Abstract:
We address the problem of finding optimal CPTP (completely positive, trace preserving) maps between a set of binary pure states and another set of binary generic mixed state in a two dimensional space. The necessary and sufficient conditions for the existence of such CPTP maps can be discussed within a simple geometrical picture. We exploit this analysis to show the existence of an optimal quantum repeater which is superior to the known repeating strategies for a set of coherent states sent through a lossy quantum channel. We also show that the geometrical formulation of the CPTP mapping conditions can be a simpler method to derive a state-dependent quantum (anti) cloning machine than the study so far based on the explicit solution of several constraints imposed by unitarity in an extended Hilbert space.

Abstract:
This article is an introduction to quant-ph/0302092. We propose to quantify how "quantum" a set of quantum states is. The quantumness of a set is the worst-case difficulty of transmitting the states through a classical communication channel. Potential applications of this measure arise in quantum cryptography, where one might like to use an alphabet of states most sensitive to quantum eavesdropping, and in lab demonstrations of quantum teleportation, where it is necessary to check that entanglement has indeed been used.

Abstract:
In this paper we propose a general method to quantify how "quantum" a set of quantum states is. The idea is to gauge the quantumness of the set by the worst-case difficulty of transmitting the states through a purely classical communication channel. Potential applications of this notion arise in quantum cryptography, where one might like to use an alphabet of states that promises to be the most sensitive to quantum eavesdropping, and in laboratory demonstrations of quantum teleportation, where it is necessary to check that quantum entanglement has actually been used in the protocol.

Abstract:
Universal Lax pairs of the root type with spectral parameter and independent coupling constants for twisted non-simply laced Calogero-Moser models are constructed. Together with the Lax pairs for the simply laced models and untwisted non-simply laced models presented in two previous papers, this completes the derivation of universal Lax pairs for all of the Calogero-Moser models based on root systems. As for the twisted models based on B_n, C_n and BC_nroot systems, a new type of potential term with independent coupling constants can be added without destroying integrability. They are called extended twisted models. All of the Lax pairs for the twisted models presented here are new, except for the one for the F_4 model based on the short roots. The Lax pairs for the twisted G_2 model have some novel features. Derivation of various functions, twisted and untwisted, appearing in the Lax pairs for elliptic potentials with the spectral parameter is provided. The origin of the spectral parameter is also naturally explained. The Lax pairs with spectral parameter, twisted and untwisted, for the hyperbolic, the trigonometric and the rational potential models are obtained as degenerate limits of those for the elliptic potential models.

In the integer and fractional quantum Hall effects, the electric
current flows through a thin layer under the strong magnetic field. The
diagonal resistance becomes very small at integer and specific fractional
filling factors where the electron scatterings are very few. Accordingly the
coherent length is large and therefore a tunneling effect of electrons may be
observed. We consider a new type of a quantum Hall device which has a narrow
potential barrier in the thin layer. Then the electrons flow with tunneling
effect through the potential barrier. When the oscillating magnetic field is applied
in addition to the constant field, the voltage steps may appear in the curve of
voltage V versus electric current I. If the voltage steps are
found in the experiment, it is confirmed that the 2D electron system yields the
same phenomenon as that of the ac-Josephson effect in a superconducting system.
Furthermore the step V is related to the transfer charge Q as V = (hf)/Q where f is the frequency of the
oscillating field and h is the Planck
constant. Then the detection of the step V determines the transfer charge Q. The
ratio Q/e (e is the
elementary charge) clarifies the

In this article, we study the string equation of type (2,5), which is
derived from 2D gravity theory or the string theory. We consider the equation
as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it
concretely by use of the Weierstrass’ elliptic function.

Abstract:
In this note, we analyze a few major claims about . As a consequence, we rewrite a major theorem, nullify its proof and one remark of importance, and offer a valid proof for it. The most important gift of this paper is probably the reasoning involved in all: We observe that a constant, namely t, has been changed into a variable, and we then tell why such a move could not have been made, we observe the discrepancy between the claimed domain and the actual domain of a supposed function that is created and we then explain why such a function could not, or should not, have been created, along with others.

This article concerns the quantum
superintegrable system obtained by Tremblay and Winternitz, which allows the separation
of variables in polar coordinates and possesses three conserved quantities with
the potential described by the sixth Painlevé equation. The degeneration procedure
from the sixth Painlvé equation to the fifth one yields another new
superintegrable system; however, the Hermitian nature is broken.

Abstract:
The spin polarization of a fractional quantum Hall state shows very
interesting properties. The curve of polarization versus magnetic field has
wide plateaus. The fractional quantum Hall effect is caused by the Coulomb
interaction because the 2D electron system without the Coulomb interaction yields
no energy gap at the fractional filling factor. Therefore, the wide plateau in
the polarization curve is also caused by the Coulomb interaction. When the
magnetic field is weak, some electrons have up-spins and the others down-spins.
Therein the spin-exchange transition occurs between two electrons with up and
down spins via the Coulomb interaction. Then the charge distribution before the
transition is the same as one after the transition. So these two states have
the same classical Coulomb energy. Accordingly, the partial Hamiltonian
composed of the spin exchange interaction should be treated exactly. We have
succeeded in diagonalizing the spin exchange interaction for the first and
second nearest electron pairs. The theoretical results reproduce the wide
plateaus very well. If the interval modulations between Landau orbitals are
taken into the Hamiltonian, the total energy has the Peierls instability. We
can diagonalize the Hamiltonian with the interval modulation. The results
reproduce wide plateaus and small shoulders which are in good agreement with
the experimental data.

Abstract:
We have investigated the Fractional Quantum Hall Effect (FQHE) on the fundamental Hamiltonian with the Coulomb interactions between normal electrons without any quasi particle. The electron pairs placed in the Landau orbitals can transfer to many empty orbitals. The number of the quantum transitions decreases discontinuously when the filling factor v deviates from the specific fractional number of v_{0}. The discontinuous decreasing produces the energy valley at the specific filling factors v_{0} = 2/3, 4/5, 3/5, 4/7, 3/7, 2/5, 1/3 and so on. The diagonal elements of the total Hamiltonian and the number of the quantum transitions give the total energy of the FQH states. The energy per electron has the discontinuous spectrum depending on the filling factor v. We obtain the function form of the energy per electron in the quantum Hall system. Then the theoretical Hall resistance curve is calculated near several filling factors. Therein the quantum Hall plateaus are derived from the energy valleys. The depths of the energy valleys are compared with the widths of the quantum Hall plateaus appearing in the experimental data of the Hall resistance. Our theoretical results are in good agreement with the experimental results.