CMS-MITACS Joint Conference 2007

In quantum information theory entanglement arises due to the restriction to local operations and classical communication (LOCC). In particular, entanglement can be considered as a quantum resource with which spatially separated parties can overcome or at least partly overcome the limitation of LOCC. Clearly, different types of restrictions corresponds to different kinds of quantum resource theories (QRTs). In this talk I will discuss the QRTs that emanate from various natural constraints. I will focus on QRTs that follow from the presence of super-selection rules or the absence of shared reference frames. In particular, I will discuss the analogies and distinctions between and among the different QRTs and show that, in general, QRTs in many aspects are very similar to entanglement theory. Such comparisons provide a much broader perspective on all of these resource theories and allow us to use the insights gained from one QRT to solve the problems that arise in the context of another QRT.

Joint work with Rob Spekkens.

A reverse Shannon theorem characterizes the resources required to simulate a given noisy channel. The quantum reverse Shannon theorem for single-sender/single-receiver channels, a joint effort of Bennett, Devetak, Harrow, Shor and Winter, established that a single number can be used to characterize the strength of such channels in the presence of free entanglement. In this talk I'll explain how to prove an optimal reverse Shannon theorem for quantum channels with a single sender but many receivers, known as broadcast channels. Surprisingly, the simulation cost for a broadcast channel can be characterized by a simple, tractable optimization problem even though no such simple solution has been found for the capacity region itself, even in the classical case.

Joint work with Frederic Dupuis.

We will discuss a construction based on quantum list-codes, and its application to adversarial quantum channels.

Joint work with Graeme Smith.

We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases, being equivalent whenever the design is composed of d+1 bases in dimension d. We show that, for the purpose of quantum state determination, these designs specify an optimal collection of orthogonal measurements. Using highly nonlinear functions on abelian groups, we construct explicit examples from d+2 orthonormal bases whenever d+1 is a prime power, covering dimensions d+d=6, 10, and 12, for example, where no complete sets of mutually unbiased bases have thus far been found.

This is joint work with Andrew Scott.

Quantum key distribution must overcome two important hurdles: authentication to avoid the man-in-the-middle attack and relays or repeaters to allow long-distance communication. Current feasible approaches suggest complete trust of intermediate nodes in a network. We show that, in a network of partially trusted nodes (even with a low level of trust), our scheme enables probabilistic information-theoretic secure authentication and long-distance key distribution based on existing quantum key distribution technology, thus making our approach feasible now without reliance on total trust of intermediate nodes.