296 research outputs found
Inverse spin-s portrait and representation of qudit states by single probability vectors
Using the tomographic probability representation of qudit states and the
inverse spin-portrait method, we suggest a bijective map of the qudit density
operator onto a single probability distribution. Within the framework of the
approach proposed, any quantum spin-j state is associated with the
(2j+1)(4j+1)-dimensional probability vector whose components are labeled by
spin projections and points on the sphere. Such a vector has a clear physical
meaning and can be relatively easily measured. Quantum states form a convex
subset of the 2j(4j+3) simplex, with the boundary being illustrated for qubits
(j=1/2) and qutrits (j=1). A relation to the (2j+1)^2- and
(2j+1)(2j+2)-dimensional probability vectors is established in terms of spin-s
portraits. We also address an auxiliary problem of the optimum reconstruction
of qudit states, where the optimality implies a minimum relative error of the
density matrix due to the errors in measured probabilities.Comment: 23 pages, 4 figures, PDF LaTeX, submitted to the Journal of Russian
Laser Researc
Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics
Symmetric informationally complete positive operator valued measures
(SIC-POVMs) are studied within the framework of the probability representation
of quantum mechanics. A SIC-POVM is shown to be a special case of the
probability representation. The problem of SIC-POVM existence is formulated in
terms of symbols of operators associated with a star-product quantization
scheme. We show that SIC-POVMs (if they do exist) must obey general rules of
the star product, and, starting from this fact, we derive new relations on
SIC-projectors. The case of qubits is considered in detail, in particular, the
relation between the SIC probability representation and other probability
representations is established, the connection with mutually unbiased bases is
discussed, and comments to the Lie algebraic structure of SIC-POVMs are
presented.Comment: 22 pages, 1 figure, LaTeX, partially presented at the Workshop
"Nonlinearity and Coherence in Classical and Quantum Systems" held at the
University "Federico II" in Naples, Italy on December 4, 2009 in honor of
Prof. Margarita A. Man'ko in connection with her 70th birthday, minor
misprints are corrected in the second versio
Piecewise Smooth Dynamical Systems Theory: The Case of the Missing Boundary Equilibrium Bifurcations
Theory of differential inclusions and its application in mechanics
The following chapter deals with systems of differential equations with
discontinuous right-hand sides. The key question is how to define the solutions
of such systems. The most adequate approach is to treat discontinuous systems
as systems with multivalued right-hand sides (differential inclusions). In this
work three well-known definitions of solution of discontinuous system are
considered. We will demonstrate the difference between these definitions and
their application to different mechanical problems. Mathematical models of
drilling systems with discontinuous friction torque characteristics are
considered. Here, opposite to classical Coulomb symmetric friction law, the
friction torque characteristic is asymmetrical. Problem of sudden load change
is studied. Analytical methods of investigation of systems with such
asymmetrical friction based on the use of Lyapunov functions are demonstrated.
The Watt governor and Chua system are considered to show different aspects of
computer modeling of discontinuous systems
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Synthesis and solution properties of a temperature-responsive PNIPAM–b-PDMS–b-PNIPAM triblock copolymer
In this paper, we report the synthesis and self-assembly of a novel thermoresponsive PNIPAM60–b-PDMS70–b-PNIPAM60 triblock copolymer in aqueous solution. The copolymer used a commercially available precursor modified with an atom transfer radical polymerization (ATRP) initiator to produce an ABA triblock copolymer via ATRP. Small-angle neutron scattering (SANS) was used to shed light on the structures of nanoparticles formed in aqueous solutions of this copolymer at two temperatures, 25 and 40 °C. The poly(dimethylsiloxane) block is very hydrophobic and poly(N-isopropylacrylamide) (PNIPAM) is thermoresponsive. SANS data at 25 °C indicates that the solutions of PNIPAM–b-PDMS–b-PNIPAM copolymers form well-defined aggregates with presumably core–shell structures below cloud point temperature. The scattering curves originating from nanoparticles formed at 40 °C in 100% D2O or 100% H2O were successfully fitted with the Beaucage model describing aggregates with hierarchical structure
Quasi-Linear Differential-Deference Game of Approach
This is a post-peer-review, pre-copyedit version of a book chapter that is part of “V. A. Sadovnichiy, M. Z. Zgurovsky (eds.). Modern Mathematics and Mechanics. Understanding Complex Systems”. The final authenticated version is available online at: https://link.springer.com/chapter/10.1007/978-3-319-96755-4_26The paper is devoted to the games of approach. We consider a controlled object whose dynamics is described by the linear differential system with a pure time delay or the differential-difference system with commutative matrices in Euclidean space. The approaches to the solutions of these problems are proposed which based on the Method of Resolving Functions and the First Direct Method of L.S. Pontryagin. The guaranteed times of the game termination are found, and corresponding control laws are constructed. The results are illustrated by a model example
The regularized visible fold revisited
The planar visible fold is a simple singularity in piecewise smooth systems.
In this paper, we consider singularly perturbed systems that limit to this
piecewise smooth bifurcation as the singular perturbation parameter
. Alternatively, these singularly perturbed systems can
be thought of as regularizations of their piecewise counterparts. The main
contribution of the paper is to demonstrate the use of consecutive blowup
transformations in this setting, allowing us to obtain detailed information
about a transition map near the fold under very general assumptions. We apply
this information to prove, for the first time, the existence of a locally
unique saddle-node bifurcation in the case where a limit cycle, in the singular
limit , grazes the discontinuity set. We apply this
result to a mass-spring system on a moving belt described by a Stribeck-type
friction law
A comparative study of qualitative and quantitative dynamic models of biological regulatory networks
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