular ion H_2^+. In the case of
Si, heteropolar configurations such as PSb^+ pairs are also considered. For
both homopolar and heteropolar pairs, the multivalley conduction band structure
of Si leads to short-period oscillations of the tunnel-coupling strength as a
function of the inter-donor relative position. However, for any fixed donor
configuration, the response of the bound electron to a uniform electric field
in Si is qualitatively very similar to the GaAs case, with no valley quantum
interference-related effects, leading to the conclusion that electric field
driven coherent manipulation of donor-based charge qubits is feasible in
semiconductors
Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities
Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number
of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n,
k,n=0,1,2,3,..., PT denoting partial transpose, for both generic
(9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit
(alpha=1) density matrices rho. The results are, then, incorporated by Dunkl
into a general formula (Appendix D6), parameterized by k, n and alpha, with the
case alpha=2, presumptively corresponding to generic (27-dimensional)
quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the
induced univariate moments (|rho| |rho^{PT}|)^n and |rho^{PT}|^n are inputted
into a Legendre-polynomial-based (least-squares) probability-distribution
reconstruction algorithm of Provost (Mathematica J., 9, 727 (2005)), yielding
alpha-specific separability probability estimates. Since, as the number of
inputted moments grows, estimates based on |rho| |rho^{PT}| strongly decrease,
while ones employing |rho^{PT}| strongly increase (and converge faster), the
gaps between upper and lower estimates diminish, yielding sharper and sharper
bounds. Remarkably, for alpha = 2, with the use of 2,325 moments, a
separability-probability lower-bound 0.999999987 as large as 26/323 = 0.0804954
is found. For alpha=1, based on 2,415 moments, a lower bound results that is
0.999997066 times as large as 8/33 = 0.242424, a (simpler still) fractional
value that had previously been conjectured (J. Phys. A, 40, 14279 (2007)).
Furthermore, for alpha = 1/2, employing 3,310 moments, the lower bound is
0.999955 times as large as 29/64 = 0.453125, a rational value previously
considered (J. Phys. A, 43, 195302 (2010)).Comment: 47 pages, 12 figures; slightly expanded and modified for journal
publication; this paper incorporates and greatly extends arXiv:1104.021
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