Geometry of the 3-Qubit State, Entanglement and Division Algebras

We present a generalization to 3-qubits of the standard Bloch sphere representation for a single qubit and of the 7-dimensional sphere representation for 2 qubits presented in Mosseri {\it et al.}\cite{Mosseri2001}. The Hilbert space of the 3-qubit system is the 15-dimensional sphere $S^{15}$, which allows for a natural (last) Hopf fibration with $S^8$ as base and $S^7$ as fiber. A striking feature is, as in the case of 1 and 2 qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling 3 qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the 3-qubit state. Conjectures on the possibility to generalize the construction for higher qubit states are also discussed.Comment: 12 pages, 2 figures, final versio

Using of small-scale quantum computers in cryptography with many-qubit entangled states

We propose a new cryptographic protocol. It is suggested to encode information in ordinary binary form into many-qubit entangled states with the help of a quantum computer. A state of qubits (realized, e.g., with photons) is transmitted through a quantum channel to the addressee, who applies a quantum computer tuned to realize the inverse unitary transformation decoding of the message. Different ways of eavesdropping are considered, and an estimate of the time needed for determining the secret unitary transformation is given. It is shown that using even small quantum computers can serve as a basis for very efficient cryptographic protocols. For a suggested cryptographic protocol, the time scale on which communication can be considered secure is exponential in the number of qubits in the entangled states and in the number of gates used to construct the quantum network

Entangled photons from a strongly coupled quantum dot-cavity system

A quantum dot strongly coupled to a photonic crystal has been recently proposed as a source of entangled photon pairs [R. Johne et al., Phys. Rev. Lett. 100, 240404 (2008)]. The biexction decay via intermediate polariton states can be used to overcome the natural splitting between the exciton states coupled to the horizontally and vertically polarized light modes, so that high degrees of entanglement can be expected. We investigate theoretically the features of realistic dot-cavity systems, including the effect of the different oscillator strength of excitons resonances coupled to the different polarizations of light. We show that in this case, an independent adjustment of the cavity resonances is needed in order to keep a high entanglement degree. We also consider the case when the biexciton-exciton transition is also strongly coupled to a cavity mode. We show that a very fast emission rate can be achieved allowing the repetition rates in the THz range. Such fast emission should however be paid for by a very complex tuning of the many strongly coupled resonances involved and by a loss of quantum efficiency. Altogether a strongly coupled dot-cavity system seems to be very promising as a source of entangled photon pairs.Comment: 7 pages, 5 figure

Quantum Cryptography Based on the Time--Energy Uncertainty Relation

A new cryptosystem based on the fundamental time--energy uncertainty relation is proposed. Such a cryptosystem can be implemented with both correlated photon pairs and single photon states.Comment: 5 pages, LaTex, no figure

Remarks on the Central Limit Theorem for Non-Convex Bodies

In this note, we study possible extensions of the Central Limit Theorem for non-convex bodies. First, we prove a Berry-Esseen type theorem for a certain class of unconditional bodies that are not necessarily convex. Then, we consider a widely-known class of non-convex bodies, the so-called p-convex bodies, and construct a counter-example for this class

Neutrinos in a spherical box

In the present paper we study some neutrino properties as they may appear in the low energy neutrinos emitted in triton decay with maximum neutrino energy of 18.6 keV. The technical challenges to this end can be achieved by building a very large TPC capable of detecting low energy recoils, down to a a few tenths of a keV, within the required low background constraints. More specifically We propose the development of a spherical gaseous TPC of about 10-m in radius and a 200 Mcurie triton source in the center of curvature. One can list a number of exciting studies, concerning fundamental physics issues, that could be made using a large volume TPC and low energy antineutrinos: 1) The oscillation length involving the small angle of the neutrino mixing matrix, directly measured in this disappearance experiment, is fully contained inside the detector. Measuring the counting rate of neutrino-electron elastic scattering as a function of the distance of the source will give a precise and unambiguous measurement of the oscillation parameters free of systematic errors. In fact first estimates show that even with a year's data taking a sensitivity of a few percent for the measurement of the above angle will be achieved. 2) The low energy detection threshold offers a unique sensitivity for the neutrino magnetic moment which is about two orders of magnitude beyond the current experimental limit. 3) Scattering at such low neutrino energies has never been studied and any departure from the expected behavior may be an indication of new physics beyond the standard model. In this work we mainly focus on the various theoretical issues involved including a precise determination of the Weinberg angle at very low momentum transfer.Comment: 16 Pages, LaTex, 7 figures, talk given at NANP 2003, Dubna, Russia, June 23, 200

Tendency to Maximum Complexity in a Non-Equilibrium Isolated System

The time evolution equations of a simplified isolated ideal gas, the "tetrahe- dral" gas, are derived. The dynamical behavior of the LMC complexity [R. Lopez-Ruiz, H. L. Mancini, and X. Calbet, Phys. Lett. A 209, 321 (1995)] is studied in this system. In general, it is shown that the complexity remains within the bounds of minimum and maximum complexity. We find that there are certain restrictions when the isolated "tetrahedral" gas evolves towards equilibrium. In addition to the well-known increase in entropy, the quantity called disequilibrium decreases monotonically with time. Furthermore, the trajectories of the system in phase space approach the maximum complexity.Comment: 22 pages, 0 figures. Published in Phys. Rev. E 63, 066116(9) (2001

Separable approximation for mixed states of composite quantum systems

We describe a purely algebraic method for finding the best separable approximation to a mixed state of a composite 2x2 quantum system, consisting of a decomposition of the state into a linear combination of a mixed separable part and a pure entangled one. We prove that, in a generic case, the weight of the pure part in the decomposition equals the concurrence of the state.Comment: 13 pages, no figures; minor changes; accepted for publication in PR

Multiphoton localization and propagating quantum gap solitons in a frequency gap medium

The many-particle spectrum of an isotropic frequency gap medium doped with impurity resonance atoms is studied using the Bethe ansatz technique. The spectrum is shown to contain pairs of quantum correlated ``gap excitations'' and their heavy bound complexes (``gap solitons''), enabling the propagation of quantum information within the classically forbidden gap. In addition, multiparticle localization of the radiation and the medium polarization occurs when such a gap soliton is pinned to the impurity atom.Comment: 8 pages, RevTEX, to appear in Phys. Rev. Let

On the Rees-Sciama effect: maps and statistics

Small maps of the Rees-Sciama (RS) effect are simulated by using an appropriate N-body code and a certain ray-tracing procedure. A method designed for the statistical analysis of cosmic microwave background (CMB) maps is applied to study the resulting simulations. These techniques, recently proposed --by our team-- to consider lens deformations of the CMB, are adapted to deal with the RS effect. This effect and the deviations from Gaussianity associated to it seem to be too small to be detected in the near future. This conclusion follows from our estimation of both the RS angular power spectrum and the RS reduced n-direction correlation functions for n<7.Comment: 11 pages, 13 figures, to appear in MNRA