Two aspects of the relativistic version of the Einstein-Podolsky-Rosen-Bohm (EPRB) experiment with massive particles are discussed: (a) a possibility of using the experiment as an implicit test of a relativistic center-of-mass concept, and (b) influence of the relativistic effects on degree of violation of the Bell inequality. The nonrelativistic singlet state average ψ| a· σ ⊗ b· σ|ψ = − a· b is relativistically generalized by defining spin via the relativistic center-of-mass operator. The corresponding EPRB average contains relativistic corrections which are stronger in magnitude than standard relativistic phenomena such as the time delay, and can be measured in Einstein-Podolsky-Rosen-Bohm-type experiments with relativistic massive spin-1/2 particles. The degree of violation of the Bell inequality is shown to depend on the velocity of the pair of spin-1/2 particles with respect to laboratory. Experimental confirmation of the relativistic formula would indicate that for relativistic nonzerospin particles centers of mass and charge do not coincide. The result may have implications for quantum cryptography based on massive particles.

Modern approaches to semanic analysis if reformulated as Hilbert-space problems reveal formal structures known from quantum mechanics. Similar situation is found in distributed representations of cognitive structures developed for the purposes of neural networks. We take a closer look at similarites and differences between the above two fields and quantum information theory.

Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters `plus' or `times' one has certain freedom of interpreting this operation. This leads to some freedom in definitions of derivatives, integrals and, thus, practically all equations occurring in natural sciences. A change of realization of arithmetic, without altering the remaining structures of a given equation, plays the same role as a symmetry transformation. An appropriate construction of arithmetic turns out to be particularly important for dynamical systems in fractal space-times. Simple examples from classical and quantum, relativistic and nonrelativistic physics are discussed, including the eigenvalue problem for a quantum harmonic oscillator. It is explained why the change of arithmetic is not equivalent to the usual change of variables, and why it may have implications for the Bell theorem

The Bennett-Brassard cryptographic scheme (BB84) needs two bases, at least one of them linearly polarized. The problem is that linear polarization formulated in terms of helicities is not a relativistically covariant notion: State which is linearly polarized in one reference frame becomes depolarized in another one. We show that a relativistically moving receiver of information should define linear polarization with respect to projection of Pauli-Lubanski's vector in a principal null direction of the Lorentz transformation which defines the motion, and not with respect to the helicity basis. Such qubits do not depolarize. PACS numbers: PACS numbers: 03.67.Dd, 03.65.UdIn non-relativistic quantum mechanics a generic state of a free particle with spin takes the form where spin and momentum degrees of freedom are non-entangled, i.e.This is the reason why it is possible to base the concept of a non-relativistic qubit on a 2-dimensional Hilbert space.In particular, observables asociated with spin are always of the form A⊗1, where 1 = d 3 p|p p| is the identity in momentum space and A stands for a spin operator. The formula Tr ρ(A ⊗ 1) = Tr r ρ r A. allows to define states of qubits in terms of 2 × 2 reduced density matrices.In relativistic quantum mechanics a generic state satisfiesThe origin of this property is very deeply rooted in the structure of unitary representations of the Poincaré group. A qubit which in one reference frame takes the form (1) will be seen in a form (2) by another observer. A Poincaré transformation necessarily involves multiplication by p-dependent SU (2) matrices, a fact making the form (1) non-covariant. Definitions of qubits in terms of reduced density matrices with traced-out momenta are no longer justified. This is why quantum information theory based on such a formal notion of qubit [1,2,3,4] is in danger of internal physical inconsistency. Constructing nonzero-spin unitary representations of the Poincaré group we always encounter certain spinor structure. The simplest representation corresponds to mass m and spin 1/2. Whenever we write the state in a form (2) we implicitly choose a 'spin quantization axis' and spin is here associated with the second Casimir invariant of the group, W a W a , where W a is the Pauli-Lubanski (PL) vector.The most popular choice of quantization axis corresponds to a timelike direction t a = (1, 0, 0, 0). The resulting spin operator t a W a is proportional to the helicity (in order to obtain directly the helicity one should choose t a = (1/|p|, 0, 0, 0)). In application to quantum cryptography we need several different yes-no observables and helicity eigenstates are not sufficient. Natural candidates for such yes-no observables are projectors on linear combinations of opposite helicities, i.e. linear polarizations. The problem with linear polarizations defined in terms of helicities is that different momentum components undergo different SU (2) transformations. In the photon case the SU (2) transformations are diagonal and multiply opposite helicities by phas...