Quantum Physics

In this talk I will present several new results from joint work with Dmitry Gavinsky, Oded Regev and Ronald de Wolf, relating to the model of one-way communication and the simultaneous model of communication. I will describe several separations between various resources (entanglement versus event coin, quantum communication versus classical communication), showing in particular that quantum communication cannot simulate a public coin and that entanglement can be much more powerful than a public coin, even if communication is quantum. I will also present a characterization of the quantum fingerprinting technique.

How should we think about quantum computing? The usual answer to this question is based on ideas inspired by computer science, such as qubits, quantum gates, and quantum circuits. In this talk I will explain an alternate geometric approach to quantum computation. In the geometric approach, an optimal quantum computation corresponds to "free falling" along the minimal geodesics of a certain Riemannian manifold. This reformulation opens up the possibility of using tools from geometry to understand the strengths and weaknesses of quantum computation, and perhaps to understand what makes certain physical operations difficult (or easy) to synthesize.

In order to predict the future state of a quantum system, we generally do not need to know the past state of the entire universe, but only the state of a finite neighborhood of the system. This locality is best expressed as a restriction on how information "flows" between systems. In this talk I will describe some recent work, inspired by quantum cellular automata, about the information strucutre of local quantum dynamics. Issues to be discussed include the definition of "locality", some characterization theorems, connections between classical and quantum locality for reversible maps, the relation between local and global dynamics, and the dissection of CNOT.

In this talk, I will outline a quantum generalization of causal networks that are used to analyze complex probabilistic inference problems involving large numbers of correlated random variables. I will review the framework of classical causal networks and the graph theoretical constructions that are abstracted from them, including entailed conditional independence, d-separation and Markov equivalence. I will show how to generalize the definition of causal networks to the quantum case, such that the same graph theoretic constructions apply, and give an explicit representation of the states supported on the graph as the Gibbs states of certain classes of Hamiltonians.

Not only general relativity but also quantum theory plays important roles in current cosmology. Quantum fluctuations of matter fields are supposed to have provided the initial seeds of all the structure of the current universe, and quantum gravity is assumed to have been essential in the earliest stages. Both issues are not fully understood, although several heuristic effects have been discussed. In this talk, implications of an effective framework taking into account the coupling of matter and gravity are discussed. This touches on interpretational issues of quantum mechanics, cosmological observations and properties of quantum gravity.

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.