PSI and the Combination Problem

Doron Kwiat

Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. Quantum mechanics is described with real fields and real operators. Schrodinger and Dirac equations then are solved. The solution to Dirac equation gives four, real, 2-vectors solutions ψ 1 =( U 1 D 1 ) ψ 2 =( U 2 D 2 ) ψ 3 =( U 3 D 3 ) ψ 4 =( U 4 D 4 ) where ( ψ 1 , ψ 4 ) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, ( ψ 2 , ψ 3 ) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. For an incoming entangled pair of fermions, the combined solution is Ψ in = c 1 ψ 1 + c 4 ψ 4 where c 1 and c 4 are some hidden variables. By applying a magnetic field in +Z and +x the theoretical results of a triple Stern-Gerlach experiment are predicted correctly. Then, by repeating Bell’s and Mermin Gedanken experiment with three magnetic filters σ θ , at three different inclination angles θ , the violation of Bell’s inequality is proven. It is shown that all fermions are in a mixed state of spins and the ratio between spin-up to spin-down depends on the hidden variables.

Giacomo Mauro D’Ariano

Liu Lian

In this paper,we give a structure feature of convex combination preserving bijective maps which send separable states to separable states of bipartite system.It is shown that such a map Φ has the form of Φ(ρσ)=(ST)Ψ(ρσ)(ST)*Tr((ST)Ψ(ρσ)(ST)*),where Ψ is one of the identity,transpose,partial transpose,or a combination of above mentioned map with the swap.

Stephen Bruce Sontz

Oliver B. Downs, Hagai T. Attias, Chris J. C. Burges +1 more

In this work we present a novel method for global optimization, exploiting the mathematics of quantum mechanics, and in particular the tunnelling phenomenon. We consider a quantum mechanical system with a single particle in a potential specified by the cost function of our optimization problem. We assume the groundstate of the system is localised to the global minimum of the potential function, a condition which can be assured by choosing the particle mass sufficiently large. We then approximate this groundstate using a variational approach to optimise an expansion in terms of a truncated basis set of harmonic oscillator wavefunctions. We show how to encode integer factoring and travelling salesman problems in terms of finding the global minimum of a quartic polynomial cost function. We demonstrate our quantum algorithm on oneand two-dimensional toy problems, and apply it to factoring biprimes with 10 bits. Introduction The state of the art in global optimization for general problems is simulated annealing and its derivatives. The algorithm can be viewed in the physical context of a classical thermal system. This inspires the novel approach presented here, in which we draw upon intuition from non-classical physical systems which obey quantum mechanics (QM), and in particular exhibit the phenomenon of quantum tunnelling. In QM, a potential function V (x) describes the energy terrain experienced by a particle, which may contain multiple minima separated by barriers. Combined with a term describing the kinetic energy of the particle, it defines the Hamiltonian operator: Ĥ = − 2 2m ∇2 + V (1) QM captures all available information on the state of the particle in a (complex) wave function ψ(x). For instance, |ψ(x)|2 is the probability distribution over the position x of the particle. To obtain ψ(x) one solves the eigenvalue problem Ĥψ = Eψ (2) known as the Schroedinger equation, where E is the energy. The solution is a set of wave functions termed eigenstates, and a matching set of energies. The eigenstate with the lowest energy is the ground state, ψ. By choosing the particle’s mass sufficiently large, the probability density associated with the ground state wave function becomes localized to the global minimum of the potential. Hence, a given optimization problem involving the minimization of a cost function can be restated as the problem of finding the ground state of the QM system with a potential specified by that cost function. In particular, computationally ‘hard’ integer problems, including factoring [1] and the travelling salesman problem, can be cast as QM problems involving a quartic polynomial potential, [3]. However, solving the Schroedinger equation exactly for these potentials is generally an analytically intractable problem. In our approach we approximate the groundstate of such a system using a method based on perturbation theory.