Shor and Preskill's and Mayers's security proof for the BB84 quantum key distribution protocol

We review two security proofs for the BB84 quantum key distribution protocol: Mayers's security proof and the more recent proof of Shor and Preskill. We focus on the basic principles and the intuition in Mayers's proof instead of technical details. We present a variation on Shor's and Preskill's proof which is convenient for purpose of comparison. We explain the connection between these two proofs. Received 14 July 2001     

Measurement of Magnetic Properties of an Inertial Sensor with a Torsion Balance

A torsion balance is constructed to accurately measure the magnetic properties of the real proof mass of the inertial sensor. The magnetic susceptibility and remnant moment of a titanium proof mass are measured in the first experiment, and further improvements and the potential sensitivity are also discussed. This is significant for choosing the material of the proof mass and investigating the magnetic disturbances acting on the proof mass.     

Practical scheme for quantum computation with any two-qubit entangling gate

Which gates are universal for quantum computation? Although it is well known that certain gates on two-level quantum systems (qubits), such as the controlled-not, are universal when assisted by arbitrary one-qubit gates, it has only recently become clear precisely what class of two-qubit gates is universal in this sense. We present an elementary proof that any entangling two-qubit gate is universal for quantum computation, when assisted by one-qubit gates. A proof of this result for systems of arbitrary finite dimension has been provided by Brylinski and Brylinski; however, their proof relies on a long argument using advanced mathematics. In contrast, our proof provides a simple constructive procedure which is close to optimal and experimentally practical.     

Geometric methods in multiparticle quantum systems

Technically simple proofs are given of the HVZ theorem on the bottom of the essential spectrum of multiparticle systems and of Combes' result on completeness below the lowest three body threshold. The first proof is a variant of a proof of Enss and a decendent of Zhislin's original proof. Finally, we apply our methods to the bound state spectrum.On leave from Princeton University. Research support in part by U.S. National Science Foundation under grants MPS-75-11864 and MPS-75-20638     

Energy and Volume: A Proof of the Positivity of ADM Energy Using the Yamabe Invariant of Three-Manifolds

We give a new proof of the positivity (non-negativity) of ADM energy¹ using the Yamabe invariant of three-manifolds. From a physical point of view, the new proof is motivated by a formula (explicitly non-negative) for the total ADM energy of emerging (asymptotically flat) stationary solutions on maximally expanding compact cosmologies. Mathematically, the proof is an application of the Thurston Geometrization of three-manifolds. ¹ Properly speaking, we give a new proof of the Riemannian positive energy Theorem. Namely, we prove that an asymptotically flat Riemannian three-manifold with non-negative scalar curvature cannot have negative mass.     

Determinism,realism, and Stapp's 1985 proof of nonlocality

Two recent criticisms of Stapp's 1985 proof of an incompatibility between locality and quantum mechanics are clarified. One of these criticisms (charging that the proof requires determinism) is supported by a detailed logical analysis of the locality conditions in the proof. The other criticism (charging that the proof assumes realism) is shown to be based upon a misinterpretation of the rôle of counterfactuals in the proof. Further, Stapp's own replies to these criticisms are shown to be inadequate.     

The positivity of gravitational energy and global supersymmetry

The concept of gravitational energy and the proof of its positivity are reviewed. The relationship between Witten's proof of the positivity of mass and supergravity is explained with reference to the group of global supersymmetries of a spacetime. A formula for the mass is given, in terms of the change of the supercharge under global supersymmetry, which has a simple positivity theorem and which reduces to Witten's expression. An interpretation of Witten's constraint on the spinors used in his proof is given.     

The Shannon-McMillan Theorem for AF C*-systems

We give a new proof of quantum Shannon-McMillan theorem, extending it to AF C*-systems. Our proof is based on the variational principle, instead of the classical Shannon-McMillan theorem.     

A simple proof of the perturbative uniqueness of the Kerr solution

We present a mathematically simple and coordinate-free proof, based on the GHP spin-coefficient formalism, that a small stationary perturbation of a Kerr solution results in a new Kerr solution. Thus, under physically reasonable assumptions, we give a new and simple proof that a stationary black hole is given by a Kerr solution.     

On stochastic Einstein locality in algebraic relativistic quantum field theory

This paper assesses Miklós Rédei's [1991] proof of the proposition that algebraic relativistic quantum field theory is stochastic Einstein local. The conclusion is that either Rédei's proof is spurious, in that it does not really prove what it intends to establish, or that the proof is fallacious. The paper is self-contained in the sense that the few ingredients of algebraic quantum theory that go into Rédei's proof are first summed up. Then Hellman's definition of stochastic Einstein locality is discussed, a detailed exposition is offered of Rédei's proof, and finally the author's refutation is explicated.     

An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction

Using fixed point theorems for local contractions in Banach spaces, an existence and uniqueness proof for the Hartree-Fock time-dependent problem is given in the case of a finite Fermi system interacting via a bounded two-body potential. The existence proof for the “strong” solution of the evolution problem is obtained under suitable conditions on the initial state.     

Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model

A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.     

A new proof for the convergent iterative solution of the degenerate quantum double-well potential and its generalization

We present a new and simpler proof for the convergent iterative solution of the one-dimensional degenerate double-well potential. This new proof depends on a general theorem, called the hierarchy theorem, that shows the successive stages in the iteration to form a monotonically increasing sequence of approximations to the energy and to the wavefunction at any point x. This important property makes possible a much simpler proof of convergence than the one given before in the literature. The hierarchy theorem proven in this paper is applicable to a much wider class of potentials which includes the quartic potential.     

On the BKP Hierarchy: Additional Symmetries,Fay Identity and Adler–Shiota–van Moerbeke Formula

We give an alternative proof of the Adler–Shiota–van Moerbeke formula for the BKP hierarchy. The proof is based on a simple expression for the generator of additional symmetries and the Fay identity of the BKP hierarchy.      

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. Received: 29 January 2001 / Accepted: 8 March 2001     

Carathéodory's principle and the existence of global integrating factors

A proof is given of a theorem on the integrability of Pfaffian forms which is used in Carathéodory's approach to thermodynamics. It is pointed out that Carathéodory's original proof of the existence of entropy and of absolute temperature is incomplete, since it fails to take into account the local nature of this theorem. By combining the theorem with the results ofBuchdahl andGreve on the existence of continuous empirical entropy functions, it is shown that the First and Second Laws of Thermodynamics imply the existence of a globally defined differentiable empirical entropy function for every simple thermodynamic system. This result supplies the missing step in Carathéodory's argument and makes a separate proof of the principle of increase of entropy unnecessary.     

Multiple-conjugation expansion in distortion elimination by phase conjugation

The multiple-conjugation expansion results from a method of phase-conjugate scattering analysis that is convenient in connection with strong aberrators and weak phase conjugators. Based on this expansion, a new proof is obtained of a recently established theorem according to which a complete cancellation of distortion is possible in the absence of gains or losses. The proof makes use of the reciprocity property and of the generalized optical theorem of potential scattering.     

A Proof of the Exponentially Small Transversality of the Separatrices for the Standard Map

In 1984 V. F. Lazutkin [Laz84, LST89] obtained an asymptotic formula for the separatrix splitting angle for the standard map. The difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter. Lazutkin's proof was based on two conjectures. Probably, the original form of those conjectures was incorrect, but Lazutkin's method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices. The consequent works [Laz91, Laz92, GLS94] and [Gel96] prepared the base to fill all the gaps of the original proof. The present paper contains a complete and self-contained proof of a refined version of the original formula (formula (1.7) of the present paper). In this form the formula was obtained in [GLS94]. The proof is inspired by the ideas of Lazutkin's original paper [Laz84]. Received: 16 February 1998 / Accepted: 20 August 1998     

Generalized Boundary Conditions in an Existence and Uniqueness Proof for the Solution of the Non-Stationary Electron Boltzmann Equation by Means of Operator-Semigroups

Recently, based on the semigroup approach a new proof was presented of the existence of a unique solution of the non-stationary Boltzmann equation for the electron component of a collision dominated plasma. The proof underlies some restrictions which should be overcome to extend the validity range to other problems of physical interest. One of the restrictions is the boundary condition applied. The choice of the boundary condition is essential for the proof because it determines the range of definition of the infinitesimal generator and thus the operator semigroup itself. The paper proves the existence of a unique solution for generalized boundary conditions, this solution takes non-negative values, which is necessary for a distribution function from the physical point of view.     

BPHZ renormalization of λφ4 field theory in curved spacetime

A proof is given to all orders in perturbation theory of the renormalizability of λφ⁴ field theory in curved spacetime. The proof is based on the BPHZ definition of a renormalized Feynman integrand and uses dimensional regularization to ensure that products of Feynman propagators are well-defined distributions. The explicit structure of the pole terms in the Feynman integrand is obtained using a local momentum space representation of the Feynman propagator and is shown to be of a form which can be cancelled by counterterms in the scalar field Lagrangian. The proof given is, technically, only valid for metrics which have been analytically continued to Euclidean (++++) signature.