Representations of the Cuntz algebras, or the Cuntz relations, play a key role in analysis/synthesis filters in signal processing, both for transmission of speech and of images. They are used in compression in wavelet algorithms, and, at the same time in quantum programs. While the factoring algorithm of P. Shor, and the search algorithm of L. Grover are the known quantum algorithms which are closest to being "practical", and at the same time in showing dramatic speedup compared to the corresponding classical algorithms, there are others, and the wavelet algorithm is one. In the talk we will compare the wavelet algorithms in the two cases, classical and quantum. The role played by quantum error-correction codes will be touched on.

Maxim Raginsky Quantum operations, Radon-Nikodym and all that

Given two completely positive (CP) maps, a theorem of Radon-Nikodym type gives necessary and sufficient conditions for their difference to be CP as well. I will discuss the significance and applications of this Radon-Nikodym theorem, as well as some of its consequences, such as the Lebesgue decomposition of one unital CP map with respect to another, in the context of quantum information theory.

Jody Trout Asymptotic spectral measures: between quantum theory and C*-algebra E-theory

We will discuss the relationship between positive operator-valued (POV) measures in quantum measurement theory and asymptotic morphisms in the C*-algebra E-theory of Connes and Higson. The theory of "asymptotically projective" POV-measures, introduced by Martinez-Trout (CMP 226), is integrally related to positive asymptotic morphisms on locally compact spaces via an asymptotic Riesz Representation Theorem. Examples and applications from quantum physics, including quantum noise models and semiclassical limits, will also be presented.

Hans Halvorson A no bit commitment theorem for infinite quantum systems

The Hughston-Jozsa-Wootters theorem shows that any finite ensemble of quantum states can be prepared "at a distance", and it has been used to show the insecurity of bit commitment protocols based on finite quantum systems without superselection rules. We sketch the proof of a generalized HJW theorem for arbitrary measures on the state space of a hyperfinite von Neumann algebra, and we discuss the significance of this result for generalized bit commitment protocols.

Søren Eilers C*-algebras associated to shift spaces

Shift spaces are dynamical systems which may be used, among other things, to model various coding problems. Two different constructions allow the association of shift spaces with operator algebras in an invariant way, and this in turn allows the application of the quite advanced structure theory of operator algebras to such spaces. Notably, K-theory for operator algebras leads to invariants for the shift spaces.

I will emphasize a recent instance of this modus operandi to shift spaces associated to substitutions such as 0->1, 1->01, carried out by myself and Toke M. Carlsen, but will also try to give an overview of the present state of this area of work in progress.

David Kribs Quantum error correction and Young tableaux

In this talk I will discuss some of my recent work on problems motivated by experimental efforts towards physically realizing quantum computation.

Feng Xu Solitons in affine and permutation orbifolds (joint work with Roberto Longo and Victor Kac)

In this talk we will describe our recent work on a class of orbifolds of conformal field theories in the von Neumann algebraic framework.

Greg Kuperberg What is quantum memory?

Shannon's information theory provides a rigorous framework for information in non-quantum computers and communication channels. One important result in this theory is that statistical information, memory, and channel capacity can all be measured by a single unit, e.g., bits. In the familiar purely quantum model, bits are replaced by qubits, which again suffice as the fundamental unit of both entropy and memory capacity. But it is known that quantum channel capacity is not characterized by a single unit and its characterization is probably intractible.

I will discuss a mutual generalization of bits and qubits, hybrid quantum memory, defined an arbitrary finite-dimensional C*-algebra. Unlike purely classical or purely quantum memory, hybrid quantum memory is a container with a shape. But unlike a general quantum channel, its capacity can be computed. Various basic properties of hydrid quantum memory can be established with the aid of two Choi-Effros theorems, Bratteli diagrams, bin packing, the Cramer-Chernoff estimate from probability theory, and a hybrid quantum pigeonhole principle.

See: arXiv:quant-ph/0203105

Marc Rieffel Distances between states

I will discuss a variety of ways to define the distance between states of a C*-algebra. These will apply in particular to matrix algebras such as occur in quantum computing.

Eleanor Rieffel Why does it work?

A central question in quantum information processing is "Why does it work?" A common answer is "quantum entanglement." But which types of quantum entanglement, and for which tasks. The classification and quantification of various types of multipartite entanglement remain only poorly understood. Quantum measurement also contributes to quantum information processing, but its role has been de-emphasized in the traditional quantum circuit model. We will survey alternative quantum computational models including measurement-based cluster-state quantum computing and adiabatic quantum computing, and discuss how they differ strongly from each other and from the standard circuit model in their use of quantum transformations, entanglement, and measurement.

Masamichi Takesaki Conditional expectations

In this talk, after quick review of conditional expectations in the classical probability theory framework, I'll discuss how the concept of conditional expectations is adapted in the theory of operator algebras, which is regarded as non-commutative probability theory needed to describe quantum phonomenon and its use in the analysis of the algebraic structure of von Neumann algebras.

Stanislaw Szarek Separability of quantum states and geometry of Banach spaces

We connect separability of quantum states to "standard" notions and "standard" methods of asymptotic geometric analysis and geometry of Banach spaces. In particular, we investigate the effective radii (in the sense of volume) of various sets of states.

Ed Effros Some aspects of the Fock spaces

Several properties of the theory of Gaussian spaces will be considered in the more general context of free and q-random variables.