72 research outputs found
An Extension of Heron's Formula to Tetrahedra, and the Projective Nature of Its Zeros
A natural extension of Heron's 2000 year old formula for the area of a
triangle to the volume of a tetrahedron is presented. This gives the fourth
power of the volume as a polynomial in six simple rational functions of the
areas of its four faces and three medial parallelograms, which will be referred
to herein as "interior faces." Geometrically, these rational functions are the
areas of the triangles into which the exterior faces are divided by the points
at which the tetrahedron's in-sphere touches those faces. This leads to a
conjecture as to how the formula extends to -dimensional simplices for all
. Remarkably, for the zeros of the polynomial constitute a
five-dimensional semi-algebraic variety consisting almost entirely of collinear
tetrahedra with vertices separated by infinite distances, but with generically
well-defined distance ratios. These unconventional Euclidean configurations can
be identified with a quotient of the Klein quadric by an action of a group of
reflections isomorphic to , wherein four-point configurations in
the affine plane constitute a distinguished three-dimensional subset. The paper
closes by noting that the algebraic structure of the zeros in the affine plane
naturally defines the associated four-element, rank chirotope, aka affine
oriented matroid.Comment: 51 pages, 6 sections, 5 appendices, 7 figures, 2 tables, 81
references; v7 clarifies the definitions made in the text leading up to
Theorem 5.4, along with the usual miscellaneous minor corrections and
improvement
A Bloch-Sphere-Type Model for Two Qubits in the Geometric Algebra of a 6-D Euclidean Vector Space
Geometric algebra is a mathematical structure that is inherent in any metric
vector space, and defined by the requirement that the metric tensor is given by
the scalar part of the product of vectors. It provides a natural framework in
which to represent the classical groups as subgroups of rotation groups, and
similarly their Lie algebras. In this article we show how the geometric algebra
of a six-dimensional real Euclidean vector space naturally allows one to
construct the special unitary group on a two-qubit (quantum bit) Hilbert space,
in a fashion similar to that used in the well-established Bloch sphere model
for a single qubit. This is then used to illustrate the Cartan decompositions
and subalgebras of the four-dimensional special unitary group, which have
recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev.
A 67, 042313, 2003] to study the entangling capabilities of two-qubit
unitaries.Comment: 14 pages, 2 figures, in press (Proceedings of SPIE Conference on
Defense & Security
Procedures for Converting among Lindblad, Kraus and Matrix Representations of Quantum Dynamical Semigroups
Given an quantum dynamical semigroup expressed as an exponential
superoperator acting on a space of N-dimensional density operators, eigenvalue
methods are presented by which canonical Kraus and Lindblad operator sum
representations can be computed. These methods provide a mathematical basis on
which to develop novel algorithms for quantum process tomography, the
statistical estimation of superoperators and their generators, from a wide
variety of experimental data. Theoretical arguments and numerical simulations
are presented which imply that these algorithms will be quite robust in the
presence of random errors in the data.Comment: RevTeX4, 31 pages, no figures; v4 adds new introduction and a
numerical example illustrating the application of these results to Quantum
Process Tomograph
Reflection Symmetries for Multiqubit Density Operators
For multiqubit density operators in a suitable tensorial basis, we show that
a number of nonunitary operations used in the detection and synthesis of
entanglement are classifiable as reflection symmetries, i.e., orientation
changing rotations. While one-qubit reflections correspond to antiunitary
symmetries, as is known for example from the partial transposition criterion,
reflections on the joint density of two or more qubits are not accounted for by
the Wigner Theorem and are well-posed only for sufficiently mixed states. One
example of such nonlocal reflections is the unconditional NOT operation on a
multiparty density, i.e., an operation yelding another density and such that
the sum of the two is the identity operator. This nonphysical operation is
admissible only for sufficiently mixed states.Comment: 9 page
Expressing the operations of quantum computing in multiparticle geometric algebra
We show how the basic operations of quantum computing can be expressed and
manipulated in a clear and concise fashion using a multiparticle version of
geometric (aka Clifford) algebra. This algebra encompasses the product operator
formalism of NMR spectroscopy, and hence its notation leads directly to
implementations of these operations via NMR pulse sequences.Comment: RevTeX, 10 pages, no figures; Physics Letters A, in pres
A new method for building protein conformations from sequence alignments with homologues of known structure
We describe a largely automatic procedure for building protein structures from sequence alignments with homologues of known structure. This procedure uses simple rules by which multiple sequence alignments can be translated into distance and chirality constraints, which are then used as input for distance geometry calculations. By this means one obtains an ensemble of conformations for the unknown structure that are compatible with the rules employed, and the differences among these conformations provide an indication of the reliability of the structure prediction. The overall approach is demonstrated here by applying it to several Kazal-type trypsin inhibitors, for which experimentally determined structures are available. On the basis of our experience with these test problems, we have further predicted the conformation of the human pancreatic secretory trypsin inhibitor, for which no experimentally determined structure is presently available.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29506/1/0000593.pd
Quantum Process Tomography of the Quantum Fourier Transform
The results of quantum process tomography on a three-qubit nuclear magnetic
resonance quantum information processor are presented, and shown to be
consistent with a detailed model of the system-plus-apparatus used for the
experiments. The quantum operation studied was the quantum Fourier transform,
which is important in several quantum algorithms and poses a rigorous test for
the precision of our recently-developed strongly modulating control fields. The
results were analyzed in an attempt to decompose the implementation errors into
coherent (overall systematic), incoherent (microscopically deterministic), and
decoherent (microscopically random) components. This analysis yielded a
superoperator consisting of a unitary part that was strongly correlated with
the theoretically expected unitary superoperator of the quantum Fourier
transform, an overall attenuation consistent with decoherence, and a residual
portion that was not completely positive - although complete positivity is
required for any quantum operation. By comparison with the results of computer
simulations, the lack of complete positivity was shown to be largely a
consequence of the incoherent errors during the quantum process tomography
procedure. These simulations further showed that coherent, incoherent, and
decoherent errors can often be identified by their distinctive effects on the
spectrum of the overall superoperator. The gate fidelity of the experimentally
determined superoperator was 0.64, while the correlation coefficient between
experimentally determined superoperator and the simulated superoperator was
0.79; most of the discrepancies with the simulations could be explained by the
cummulative effect of small errors in the single qubit gates.Comment: 26 pages, 17 figures, four tables; in press, Journal of Chemical
Physic
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