The secure distribution of the secret random bit sequences known as `key' material, is an essential precursor to their use for the encryption and decryption of confidential communications. Quantum cryptography is an emerging technology for secure key distribution with single-photon transmissions: Heisenburg's uncertainty principle ensures that an adversary can neither successfully tap the key transmissions, nor evade detection (eavesdropping raises the key error rate above a threshold value). We have developed experimental quantum cryptography systems based on the transmission of non-orthogonal single-photon states to generate shared key material over multi-kilometer optical fiber paths and over line-of-sight links. In both cases, key material is built up using the transmission of a single- photon per bit of an initial secret random sequence. A quantum-mechanically random subset of this sequence is identified, becoming the key material after a data reconciliation stage with the sender. In our optical fiber experiment we have performed quantum key distribution over 24-km of underground optical fiber using single-photon interference states, demonstrating that secure, real-time key generation over `open' multi-km node-to-node optical fiber communications links is possible.

We present a mathematical physics analysis of entangled translucent eavesdropping in quantum cryptography, based on the recent work of Ekert, Huttner, Palma, and Peres. The key generation procedure involves the transmission, interception, and reception of two nonorthogonal photon polarization states. At the receiving end, a positive operator valued measure (POVM) is employed in the measurement process. The eavesdropping involves an information-maximizing von Neumann-type projective measurement. We propose a new design for a receiver that is an all-optical realization of the POVM, using a Wollaston prism, a mirror, two beam splitters, a polarization rotator, and three photodetectors. We present a quantitative analysis of the receiver. We obtain closed-form algebraic expressions for the error rates and mutual information, expressed in terms of the POVM-receiver error rate and the angle between the carrier polarization states. We also prove a significant result, namely, that in the entangled translucent eavesdropping approach, the unsafe error rate based on standard mutual information comparisons is equivalent to the maximum allowable error rate based on perfect mutual information for the eavesdropper. In this case, the above unsafe error rate is in fact not overly conservative.

The analysis of formation of chaotic quantum states for two (or four) orthogonally polarized modes in a tunnelly-coupled optical fiber has been carried out. A special attention is devoted both to the structure of arising polarization instabilities and to the possibility to control of such a quantum state by procedure of quantum nondemolition measurements. New effect of polarization switching is obtained and we discuss the problem of both coding of information and construction of quantum logical elements for quantum computing. The optical schemes to realize the quantum AND (NOT-AND), OR (NOT-OR) and controlled-NOT gates with quantum nondemolition filtering of the signal are proposed.

Using simple physical arguments we investigate the capabilities of a quantum computer based on cold trapped ions of the type recently proposed by Cirac and Zoller. From the limitations imposed on such a device by decoherence due to spontaneous decay, laser phase coherence times, ion heating and other possible sources of error, we derive bounds on the number of laser interactions and on the number of ions that may be used. As a quantitative measure of the possible performance of these devices, the largest number which may be factored using Shor's quantum factoring algorithm is determined for a variety of species of ion.

We first review the mathematical physics of quantum decoherence in generic single-qubit devices, including environmental interactions. Next, we formulate the persistence probability for a qubit device as the probability of measuring the qubit device in the unperturbed state without the decoherence arising from environmental interactions. The decoherence time can be obtained from the persistence probability. Drawing on recent work of Garg, and also Palma, Suomine, and Ekert, we apply the persistence probability formalism to a generic single-qubit device coupled to a thermal environment, and also apply it to a trapped-ion quantum register coupled to the ion vibrational modes.

A decoherence mechanism in the transition from pure quantum- mechanical systems to thermodynamical systems is proposed. We see that a quantum-mechanical expression of temperature is derived and the expression for boson systems and that for fermion systems are different. An application to the calculation of change of temperatures is investigated in a quasi-static process and numerical analyses for mixing of two gases with different temperatures are carried out. Considerations in terms of ultra-power representations of nonstandard-analysis are also presented.

Both quantum computing and tests of certain theories of quantum measurement require tight control over entanglement of the degrees of freedom of the system with those of the environment. In particular, one wishes to avoid entanglement-induced decoherence due to the confining of the system by the macroscopic apparatus. Nonetheless, when the system interacts with its walls (treated quantum mechanically), there is an inevitable intertwining of degrees of freedom. We show that this need not lead to entanglement, hence decoherence. It will generally lead to error. The wave function optimization required to avoid decoherence is also examined.

We argue that the analog nature of quantum computing makes the usual design approach of constructing complicated logical operations from many simple gates inappropriate. Instead, we propose to design multi-spin quantum gates in which the input and output two-state systems (spins) are not necessarily identical. We outline the design criteria for such devices and then review recent results for single-unit Hamiltonians that accomplish the NOT and XOR functions.

In our paper we consider the behavior of both the Hermitian (observable) Stokes parameters of light and the phases difference that describes the polarization state of optical field. In general case we introduce two pairs of phase operators associated with the phase angles in 3D picture for the polarization Stokes parameters of light on the Poincare sphere. The special eight-port polarization interferometer is presented for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phases and also of their fluctuations. It is shown that an anisotropic (spatio-periodically) Kerr-like nonlinear medium placed in the polarization interferometer can be used for generation and observation of the polarization-squeezed phase states of light.

Raman transitions can be induced in quantum dots structures by impinching the crystal with a coherent laser beam whose frequency resonates with a Raman frequency of the crystal. The discrete nature of the frequencies and the fact that a set of Raman frequencies can be mapped unto itself through energy exchanges presents a framework for all-optical quantum computing in quantum dots. In this paper we discuss how the distributive, associative, and communicative laws of logic can be applied to a `Raman Set'. We also present the description and implementation of some basic logic operations such as negation, union and intersection on this set.

These notes begin a study of test-spaces equipped with a group action. Several notions of irreducibility and decomposability for such G-test spaces are discussed and compared. We introduce classes of test-spaces having strong symmetry properties and investigate their properties. We also discuss induced and co-induced test-spaces in this context, and prove an imprimitivity theorem characterizing the latter.

The forcing process from mathematical logic offers a promising framework for studying the feasibility of quantum computation on a practical scale, since decoherence is a serious concern and, so far, questions of control, communication, and the implementation of operations that are important for a working computational system have received less attention than mathematical research on algorithms and basic physical investigations of creating simple gates and storing mixed states. Using forcing in this way is a new application of areas of model theory in which propositions and predicates take values in a lattice. Takeuti develops set theory for any universe built on a Boolean algebra generated by commutable projection operators on a Hilbert space, each such universe being a elementary topos of set- valued sheaves on the algebra, which thus is the lattice of truth values for the topos. Since it has a natural numbers object, it thus supports the general forcing method of Scedrov, but the idea of forcing is demonstrated by an example in the simpler context of partially ordered sets. Stout's lamination construction assembles toposes into objects with truth-value lattices that are orthomodular, like the lattice of all projection operators on Hilbert space, and clarifies some difficulties identified by Takeuti in the case where truth values are noncommutable operators. A substantial body of existing sheaf and topos theory thus is potentially relevant to quantum computation, and further work may provide guidance for system development.

A degree of violation of the Bell inequality depends on momenta of massive particles with respect to a laboratory if spin plays a role of a `yes-no' observable. For ultra- relativistic particles a standard Ekert test has to take into account this velocity dependent suppression of the degree of violation of the inequality. Otherwise Alice and Bob may `discover' a nonexisting eavesdropper.

The possibility of basing quantum computations on the laws of genuine quantum (non-Boolean) logic is shown. We investigate the new options provided by this approach and consider its relevance to the existing computational problems. A simple example of quantum adding machine that works according to this idea is studied. This example shows that the proposed new way of making quantum computations in the natural way utilizes `probabilistic' (or `fuzzy') features of quantum objects and allows to deal with non- exact data.

The simulation of the behavior of physical systems on the coarsest level, namely, restoring the logical structure of propositions about the system is suggested. To realize this simulation, the special class of finite automata, called normalized, is introduced. Relevant graph-theoretical techniques are considered.

A possibility of formation of nonclassical polarization states of light in the case of interaction of two orthogonally polarized modes in a spatio-inhomogeneous nonlinear medium is considered. A procedure of the quantum nondemolition (QND) measurement both of the Stokes parameters and the phase difference of two orthogonally polarized modes of optical field is presented and analyzed for the first time. We also introduce a new nonlinear parameter to describe the light polarization (depolarization) in quantum optics. The general approach to the measurement procedure under study allows to propose the new scheme for the QND-measurement of the angular momentum of atomic system as well.

We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.

The paper deals with principles of construction, theory and fundamental properties of a new class of optical instruments, i.e. 1 and 2 coordinate polypupil systems on the basis of geometry of linear-homogeneous zones inside segmented optical macroelements in the field of sequential multiple reflections from a curvilinear surface. Discussed in the paper are: optical synchrogenerators creating discrete sets of light delays of coherent radiation pulses with fractionary-rational step of change of delay time; hybrid devices of telescopes and objective lenses, built around decentered segments (both solid and mirror ones) of polyscopic cylindrical astigmats; devices for internal scanning by focal matrices of radiation receivers within angular sectors from 5' X 5' to 120 degree(s) X 120 degree(s); and digital control methods by systems of space object light illumination with laser sources in specified spatial angles, discrete and continuous adjustments within scan sectors being made by slight mechanical removal of the scanning element by means of the principles of optical reduction.

A new method for solving Wigner-Liouville's type equations and studying dynamics of quantum particles in composite materials and disordered system of scatterers have been developed. The approach combines both molecular dynamics and Monte Carlo methods. Numerical results have been obtained for the frequency dependencies of the tensors of electron conductivity and permittivity according to quantum Kubo formulas and for the flux- position time correlation function characterizing the energy level structure and absorption spectra of electron in potential well. Possibility of applying the developed approach to the theory of classical wave propagation in random media have been also considered.

A quantum mechanical system is presented for which a multiple-valued quantum algebra and logic are derivable. The system is distinguished from previous quantum computational proposals by the definition of higher order quantum algebras and logics derived from multi-level quantum spin systems.

We consider the use of Electron-Nuclear Double Resonance (ENDOR) techniques in quantum computing. ENDOR resolution as a possible limiting factor is discussed. It is found that ENDOR and double-ENDOR techniques have sufficient resolution for quantum computing applications.

We sketch the so called convex, or operational, framework for statistical physical theories, based on the convex set formed by the states of the physical system. A notion of observable is introduced which encompasses the ones usually adopted in classical and in quantum mechanics. In this framework a structure of effect algebra naturally arises. We also discuss a notion of extension of a descriptive physical model, pointing at the possibility of constructing classical extension of quantum mechanics.