A mathematical foundation of quantum information and quantum computer -on quantum mutual entropy and entanglement-

The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entangled state or quantum entanglement, are used to study quntum information, in part icular, quantum computation, quantum teleportation, quantum cryptography. In this paper, we mainly discuss three things below: (1) We point out the di fference between the capacity of quantum channel and that of classical-quant um-classical channel. (2) So far the entangled state is merely defined as a non-separable state, we give a wider definition of the entangled state and c lassify the entangled states into three categories. (3) The quantum mutual e ntropy for an entangled state is discussed. The above (2) and (3) are a join t work with Belavkin.Comment: 10 pages, Latex2

Numerical approach to SUSY quantum mechanics and the gauge/gravity duality

We demonstrate that Monte-Carlo simulation is a practical tool to study nonperturbative aspects of supersymmetric quantum mechanics. As an example we study D0-brane quantum mechanics in the context of superstring theory. Numerical data nicely reproduce predictions from gravity side, including the coupling constant dependence of the string alpha' correction. This strongly suggests the duality to hold beyond the supergravity approximation. Although detail of the stringy correction cannot be obtained by state-of-the-art techniques in gravity side, in the matrix quantum mechanics we can obtain concrete values. Therefore the Monte-Carlo simulation combined with the duality provides a powerful tool to study the superstring theory.Comment: 16 pages, 4 figures. Based on an invited talk at the workshop "Supersymmetric Quantum Mechanics and Spectral Design", Benasque, Spain, 2010, Jul 18 -- Jul 3

Relation between high-energy quasiparticles of quasi-one-dimensional antiferromagnets in a magnetic field and a doublon of a Hubbard chain

In spin-1/2 one-dimensional Heisenberg antiferromagnets and anisotropic triangular Heisenberg antiferromagnets, high-energy states carrying considerable spectral weights have been observed in a magnetic field using inelastic neutron scattering. Such high-energy properties cannot be explained in terms of either Nambu-Goldstone bosons due to spontaneous breaking of continuous symmetries or quasiparticles in a Tomonaga-Luttinger liquid. Here, we show that the mechanism causing the high-energy states is analogous to that of the upper Hubbard band in the one-dimensional Hubbard model, by theoretically tracing the origin of the high-energy states back to string solutions of the Bethe ansatz.Comment: 6 pages, 1 figur

On the ampleness of positive CR line bundles over Levi-flat manifolds

We give an example of a compact Levi-flat CR 3-manifold with a positive-along-leaves CR line bundle which is not ample with respect to transversely infinitely differentiable CR sections. This example shows that we cannot improve the regularity of Kodaira type embedding theorem for compact Levi-flat CR manifolds obtained by Ohsawa and Sibony.Comment: 12 pages, final version, to appear in Publications of the Research Institute for Mathematical Science

Fermi-LAT observations of GRBs with weak LAT emission

We present the analysis results of three Gamma-Ray Bursts (GRBs) detected by the Gamma-ray Burst Monitor (GBM) and the Large Area Telescope (LAT) onboard Fermi: the two long GRB 080825C and GRB 090217, and the first short burst with GeV photons GRB 081024B. The emission from GRB 081024B observed by the LAT above 100 MeV is delayed with respect to the GBM trigger, and significantly extends after the low-energy episode. Some hints for spectral hardening was observed in this burst as well as in GRB 080825C, possibly related to a separate and harder component showing up at late times. Conversely, GRB 090217 does not exhibit any noticeable feature. Together with the other bright LAT detected bursts (e.g. GRB 080916C and GRB 090510), these observations help to classify the GRB properties and give new insight on the acceleration mechanisms responsible for their emission at the highest energies.Comment: 2009 Fermi Symposium. eConf Proceedings C09112

Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals

We introduce the concept of index for regular Dirichlet forms by means of energy measures, and discuss its properties. In particular, it is proved that the index of strong local regular Dirichlet forms is identical with the martingale dimension of the associated diffusion processes. As an application, a class of self-similar fractals is taken up as an underlying space. We prove that first-order derivatives can be defined for functions in the domain of the Dirichlet forms and their total energies are represented as the square integrals of the derivatives.Comment: 33 pages, 3 figure

Large-Nc equivalence and the sign problem at finite baryon density

QCD with a finite baryon chemical potential, despite its importance, is not well understood because the standard lattice QCD simulation is not applicable due to the sign problem. Although QCD-like theories which do not suffer from the sign problem have been studied intensively, relation to QCD with a finite baryon chemical potential was not clear. This paper introduces large-$N_c$ equivalences between QCD and various QCD-like theories. These equivalences lead us to a unified viewpoint for QCD with baryon and isospin chemical potentials, $SO(2N_c)$ and $Sp(2N_c)$ gauge theories, QCD with adjoint matters and two-color QCD. In particular QCD with the baryon chemical potential is large-$N_c$ equivalent to its phase quenched version in a certain parameter region, which is relevant for heavy ion collision experiments. All previous simulation results which study the effect of the phase confirm the phase quench approximation is quantitatively good already at Nc=3; it is so good that often two theories give the same value within error. Therefore the phase quenched simulation is the best strategy for the QCD critical point search. At small volume one can study a tiny 1/Nc effect by the phase reweighting; the large-Nc equivalence guarantees that the phase reweighing method works without suffering from the overlapping problem.Comment: Slightly modified version of the article submitted to Seitaro Nakamura competition. 23 pages, 9 figures. v2: 25 pages, 9 figure

On dlog image of K_2 of elliptic surface minus singular fibers

Modular section addedComment: 60 page

Ground State Properties of the Two-Dimensional t-J Model

The two-dimensional $t$-$J$ model in the ground state is investigated by the power Lanczos method. The pairing-pairing correlation function for $d_{x^2-y^2}$-wave symmetry is enhanced in the realistic parameter regime for high-$T_c$ superconductors. The charge susceptibility $\chi_c$ shows divergent behavior as $\chi_c\propto\delta^{-1}$ near half-filling for the doping concentration $\delta$, indicating that the value of the dynamical exponent $z$ is four under the assumption of hyperscaling. The peak height of the spin structure factor $S_{max}(Q)$ also behaves as $S_{max}(Q) \propto\delta^{-1}$ near half-filling, which leads to the divergence of the antiferromagnetic correlation length $\xi_m$ as $\xi_m \propto\delta^{-1/2}$. The boundary of phase separation is estimated on the basis of the Maxwell construction. Numerical results are compared with experimental features observed in high-$T_c$ cuprates.Comment: 15 pages, RevTex, 12 PostScript figures, to appear in Phys.Rev.

Proposed experiment on the continuity of quantum entanglement

We propose experiments on quantum entanglement for investigating the Einstein Podolsky Rosen (EPR) problem with the polarization directions of photons. These experiments are performed to investigate whether the defined polarization directions in an entangled state are teleported between entangled photons. EPR-type sequential experiments are performed using a twin-photon beam and two pairs of linear polarization analyzers under the cross-Nicol condition (i.e., orthogonal to each other). If the third filter whose polarization angle is 45 degrees is set between the first cross-Nicol filters, the beam intensity is changed from 0 to 12.5 %, and at the second cross-Nicol filters, the beam intensity is changed from 0 to 25 %. In this experiment, we predict that the "continuity of quantum entanglement" under a pure Hamiltonian evolution is detected.Comment: 6 pages, 5 figure