Abstract: We present an elementary proof of the quantum de Finetti representation theorem, a quantum analog of de Finetti’s classical theorem on exchangeable probability assignments. This contrasts with the original proof of Hudson and Moody [Z. Wahrschein. verw. Geb. 33, 343 (1976)], which relies on advanced mathematics and does not share the same potential for generalization. The classical de Finetti theorem provides an operational definition of the concept of an unknown probability in Bayesian probability theory, where probabilities are taken to be degrees of belief instead of objective states of nature. The quantum de Finetti theorem, in a closely analogous fashion, deals with exchangeable density-operator assignments and provides an operational definition of the concept of an “unknown quantum state” in quantum-state tomography. This result is especially important for information-based interpretations of quantum mechanics, where quantum states, like probabilities, are taken to be states of knowledge rather than...

Abstract: Characteristic Times in One-Dimensional Scattering.- The Time-Energy Uncertainty Relation.- Jump Time and Passage Time: The Duration ofs a Quantum Transition.- Bohm Trajectory Approach to Timing Electrons.- Decoherent Histories for Space-Time Domains.- Quantum Traversal Time, Path Integrals and "Superluminal" Tunnelling.- Quantum Clocks and Stopwatches.- The Local Larmor Clock, Partial Densities of States, and Mesoscopic Physics.- "Standard" Quantum-Mechanical Approach to Times of Arrival.- Experimental Issues in Quantum-Mechanical Time Measurement.- Microwave Experiments on Tunneling Time.- The Two-State Vector Formalism: An Updated Review.

Abstract: In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. In this paper, we show that, despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally, we give a Bayesian formulation of quantum-state tomography.

Abstract: In this paper, I try once again to cause some good-natured trouble The issue remains, when will we ever stop burdening the taxpayer with conferences devoted to the quantum foundations? The suspicion is expressed that no end will be in sight until a means is found to reduce quantum theory to two or three statements of crisp physical (rather than abstract, axiomatic) significance In this regard, no tool appears better calibrated for a direct assault than quantum information theory Far from a strained application of the latest fad to a time-honored problem, this method holds promise precisely because a large part--but not all--of the structure of quantum theory has always concerned information It is just that the physics community needs reminding This paper, though taking quant-ph/0106166 as its core, corrects one mistake and offers several observations beyond the previous version In particular, I identify one element of quantum mechanics that I would not label a subjective term in the theory--it is the integer parameter D traditionally ascribed to a quantum system via its Hilbert-space dimension

Abstract: Quantum trajectory theory, developed largely in the quantum optics community to describe open quantum systems subjected to continuous monitoring, has applications in many areas of quantum physics. I present a simple model, using two-level quantum systems (q-bits), to illustrate the essential physics of quantum trajectories and how different monitoring schemes correspond to different “unravelings” of a mixed state master equation. I also comment briefly on the relationship of the theory to the consistent histories formalism and to spontaneous collapse models.

Abstract: The concepts of complementarity and entanglement are considered with respect to their significance in and beyond physics. A formally generalized, weak version of quantum theory, more general than ordinary quantum theory of physical systems, is outlined and tentatively applied to two examples.

Abstract: We derive noncommutative multiparticle quantum mechanics from noncommutative quantum field theory in the nonrelativistic limit. Particles of opposite charges are found to have opposite noncommutativity. As a result, there is no noncommutative correction to the hydrogen atom spectrum at the tree level. We also comment on the obstacles to take noncommutative phenomenology seriously and propose a way to construct noncommutative SU(5) grand unified theory.

Abstract: Preface. 1. Standard and generalized formalisms of quantum mechanics. 2. Empiricist and realist interpretations of quantum mechanics. 3. Quantum mechanical description of measurement, and the 'measurement problem'. 4. The Copenhagen interpretation. 5. The Einstein-Podolsky-Rosen problem. 6. Individual-particle and ensemble interpretations of quantum mechanics. 7. Generalized quantum mechanics. 8. Applications of generalized quantum mechanics. 9. The Bell inequality in quantum mechanics. 10. Subquantum or hidden-variables theories. A: Mathematical appendix. Bibliography. Index.

Abstract: The application of the methods of quantum mechanics to game theory provides us with the ability to achieve results not otherwise possible. Both linear superpositions of actions and entanglement between the players' moves can be exploited. We provide an introduction to quantum game theory and review the current status of the subject.

Abstract: We derive a necessary and sufficient condition for a sequence of quantum measurements to achieve the optimal performance in quantum hypothesis testing. We discuss what quantum measurement we should perform in order to attain the optimal exponent of the second error probability under the condition that the first error probability goes to 0. As an asymptotically optimal measurement, we propose a projection measurement characterized by the irreducible representation theory of the special linear group SL(). Especially, in the spin-1/2 system, it is realized by the simultaneous measurement of the total momentum and a momentum of a specified direction. As a by-product, we obtain another proof of quantum Stein's lemma. In addition, an asymptotically optimal measurement is constructed in the quantum Gaussian case, and it is physically meaningful.

Abstract: An anomaly is said to occur when a symmetry that is valid classically becomes broken as a result of quantization. Although most manifestations of this phenomenon are in the context of quantum field theory, there are at least two cases in quantum mechanics—the two-dimensional delta function interaction and the 1/r2 potential. The former has been treated in this journal; in this article we discuss the physics of the latter together with experimental consequences.

Abstract: An attempt is made to formulate quantum mechanics (QM) in physical rather than in mathematical terms. It is argued that the appropriate conceptual framework for QM is `contextual objectivity', which includes an objective definition of the quantum state. This point of view sheds new light on topics such as the reduction postulate and the quantum measurement process.

Abstract: Quantum adiabatic evolution provides a general technique for the solution of combinatorial search problems on quantum computers. We present the results of a numerical study of a particular application of quantum adiabatic evolution, the problem of finding the largest clique in a random graph. An n-vertex random graph has each edge included with probability 1/2, and a clique is a completely connected subgraph. There is no known classical algorithm that finds the largest clique in a random graph with high probability and runs in a time polynomial in n. For the small graphs we are able to investigate (n ≤ 18), the quantum algorithm appears to require only a quadratic run time.

Abstract: We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the others is returned to the agent. If the rules of rationality are followed one obtains the peculiarities of quantum probability, the uncertainty relations and the EPR paradox among others. The consequences of this approach for hidden variables and quantum logic are analyzed.

Abstract: Nonequilibrium Dynamics.- Some Recent Advances in Classical Statistical Mechanics.- Deterministic Thermostats and Flctuation Relations.- What Is the Microscopic Response of a System Driven Far From Equilibrium?.- Non-equilibrium Statistical Mechanics of Classical and Quantum Systems.- Dynamics of Relaxation and Chaotic Behaviour.- Dynamical Theory of Relaxation in Classical and Quantum Systems.- Relaxation and Noise in Chaotic Systems.- Fractal Structures in the Phase Space of Simple Chaotic Systems with Transport.- Dynamical Semigroups.- Markov Semigroups and Their Applications.- Invitation to Quantum Dynamical Semigroups.- Finite Dissipative Quantum Systems.- Complete Positivity in Dissipative Quantum Dynamics.- Quantum Stochastic Dynamical Semigroup.- Driving, Dissipation and Control in Quantum Systems.- Driven Chaotic Mesoscopic Systems, Dissipation and Decoherence.- Quantum State Control in Cavity QED.- Solving Schrodinger's Equation for an Open System and Its Environment.- Dynamics of Large Systems.- Thermodynamic Behavior of Large Dynamical Systems.- Coherent and Dissipative Transport in Aperiodic Solids: An Overview.- Scaling Limits of Schrodinger Quantum Mechanics.

Abstract: We have presented a simple approach to quantum theory of Brownian motion and barrier crossing dynamics. Based on an initial coherent state representation of bath oscillators and an equilibrium canonical distribution of quantum-mechanical mean values of their co-ordinates and momenta we have derived a c number generalized quantum Langevin equation. The approach allows us to implement the method of classical non-Markovian Brownian motion to realize an exact generalized non-Markovian quantum Kramers' equation. The equation is valid for arbitrary temperature and friction. We have solved this equation in the spatial diffusion-limited regime to derive quantum Kramers' rate of barrier crossing and analyze its variation as a function of the temperature and friction. While almost all the earlier theories rest on quasiprobability distribution functions (e.g., Wigner function) and path integral methods, the present work is based on true probability distribution functions and is independent of path integral techniques. The theory is a natural extension of the classical theory to quantum domain and provides a unified description of thermally activated processes and tunneling.

Abstract: The usual formulation of quantum theory is rather abstract. In recent work I have shown that we can, nevertheless, obtain quantum theory from five reasonable axioms. Four of these axioms are obviously consistent with both classical probability theory and quantum theory. The remaining axiom requires that there exists a continuous reversible transformation between any two pure states. The requirement of continuity rules out classical probability theory. In this paper I will summarize the main points of this new approach. I will leave out the details of the proof that these axioms are equivalent to the usual formulation of quantum theory (for these see reference [1]).

Abstract: White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Ito type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Ito theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

Abstract: All current approaches to quantum gravity employ essentially standard quantum theory including, in particular, continuum quantities such as the real or complex numbers. However, I wish to argue that this may be fundamentally wrong in so far as the use of these continuum quantities in standard quantum theory can be traced back to certain {\em a priori} assumptions about the nature of space and time: assumptions that may be incompatible with the view of space and time adopted by a quantum gravity theory. My conjecture is that in, some yet to be determined sense, to each type of space-time there is associated a corresponding type of quantum theory in which continuum quantities do not necessarily appear, being replaced with structures that are appropriate to the specific space-time. Topos theory then arises as a possible tool for `gluing' together these different theories associated with the different space-times. As a concrete example of the use of topos ideas, I summarise recent work applying presheaf theory to the Kochen-Specher theorem and the assignment of values to physical quantities in a quantum theory.

Abstract: This talk is inspired by two previous ICM talks, by V.Drinfeld (1986) and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is that the quantum Yang-Baxter equation (QYBE), which is an important equation arising in quantum field theory and statistical mechanics, is best understood within the framework of Hopf algebras, or quantum groups. On the other hand, in Felder's talk, it is explained that another important equation of mathematical physics, the star-triangle relation, may (and should) be viewed as a generalization of QYBE, in which solutions depend on additional ``dynamical'' parameters. It is also explained there that to a solution of the quantum dynamical Yang-Baxter equation one may associate a kind of quantum group. These ideas gave rise to a vibrant new branch of ``quantum algebra'', which may be called the theory of dynamical quantum groups. My goal in this talk is to give a bird's eye review of some aspects of this theory and its applications.

Abstract: These notes give an introduction to some aspects of quantum Markov processes Quantum Markov processes come into play whenever a mathematical description of irreversible time behaviour of quantum systems is aimed at Indeed, there is hardly a book on quantum optics without having at least a chapter on quantum Markov processes However, it is not always easy to recognize the basic concepts of probability theory in families of creation and annihilation operators on Fock space Therefore, in these lecture notes much emphasis is put on explaining the intuition behind the mathematical machinery of classical and quantum probability The lectures start with describing how probabilistic intuition is cast into the mathematical language of classical probability (Sects 41-43) Later on, we show how this formulation can be extended such as to incorporate the Hilbert space formulation of quantum mechanics (Sects 44,45) Quantum Markov processes are constructed and discussed in Sects 46,47, and we add some further discussions and examples in Sects 48-411

Abstract: Any real interaction process produces many incompatible system versions, or realisations, giving rise to omnipresent dynamic randomness and universally defined complexity (arXiv:physics/9806002) Since quantum behaviour dynamically emerges as the lowest complexity level (arXiv:quant-ph/9902016), quantum interaction randomness can only be relatively strong, which reveals the causal origin of quantum indeterminacy (arXiv:quant-ph/9511037) and true quantum chaos (arXiv:quant-ph/9511035), but rigorously excludes the possibility of unitary quantum computation, even in an "ideal", noiseless system Any real computation is an internally chaotic (multivalued) process of system complexity development occurring in different regimes Unitary quantum machines, including their postulated "magic", cannot be realised as such because their dynamically single-valued scheme is incompatible with the irreducibly high dynamic randomness at quantum complexity levels and should be replaced by explicitly chaotic, intrinsically creative machines already realised in living organisms and providing their quite different, realistic kind of magic The related concepts of reality-based, complex-dynamical nanotechnology, biotechnology and intelligence are outlined, together with the ensuing change in research strategy The unreduced, dynamically multivalued solution to the many-body problem reveals the true, complex-dynamical basis of solid-state dynamics, including the origin and internal dynamics of macroscopic quantum states The critical, "end-of-science" state of unitary knowledge and the way to positive change are causally specified within the same, universal concept of complexity

Abstract: Quantum Finance represents the synthesis of the techniques of quantum theory (quantum mechanics and quantum field theory) to theoretical and applied finance. After a brief overview of the connection between these fields, we illustrate some of the methods of lattice simulations of path integrals for the pricing of options. The ideas are sketched out for simple models, such as the Black-Scholes model, where analytical and numerical results are compared. Application of the method to nonlinear systems is also briefly overviewed. More general models, for exotic or path-dependent options are discussed.

Abstract: Given a quantum mechanical observable and a state, one can construct a classical observable, that is, a real function on the configuration space, such that it is the optimal estimate of the quantum observable, in the sense of minimum variance. This optimal estimate turns out to be the quantum mechanical local value, which arises from several contexts such as de Broglie–Bohm's casual approach to quantum mechanics, instantaneous frequency in time–frequency analysis, Nelson's quantum fluctuations formalism, and phase-space approach to quantum mechanics. Accordingly, any observable can be decomposed into a local value part and a quantum fluctuation part, which are independent, both geometrically and statistically. Furthermore, the current density in quantum mechanics, the osmotic velocity in stochastic mechanics, and the Fisher information in classical statistical inference, arise naturally in connection with local value. In particular, Heisenberg uncertainty principle can be quantified more precisely by virtue of local value.

Abstract: A notion of quantum multipole (in particular, dipole) noise is considered. Quantum dipole noise is an analogue of quantum white noise but it acts in a Fock space with indefinite metric. Quantum {\it white} noise describes the leading term in the stochastic limit approximation to quantum dynamics while quantum {\it multipole} noise describes the corrections to the leading term. We obtain and study the generalized quantum stochastic equations describing corrections to the stochastic limit which include quantum dipole noise.