On observability of Renyi's entropy

Despite recent claims we argue that Renyi's entropy is an observable quantity. It is shown that, contrary to popular belief, the reported domain of instability for Renyi entropies has zero measure (Bhattacharyya measure). In addition, we show the instabilities can be easily emended by introducing a coarse graining into an actual measurement. We also clear up doubts regarding the observability of Renyi's entropy in (multi--)fractal systems and in systems with absolutely continuous PDF's.Comment: 18 pages, 1 EPS figure, REVTeX, minor changes, accepted to Phys. Rev.

Lower bound of minimal time evolution in quantum mechanics

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S^2, must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2\times 2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector \bf P of the evolving quantum state \rho={1/2}(\bf 1+ \bf{P}\cdot\boldsymbol{\sigma}) with the vector \boldsymbol{\mathcal O}(\Theta) of the 2\times 2 hermitian Hamiltonians H ={1/2}({\mathcal O}_0\boldsymbol{1}+ \boldsymbol{\mathcal O}(\Theta)\cdot\boldsymbol{\sigma}) and it is shown that the Hamiltonian H can be parameterized by two independent parameters {\mathcal O}_0 and \Theta.Comment: 4 pages, no figure, revtex

Monte-Carlo sampling of energy-constrained quantum superpositions in high-dimensional Hilbert spaces

Recent studies into the properties of quantum statistical ensembles in high-dimensional Hilbert spaces have encountered difficulties associated with the Monte-Carlo sampling of quantum superpositions constrained by the energy expectation value. A straightforward Monte-Carlo routine would enclose the energy constrained manifold within a larger manifold, which is easy to sample, for example, a hypercube. The efficiency of such a sampling routine decreases exponentially with the increase of the dimension of the Hilbert space, because the volume of the enclosing manifold becomes exponentially larger than the volume of the manifold of interest. The present paper explores the ways to optimise the above routine by varying the shapes of the manifolds enclosing the energy-constrained manifold. The resulting improvement in the sampling efficiency is about a factor of five for a 14-dimensional Hilbert space. The advantage of the above algorithm is that it does not compromise on the rigorous statistical nature of the sampling outcome and hence can be used to test other more sophisticated Monte-Carlo routines. The present attempts to optimise the enclosing manifolds also bring insights into the geometrical properties of the energy constrained manifold itself.Comment: 9 pages, 7 figures, accepted for publication in European Physical Journal

Inelastic Neutron Scattering from the Spin Ladder Compound (VO)2P2O7

We present results from an inelastic neutron scattering experiment on the candidate Heisenberg spin ladder vanadyl pyrophosphate, (VO)2P2O7. We find evidence for a spin-wave excitation gap of $E_{gap} = 3.7\pm 0.2$ meV, at a band minimum near $Q=0.8 A^{-1}$. This is consistent with expectations for triplet spin waves in (VO)2P2O7 in the spin-ladder model, and is to our knowledge the first confirmation in nature of a Heisenberg antiferromagnetic spin ladder.Comment: 11 pages and 2 figures (available as hard copy or postscript files from the authors, send request to [email protected] or [email protected]), TEX using jnl, reforder and eqnorder, ORNL-CCIP-94-05 / RAL-94-04

Electronic excitations and the tunneling spectra of metallic nanograins

Tunneling-induced electronic excitations in a metallic nanograin are classified in terms of {\em generations}: subspaces of excitations containing a specific number of electron-hole pairs. This yields a hierarchy of populated excited states of the nanograin that strongly depends on (a) the available electronic energy levels; and (b) the ratio between the electronic relaxation rate within the nano-grain and the bottleneck rate for tunneling transitions. To study the response of the electronic energy level structure of the nanograin to the excitations, and its signature in the tunneling spectrum, we propose a microscopic mean-field theory. Two main features emerge when considering an Al nanograin coated with Al oxide: (i) The electronic energy response fluctuates strongly in the presence of disorder, from level to level and excitation to excitation. Such fluctuations produce a dramatic sample dependence of the tunneling spectra. On the other hand, for excitations that are energetically accessible at low applied bias voltages, the magnitude of the response, reflected in the renormalization of the single-electron energy levels, is smaller than the average spacing between energy levels. (ii) If the tunneling and electronic relaxation time scales are such as to admit a significant non-equilibrium population of the excited nanoparticle states, it should be possible to realize much higher spectral densities of resonances than have been observed to date in such devices. These resonances arise from tunneling into ground-state and excited electronic energy levels, as well as from charge fluctuations present during tunneling.Comment: Submitted to the Physical Review

Background Independent Quantum Mechanics and Gravity

We argue that the demand of background independence in a quantum theory of gravity calls for an extension of standard geometric quantum mechanics. We discuss a possible kinematical and dynamical generalization of the latter by way of a quantum covariance of the state space. Specifically, we apply our scheme to the problem of a background independent formulation of Matrix Theory.Comment: 9 pages, LaTe

Geometrothermodynamics of black holes

The thermodynamics of black holes is reformulated within the context of the recently developed formalism of geometrothermodynamics. This reformulation is shown to be invariant with respect to Legendre transformations, and to allow several equivalent representations. Legendre invariance allows us to explain a series of contradictory results known in the literature from the use of Weinhold's and Ruppeiner's thermodynamic metrics for black holes. For the Reissner-Nordstr\"om black hole the geometry of the space of equilibrium states is curved, showing a non trivial thermodynamic interaction, and the curvature contains information about critical points and phase transitions. On the contrary, for the Kerr black hole the geometry is flat and does not explain its phase transition structure.Comment: Revised version, to be published in Gen.Rel.Grav.(Mashhoon's Festschrift

Classification of multipartite entangled states by multidimensional determinants

We find that multidimensional determinants "hyperdeterminants", related to entanglement measures (the so-called concurrence or 3-tangle for the 2 or 3 qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the $n \geq 4$ qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.Comment: revtex4, 10 pages, 4 eps figures with psfrag; v2 title changed, 1 appendix added, to appear in Phys. Rev.

Renormalization of composite operators

The blocked composite operators are defined in the one-component Euclidean scalar field theory, and shown to generate a linear transformation of the operators, the operator mixing. This transformation allows us to introduce the parallel transport of the operators along the RG trajectory. The connection on this one-dimensional manifold governs the scale evolution of the operator mixing. It is shown that the solution of the eigenvalue problem of the connection gives the various scaling regimes and the relevant operators there. The relation to perturbative renormalization is also discussed in the framework of the $\phi^3$ theory in dimension $d=6$.Comment: 24 pages, revtex (accepted by Phys. Rev. D), changes in introduction and summar

Geometric Strategy for the Optimal Quantum Search

We explore quantum search from the geometric viewpoint of a complex projective space $CP$, a space of rays. First, we show that the optimal quantum search can be geometrically identified with the shortest path along the geodesic joining a target state, an element of the computational basis, and such an initial state as overlaps equally, up to phases, with all the elements of the computational basis. Second, we calculate the entanglement through the algorithm for any number of qubits $n$ as the minimum Fubini-Study distance to the submanifold formed by separable states in Segre embedding, and find that entanglement is used almost maximally for large $n$. The computational time seems to be optimized by the dynamics as the geodesic, running across entangled states away from the submanifold of separable states, rather than the amount of entanglement itself.Comment: revtex, 10 pages, 7 eps figures, uses psfrag packag