Quantum Physics

The modern view of representing a quantum observable as a semispectral measure as opposed to the traditional approach of using only spectral measures has added a great deal to our understanding of the mathematical structures and conceptual foundations of quantum mechanics. The old questions of 1) how to determine a quantum observable from its classical counter-part (if any), 2) how much statistical information is needed to determine an observable, 3) which observables can be measured together, and 4) are there noiseless measurements, all appear in a new perspective, calling for a study of problems such as: 1) how to obtain a semispectral measure by a quantization map, 2) do the moment operators of a semispectral measure determine the operator measure, 3) are coexistent observables jointly measurable, and 4) does minimal variance occur only in the case of a spectral measure? In my talk I will survey some of the recent developments concerning these questions and problems.

Adiabatic Quantum Computation is not only a possibly more robust alternative to standard quantum computation. Since it considers a continuous-time evolution of the system, it also provides a natural bridge towards studying the dynamics of interacting many-particle quantum systems, quantum phase transitions and other issues in fundamental physics. After a brief review of adiabatic quantum computation, I will show our recent results on the dynamics of entanglement and fidelity for the search and Deutsch algorithms including several variations and optimization. I will show how these studies led to suggesting an alternative definition of entanglement and compare the results, and discuss possible implications for considering entanglement a resource. I will conclude with an outlook on further applications and extensions of adiabatic quantum computation.

Category theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, many counterintuitive features of quantum theory turn out to be properties shared by the category of Hilbert spaces and the category of cobordisms, in which objects are choices of "space" and emorphisms are choices of "spacetime". This striking fact suggests that "n-categories with duals" are a promising language for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, 3d quantum gravity coupled to point particles, and 4d BF theory coupled to strings.

Clifton, Bub, and Halvorson claim to be able to derive quantum mechanics from information-theoretic axioms. However, their derivation relies on the auxiliary assumption that the relevant probabilities for measurement outcomes can be represented by the observables (self-adjoint operators) and states of a C*-algebra. There are legitimate probability theories that are not so representable --- in particular, the nonlocal boxes of Popescu and Rohrlich. We explain the impact of nonlocal boxes on the interpretation of the CBH derivation, and we discuss possible generalizations of the CBH derivation in the framework of these more general probability theories.

This talk is concerned with the noise-insensitive transmission of quantum information. For this purpose, the sender incorporates redundancy by mapping a given initial quantum state to a messenger state on a larger-dimensional Hilbert space. This encoding scheme allows the receiver to recover part of the initial information if the messenger system is corrupted by interaction with its environment. Our noise model for the transmission leaves a part of the quantum information unchanged, that is, we assume the presence of a noiseless subsystem or of a decoherence-free subspace. We address the case when the noiseless component cannot contain all the quantum information to be transmitted, and investigate how to best spread the information in a quantum state across the noise-susceptible components. (Joint work with David Kribs and Vern Paulsen.)

We will look at the axioms of quantum mechanics as expressed, for example, in the book by M. A. Nielsen and I. L. Chung ("Quantum Computation and Quantum Information"). We then take a critical look at these axioms, raising several questions as we go. In particular, we will look at the possible informational completeness property of the family of operators that we measure. We will propose physical solutions based on the results of quantum mechanics on phase space and the measurement of quantum particles by quantum mechanical means. We illustrate this with both momentum-position measurements and spin measurements.

Inspired by the notion that the differences between quantum theory and classical physics are best expressed in terms of information theory, Hardy (2001) and Clifton, Bub, and Halvorson (2003) have constructed frameworks general enough to embrace both quantum and classical physics, within which one can invoke principles that distinguish the classical from the quantum. Independently of any view that quantum theory is essentially about quantum information, such frameworks provide a useful tool for exploring the differences between classical and quantum physics, and the relations between the various properties of quantum mechanics that distinguish it from the classical. In particular, we can ask: on which features of quantum physics do our familiar possibility/impossibility theorems depend? It turns out that it is possible to extend the no-cloning theorem and other results, such as the Holevo bound on acquisition of information by a single measurement, beyond the quantum setting.

Entanglement entropy is currently of interest in several areas in physics, such as condensed matter, field theory, and quantum information. One of the most interesting properties of the entanglement entropy is its scaling behavior, especially close to phase transitions. It was believed that for dimensions higher than 1 the entropy scales like surface area of the subsystem. We will describe a recent result for free fermions at zero temperature, where the entropy in fact scales faster. The latter problem will be related to a mathematical conjecture due to H. Widom (1982). This is a joint work with I. Klich.

While modern theories lavishly invoke several spatial dimensions within models that seek to unify relativity theory and quantum mechanics, none seems to consider the possibility that a yet-unfamiliar aspect of time may do the work. I introduce the notion of Becoming and then consider its consequences for physical theory. Becoming portrays a possible aspect of time that is "curled" very much like the extra spatial dimensions in superstring theories. Within the resulting picture of spacetime, some fundamental aspects of quantum mechanics, special and general relativity, thermodynamics and modern cosmology fit in very naturally. The proposed model is not yet a scientific theory as it still lacks a rigorous formalism and experimental predictions, yet it points out an entire family of possible theories that merit serious consideration.

Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In addition to being a potentially useful tool in the study of standard oracles, Hamiltonian oracles naturally introduce the concept of fractional queries and are amenable to study using techniques of differential equations and geometry. As an example of these ideas we shall examine the Hamiltonian oracle corresponding to the problem of oracle interrogation. This talk is intended for all those who wish to apply their knowledge of differential geometry without the risk of creating an event horizon.

Familiar textbook quantum mechanics assumes a fixed background spacetime to define states on spacelike surfaces and their unitary evolution between them. Quantum theory has been generalized as our conceptions of space and time have evolved. But quantum mechanics needs to be generalized further for quantum gravity where spacetime geometry is fluctuating and without definite value. This talk will review a fully four-dimensional, sum-over-histories, generalized quantum mechanics of cosmological spacetime geometry. In this generalization, states of fields on spacelike surfaces and their unitary evolution are emergent properties appropriate when spacetime geometry behaves approximately classically. The principles of generalized quantum theory would allow for further generalization that would be necessary were spacetime not fundamental. Emergent spacetime phenomena are discussed in general and illustrated with the examples of the classical spacetime geometries with large spacelike surfaces that emerge from the `no-boundary' wave function of the universe. These must be Lorentzian with one, and only one, time direction. The question will be raised as to whether quantum mechanics itself is emergent.