Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations

Entanglement of two parts of a quantum system is a non-local property unaffected by local manipulations of these parts. It is described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of non-local properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-bit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete. Similarly, non-local, entangling properties of two-qubit unitary gates are invariant under single-bit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-bit gates).Comment: 4 pages; minor changes; relation of our invariants and those of quant-ph/9712040 clarifie

Low-mass dileptons from nonequilibrium QGP

The rate of the emission of the high energy low-mass dileptons from the QGP is found in the first nonvanishing order with respect to strong coupling. We base on the real-time kinetic approach [2] without an explicit assumption about a complete thermal equilibrium in the emitting system. For the class of the partons distributions which may simulate that of the "hot glue scenario"[1] the rate of emission is found analytically . ( Figures can be obtained from the author )Comment: 7 pages, Preprint SUNY-NTG-93-2

Quark and gluon distributions at the earliest stage of heavy ion collision

Using the general framework of quantum field kinetics we consider new principles to compute initial distribution of quarks and gluons after the first hard interaction of heavy ions. We start by rewriting the integral equations of QCD in the form which is generalizations of the familiar QCD evolution equations. These equations describe both space-time-- and $(x,Q^2)$--evolution before the collision, and allow one to use the $ep$ DIS data without reference to parton phenomenology. New technique generate perturbation theory that avoid double count of the processes, does not contain an artificial factorization scale, and does not require low-momentum cut-offs since infrared behavior is controlled by the DIS data.Comment: 30 pages, REVTeX, 5 postscript figures appende

The Dirac field and the possible origin of gravity

The spin connections of the Dirac field have three ingredients that are connected with the Ricci rotations, the Maxwell field, and an axial field which minimally interacts with the axial current. I demonstrate that the axial field provides an effective mechanism of auto-localization of the Dirac field into compact objects. The condition that these objects are stable (the energy-momentum is self-adjoint) leads to Einstein's field equations. The Dirac field with its spin connection seem to be a natural material carrier of the space-time continuum in which compact objects are moving along geodesic lines. The long distance effect of the axial field is indistinguishable from Newton's gravity, which reveals the microscopic nature of gravity and the origin of the gravitational mass.Comment: 4 pages, Submitted to PRL; minor corrections of grammar and styl

PBW degenerate Schubert varieties: Cartan components and counterexamples

In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type $\mathrm A$ but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for $\mathfrak{sl}_6$. The counterexamples are constructed with the help of a study of the Cartan components appearing in this context

Weyl's Formula as the Brion Theorem for Gelfand-Tsetlin Polytopes

We exploit the idea that the character of an irreducible finite dimensional $\mathfrak{gl}_n$-module is the sum of certain exponents of integer points in a Gelfand-Tsetlin polytope and can thus be calculated via Brion's theorem. In order to show how the result of such a calculation matches Weyl's character formula we prove some interesting combinatorial traits of Gelfand-Tsetlin polytopes. Namely, we show that under the relevant substitution the integer point transforms of all but $n!$ vertices vanish, the remaining ones being the summands in Weyl's formula

Scenario for Ultrarelativistic Nuclear Collisions: V. Onset of Deconfinement. (How the Nuclei Get Unbound.)

We consider a Euclidean extension of the wedge form of Hamiltonian dynamics, which explicitly accounts for the strong localization of the first interaction in nuclear collisions. A new principle of the analytic continuation via the tetrad vector is introduced. We discover the existence of self-dual solutions with short life-times (ephemerons) and conjecture that these vacuum fluctuations can lower the Euclidean action of the system of the colliding nuclei, thus enforcing a breakdown of the nuclei coherence. We suggest that the ephemerons can be identified with the gluons-partons, which are resolved in high-energy nuclear collisions.Comment: 19 pages, RevTe

The wedge form of relativistic dynamics

It is commonly accepted that in hadronic or nuclear collisions at extremely high energies the shortest scales are explored. At the classical level, this property of the interaction is closely related to the Lorentz contraction of the fields of colliding particles which provides instantaneous switching the interaction on. I argue that the underlying quantum dynamics should be confined to within the light wedge of the two-dimensional plane where the first interaction takes place and suggest to include this property as the boundary condition for the quantum field theory which describes the collision process. Connection between the type of inclusive process and the temporal order of its dynamical evolution is discussed. The one-particle states and propagators of the perturbation theory for the scalar and fermion fields are found.Comment: 14 pages, REVTe

On the Origin of the Charge-Asymmetric Matter. II. Localized Dirac Waveforms

This paper continues the author's work \cite{PartI}, where a new framework of the matter-induced physical geometry was built and an intrinsic nonlinearity of the Dirac equation discovered. Here, the nonlinear Dirac equation is solved and the localized configurations are found analytically. Of the two possible types of the potentially stationary localized configurations of the Dirac field, only one is stable with respect to the action of an external field and it corresponds to a positive charge. A connection with the global charge asymmetry of matter in the Universe and with the recently observed excess of the cosmic positrons is discussed.Comment: 18 pages, published in Journal of Modern Physics, (2016) v.7, No.7. pp.662-66

Gelfand--Tsetlin degenerations of representations and flag varieties

Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such frameworks for all Gr\"obner degenerations intermediate between the flag variety and the GT toric variety. These degenerations are shown to induce filtrations on the irreducible representations and the associated graded spaces are acted upon by a certain associative algebra. To achieve our goal, we construct embeddings of our Gr\"obner degenerations into the projectivizations of said associated graded spaces in terms of this action. We also obtain an explicit description of the maximal cone in the tropical flag variety that parametrizes the Gr\"obner degenerations we consider. In an addendum we propose an alternative solution to the problem which relies on filtrations and gradings by non-abelian ordered semigroups