Ergodic Properties of the Quantum Geodesic Flow on Tori

We study ergodic averages for a class of pseudodifferential operators on the flatN-dimensional torus with respect to the Schrödinger evolution. The later can be consider a quantization of the geodesic flow on . We prove that, up to semi-classically negligible corrections, such ergodic averages are translationally invariant operators.Mathematics Subject Classifications (2000) 58J50, 58J40, 81S10.     

Implementation of a multiple round quantum dense coding using nuclear magnetic resonance

~~Implementation of a multiple round quantum dense coding using nuclear magnetic resonance1. Bennett, C. H., Wiesner, S. J., Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett., 1992, 69(20): 2881-2884. 2. Mattle, K., Weinfurter, H., Kwiat, P. G. et al., Dense coding in experimental quantum communication, Phys. Rev. Lett., 1996, 76(25): 4656-4659. 3. Fang, X. M., Zhu, X. W., Feng, M. et al., Experimental implementation of dens…     

Extended structures and new infinite-dimensional hidden symmetries for the Einstein–Maxwell-dilaton-axion theory with multiple vector fields

A so-called extended elliptical-complex (EEC) function method is proposed and used to further study the Einstein–Maxwell-dilaton-axion theory with p vector fields (EMDA-p theory, for brevity) for . An Ernst-like matrix EEC potential is introduced and the motion equations of the stationary axisymmetric EMDA-p theory are written as a so-called Hauser–Ernst-like self-dual relation for the EEC matrix potential. In particular, for the EMDA-2 theory, two Hauser–Ernst-type EEC linear systems are established and based on their solutions some new parametrized symmetry transformations are explicitly constructed. These hidden symmetries are verified to constitute an infinite-dimensional Lie algebra, which is the semidirect product of the Kac–Moody algebra and Virasoro algebra (without centre charges). These results show that the studied EMDA-p theories possess very rich symmetry structures and the EEC function method is necessary and effective.     

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.     

Information and Distinguishability of Ensembles of Identical Quantum States

We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is , where (α₁,α₂) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is . The optimal distribution is uniform over the domain in Cartesian coordinates.     

Evolution of Orbits and Quantum Correlation Functions by Quadratic Hamiltonians

Quantum systems with quadratic Hamiltonians are considered. Some results about the time evolution of homogeneous polynomials and of quantum correlation functions are given. The image of arbitrary orbit of Weyl–Heisenberg group under this time evolution is shown to be again an orbit of this group. For quantum free particle it is shown that its time evolution intersects arbitrary such orbit at most once. A result about existence of more orbits having the same dispersion of some quantum position is presented.PACS: 02.20.Qs, 02.30.Sa     

A Scheme for Physical Implementation of a Ququadrit Quantum Computation with Cooled-Trapped Ions

The physics realization of a ququadrit quantum computation with cooled trapped ¹³⁸Ba⁺ ions in a Paul trap is investigated. The ground state level 6² S_1/2(m = −1/2) and three metastable levels: 5² D_3/2(m = −1/2), 5² D_5/2(m = −1/2), and 5² D_5/2(m = 1/2), of the fine-structure of the ¹³⁸Ba⁺ ion, are used to store the quantum information of ququadrits. The use of coherent manipulation of populations in single ququadrit, being a four-dimensional Hilbert space, produces a discrete Fourier transform and the manipulation of the first red band transitions with the introduction of an ancillary quantum channel between two ququadrits generates a conditional phase gate. The combination of the both above results in a universal two-ququadrit gate, called XOR⁽⁴⁾ gate corresponding to the controlled-NOT gate operation in qubit systems. The implementation of quantum Fourier transform for n ququadrits is performed by means of the conditional phase-shift gate. The feasibility of physical realization of ququadrit quantum computation with cooled-trapped ¹³⁸Ba⁺ ions is detailed analyzed and described, and the theoretical detection method of logical states is given. Higher entanglement between ququadrits than qutrits or qubits and more security of ququadrit quantum cryptography than qutrit's or qutrit's will lead to more extensive applications ququadrits in quantum information fields. In particular, it is pointed out that this scheme should be the highest dimensional quantum computation in cooled-trapped ions, the entanglement between ququadrits should be the highest dimensional entanglement in it, and the ququadrit quantum cryptography should be the most secure cryptography protocol in it.     

Maps Between Deformed and Ordinary Gauge Fields

In this paper, we introduce a map between the q-deformed gauge fields defined on the GL_q(N) -covariant quantum hyperplane and the ordinary gauge fields. Perturbative analysis of the q-deformed QED at the classical level is presented and gauge fixing à la BRST is discussed. Another star product defined on the hybrid (q,h) -plane is explicitly constructed.     

Generalized Uncertainty Principle in a Simple Varying Speed of Light Model

In this paperwewill derive a generalized uncertainty principle (GUP) in a simple varying speed of light (VSL) model. First we will show that VSL is an immediate consequence of GUP. Then, within the framework of a simple VSL model, we will show that GUP can be expressed as a function of cosmological scale factor. This expression gives two main results: uncertainties in position and momentum are actually cosmological models dependent and these uncertainties depend on mass and momentum of the particle under consideration. The relationship between matter content of the Universe and the values of uncertainties in early stages of the evolution of the Universe will be discussed in a mini-superspace approach.