14,683 research outputs found

    Simplifying the spectral analysis of the volume operator

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    The volume operator plays a central role in both the kinematics and dynamics of canonical approaches to quantum gravity which are based on algebras of generalized Wilson loops. We introduce a method for simplifying its spectral analysis, for quantum states that can be realized on a cubic three-dimensional lattice. This involves a decomposition of Hilbert space into sectors transforming according to the irreducible representations of a subgroup of the cubic group. As an application, we determine the complete spectrum for a class of states with six-valent intersections.Comment: 19 pages, TeX, to be published in Nucl. Phys.

    Coherent spin manipulation in an exchange-only qubit

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    Initialization, manipulation, and measurement of a three-spin qubit are demonstrated using a few-electron triple quantum dot, where all operations can be driven by tuning the nearest-neighbor exchange interaction. Multiplexed reflectometry, applied to two nearby charge sensors, allows for qubit readout. Decoherence is found to be consistent with predictions based on gate voltage noise with a uniform power spectrum. The theory of the exchange-only qubit is developed and it is shown that initialization of only two spins suffices for operation. Requirements for full multi-qubit control using only exchange and electrostatic interactions are outlined.Comment: related work at http://marcuslab.harvard.ed

    Uniform Semiclassical Approximation for the Wigner 6j6j Symbol in Terms of Rotation Matrices

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    A new uniform asymptotic approximation for the Wigner 6j6j symbol is given in terms of Wigner rotation matrices (dd-matrices). The approximation is uniform in the sense that it applies for all values of the quantum numbers, even those near caustics. The derivation of the new approximation is not given, but the geometrical ideas supporting it are discussed and numerical tests are presented, including comparisons with the exact 6j6j-symbol and with the Ponzano-Regge approximation.Comment: 44 pages plus 20 figure

    The Quantum Group Structure of 2D Gravity and Minimal Models II: The Genus-Zero Chiral Bootstrap

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    The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with Liouville momenta going to \infty in a suitable way. The introduction of quantum-group-covariant operators in the three dimensional picture gives a generalisation of the quantum-group version of discrete three-dimensional gravity that includes tetrahedra associated with 3-j symbols and universal R-matrix elements. Altogether the present work gives the concrete realization of Moore and Seiberg's scheme that describes the chiral operator-algebra of two-dimensional gravity and minimal models.Comment: 56 pages, 22 figures. Technical problem only, due to the use of an old version of uuencode that produces blank characters some times suppressed by the mailer. Same content

    Racah Sum Rule and Biedenharn-Elliott Identity for the Super-Rotation 6j6-j symbols

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    It is shown that the well known Racah sum rule and Biedenharn-Elliott identity satisfied by the recoupling coefficients or by the 6j6-j symbols of the usual rotation SO(3)SO(3) algebra can be extended to the corresponding features of the super-rotation osp(12)osp(1|2) superalgebra. The structure of the sum rules is completely similar in both cases, the only difference concerns the signs which are more involved in the super-rotation case.Comment: 9 pages. Two misprints correcte

    Uniform Approximation from Symbol Calculus on a Spherical Phase Space

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    We use symbol correspondence and quantum normal form theory to develop a more general method for finding uniform asymptotic approximations. We then apply this method to derive a result we announced in an earlier paper, namely, the uniform approximation of the 6j6j-symbol in terms of the rotation matrices. The derivation is based on the Stratonovich-Weyl symbol correspondence between matrix operators and functions on a spherical phase space. The resulting approximation depends on a canonical, or area preserving, map between two pairs of intersecting level sets on the spherical phase space.Comment: 18 pages, 5 figure

    Trivalent graphs, volume conjectures and character varieties

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    The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to U_q(\fraksl_2) colored quantum invariants of the theta and tetrahedron graph. The \SL(2,\bC) character variety of the fundamental group of the complement of a trivalent graph with EE edges in S3S^3 is a Lagrangian subvariety of the Hitchin moduli space over the Riemann surface of genus g=E/3+1g=E/3+1. For the theta and tetrahedron graph, we conjecture that the configuration of the character variety is locally determined by large color asymptotics of the quantum invariants of the trivalent graph in terms of complex Fenchel-Nielsen coordinates. Moreover, the qq-holonomic difference equation of the quantum invariants provides the quantization of the character variety.Comment: 11 pages, 2 figure

    Simple operation sequences to couple and interchange quantum information between spin qubits of different kinds

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    Efficient operation sequences to couple and interchange quantum information between quantum dot spin qubits of different kinds are derived using exchange interactions. In the qubit encoding of a single-spin qubit, a singlet-triplet qubit, and an exchange-only (triple-dot) qubit, some of the single-qubit interactions remain on during the entangling operation; this greatly simplifies the operation sequences that construct entangling operations. In the ideal setup, the gate operations use the intra-qubit exchange interactions only once. The limitations of the entangling sequences are discussed, and it is shown how quantum information can be converted between different kinds of quantum dot spin qubits.Comment: 9 pages, 4 figure
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