The lagrangian approach to stochastic variational principles on curved manifolds

The classical definition of the action functional, for a dynamical system on curved manifolds, can be extended to the case of diffusion processes. For the stochastic action functional so obtained, we introduce variational principles of the type proposed by Morato. In order to generalize the class of process variations, from the flat case originally given by Morato to general curved manifolds, we introduce the notion of stochastic differential systems. These give a synthetic characterization of the process and its variations as a generalized controlled stochastic process on the tangent bundle of the manifold. The resulting programming equations are equivalent to the quantum Schrödinger equation, where the wave function is coupled to an additional vector potential, satisfying a plasma-like equation with a peculiar dissipative behavior.     

A probabilistic framework for the continuous observation of a quantum process

This paper treats quantum measurement within von Neumann's abstract framework. Specifically, observation is defined as a fixed self-adjoint operator with countable spectrum and nondegenerate eigenstates. Suppose scenarios for the observation of a quantum process over time are expanded by adding extra observations at time points interspersed among those of a previous scenario. If each observation leads to a mixture of eigenstates rather than a pure state, then the naturally defined joint probability measures on observed results are not consistent as scenarios vary. Nevertheless, we characterize the limiting subprobability measure when the times of observation become infinitely dense in any finite interval. This limiting measure corresponds to a continuous-time sub-stochastic process which decays with exponential rate out of any initial state and never reappears in any other state. Thus the process loses probability exponentially over time, and this loss occurs equally fast in the case of nonselective observation as for selective observation.Previous treatments of this problem have concentrated on the special case when Zeno's Paradox is in force, i.e. the rate of decay out of any state is zero and the process is immobilized by continuous observation. This situation exists, for instance, when the initial state is in the domain of the generator for the unitary group underlying the quantum process.     

Noiseless Subsystems for Collective Rotation Channels in Quantum Information Theory

Collective rotation channels are a fundamental class of channels in quantum computing and quantum information theory. The commutant of the noise operators for such a channel is a C*-algebra which is equal to the set of fixed points for the channel. Finding the precise spatial structure of the commutant algebra for a set of noise operators associated with a channel is a core problem in quantum error prevention. We draw on methods of operator algebras, quantum mechanics and combinatorics to explicitly determine the structure of the commutant for the class of collective rotation channels.     

Macroscopic current induced boundary conditions for Schrödinger-type operators

We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schrödinger-type operator, the spectral properties of which will be investigated.     

Hybrid unified theory

We shall investigate, in this paper, theories related to mind, consciousness, life, evolution, existence, non-existence, cosmology, quantum fields, quantum jump, Big Bang, and higher reality. A hybrid unification between ancient and modern discoveries is discussed with a view to shedding some light on the existing theories and strengthening them.     

Tensor product of quaternion hilbert modules

One of the main problems in the theory of quaternion quantum mechanics has been the construction of a tensor product of quaternion Hilbert modules. A solution to this problem is given by studying the tensor product of quaternion algebras (over the reals) and some of its quotient modules. Real, complex, and (covariant) quaternion scalar products are found in the tensor product spaces. Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The gauge transformations of a tensor product vector and the gauge fields are studied.On Sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

On the relation between the singular and the weak coupling limits

After having recalled some definitions concerning quantum stochastic processes and, in particular, quantum Brownian motions, a general scheme is introduced which allows a unified approach to the weak coupling and singular coupling limits. The analogies and differences between the two are discussed. The main difference consists of the fact that, in the singular coupling limit, the use of a Hamiltonian unbounded below seems to be unavoidable, while this is not the case for the weak coupling limit.     

Quantum Mechanical Algorithms for the Nonabelian Hidden Subgroup Problem

We provide positive and negative results concerning the standard method of identifying a hidden subgroup of a nonabelian group using a quantum computer.* Supported in part by NSF grants CCR-9820931 and CCR-0208929. Supported in part by NSF CAREER grant 0049092, the NSF Institute for Quantum Information, the Charles Lee Powell Foundation, and the Mathematical Sciences Research Institute. Supported in part by an NSF Mathematical Sciences Postdoctoral Research Fellowship and NSF grant DMS-0301320.§ Supported in part by DARPA QUIST grant F30602-01-2-0524, ARO grant DAAD19-03-1-0082, and NSF ITR grant CCR-0121555.     

Projection operators and states in the tensor product of quaternion hilbert modules

Following the construction of tensor product spaces of quaternion Hilbert modules in our previous paper, we define the analogue of a ray (in a complex quantum mechanics) and the corresponding projection operator, and through these the notion of a state and density operators. We find that there is a one-to-one correspondence between a state and an equivalence class of vectors from the tensor product space, which gives us another method to define the gauge transformations.On sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

Error Correcting Codes, Block Designs, Perfect Secrecy and Finite Fields

The ancient difficulty for establishing a common cryptographic secret key between two communicating parties Alice and Bob is nicely summarized by the Catch-22 dictum of S.J. Lomonaco [1999], to wit: “in order to communicate in secret one must first communicate in secret”. In other words, to communicate in secret, Alice and Bob must already have a shared secret key. In this paper we analyse an algorithm for establishing such a common secret key by public discussion, under the modest and practical requirement that Alice and Bob are initially in possession of keys and , respectively, of a common length which are not necessarily equal but are such that the mutual information is non-zero. This assumption is tantamount to assuming only that the corresponding statistical variables are correlated. The common secret key distilled by the algorithm will enjoy perfect secrecy in the sense of Shannon. The method thus provides a profound generalization of traditional symmetric key cryptography and applies also to quantum cryptography. Here, by purely elementary methods, we give a rigorous proof that the method proposed by Bennett, Bessette, Brassard, Salvail, and Smolin will in general converge to a non-empty common key under moderate assumptions on the choice of block lengths provided the initial bit strings are sufficiently long. Full details on the length requirements are presented. Furthermore, we consider the question of which block lengths should be chosen for optimal performance with respect to the length of the resulting common key. A new and fundamental aspect of this paper is the explicit utilization of finite fields and error-correcting codes both for checking equality of the generated keys and, later, for the construction of various hash functions. Traditionally this check has been done by performing a few times a comparison of the parity of a random subset of the bits. Here we give a much more efficient procedure by using the powerful methods of error-correcting codes. More general situations are treated in Section 8.The research of the first and second authors is supported by grants from NSERC.     

COMPARATIVE PROBABILITY ON THE C* ALGEBRA OF COMPACT OPERATORS

Abstract

Let A be a von Neumann algebra on a Hilbert space H and let P(A) denote the projections of A. A comparative probability (CP) on A (or more correctly on P(A)) is a preorder ? on P(A) satisfying:

0 ? P ? P ε P(A) with Q ≠ 0 for some Q ε P(A).

If P, Q ε P(A) then either P ? Q or Q ? P.

If P, Q and R are all in P(A) and P⊥R, Q⊥R, then P ? Q ? P + R ? Q + R.

Let τ be any of the usual locally convex topologies on A. We say ? is τ continuous if the interval topology induced on P(A) by ? is weaker than the τ topology on P(A). If μ an additive (completely additive) measure on P(A) then μ induces a uniformly (weakly) continuous CP ?μ on P(A) given by P ?_μ Q if μ(P) ? μ(Q). We show that if A is the C* algebra C(H) of compact operators on an infinite dimensional Hilbert space H, the converse is true under an extra boundedness condition on the CP which is automatically satisfied whenever the identity is present in A = P(C(H)).     

On dissipative stochastic equations in a Hilbert space

Summary A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.     

Classification of cyclic braces

Etingof, Schedler, and Soloviev have shown [P. Etingof, T. Schedler, A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999) 169-209] that T-structures on cyclic groups come from bijective 1-cocycles and thus give rise to solutions of the quantum Yang-Baxter equation. At the end of their paper, they ask for a classification of T-structures on cyclic groups, especially p-groups. We solve the latter problem by means of generalized radical rings (=braces).     

Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.     

Fisher information and kinetic energy functionals: A dequantization approach

We strengthen the connection between information theory and quantum-mechanical systems using a recently developed dequantization procedure whereby quantum fluctuations latent in the quantum momentum are suppressed. The dequantization procedure results in a decomposition of the quantum kinetic energy as the sum of a classical term and a purely quantum term. The purely quantum term, which results from the quantum fluctuations, is essentially identical to the Fisher information. The classical term is complementary to the Fisher information and, in this sense, it plays a role analogous to that of the Shannon entropy. We demonstrate the kinetic energy decomposition for both stationary and nonstationary states and employ it to shed light on the nature of kinetic energy functionals.     

Poly-Locality in Quantum Computing

A polynomial depth quantum circuit affects, by definition, a poly-local unitary transformation of a tensor product state space. It is a reasonable belief [Fe], [L], [FKW] that, at a fine scale, these are precisely the transformations which will be available from physics to solve computational problems. The poly-locality of a discrete Fourier transform on cyclic groups is at the heart of Shor's factoring algorithm. We describe a class of poly-local transformations, which include the discrete orthogonal wavelet transforms, in the hope that these may be helpful in constructing new quantum algorithms. We also observe that even a rather mild violation of poly-locality leads to a model without one-way functions, giving further evidence that poly-locality is an essential concept. March 1, 2000. Final version received: October 23, 2001.     

Derivation of Hartreeʼs theory for generic mean-field Bose systems

In this paper we provide a novel strategy to prove the validity of Hartree?s theory for the ground state energy of bosonic quantum systems in the mean-field regime. For the known case of trapped Bose gases, this can be shown using the strong quantum de Finetti theorem, which gives the structure of infinite hierarchies of k-particles density matrices. Here we deal with the case where some particles are allowed to escape to infinity, leading to a lack of compactness. Our approach is based on two ingredients: (1) a weak version of the quantum de Finetti theorem, and (2) geometric techniques for many-body systems. Our strategy does not rely on any special property of the interaction between the particles. In particular, our results cover those of Benguria–Lieb and Lieb–Yau for, respectively, bosonic atoms and boson stars.     

Power-law bounds on transfer matrices and quantum dynamics in one dimension-II

We establish quantum dynamical lower bounds for a number of discrete one-dimensional Schrödinger operators. These dynamical bounds are derived from power-law upper bounds on the norms of transfer matrices. We develop further the approach from part I and study many examples. Particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power-law bounds on transfer matrices. The models discussed in this paper include substitution models, Sturmian models, a hierarchical model, the prime model, and a class of moderately sparse potentials.     

Quantum and non-causal stochastic calculus

Summary The quantum stochastic calculus initiated by Hudson and Parthasarathy, and the non-causal stochastic calculus originating with the papers of Hitsuda and Skorohod, are two potent extensions of the Itô calculus, currently enjoying intensive development. The former provides a quantum probabilistic extension of Schrödinger's equation, enabling the construction of a Markov process for a quantum dynamical semigroup. The latter allows the treatment of stochastic differential equations which involve terms which anticipate the future. In this paper the close relationship between these theories is displayed, and a noncausal quantum stochastic calculus, already in demand from physics, is described.