Thematic Program on Mathematics in Quantum
Distinguished Lecture Series
August 17, 18, 20, 2009 -- 4:00 pm
Room 230, Fields Institute
Matthew B. Hastings
(Microsoft Research Station Q)
Communicating over Quantum Channels
Information theory started with the problem of sending information
over a noisy communication channel. In this talk, I will explain
to a general audience the type of tasks one would like to accomplish
with a quantum communication channel, either communicating classical
or quantum information and using inputs and measurements that are
either entangled or not, and I will use entropy differences to quantify
the rate at which information can be sent. After motivating the
mathematical problems in terms of communication tasks, I will describe
recent results in channel capacity, focusing on those aspects that
can be understood in terms of probability theory.
The Computational Complexity of Ground States of Quantum Systems
The Computational Complexity of Time Evolution of Quantum
A typical problem in physics is to describe the ground state or
time evolution of a system of N particles, each of which has 2 different
states. The Hilbert space for this system has dimension 2^N, and
hence a brute force approach to finding the ground state is completely
impractical for even modest N. In these talks I will discuss recent
results on both of these problems, including "easiness results"
(such as the result that gapped one dimensional quantum systems
are in NP) and "hardness results" (such as the difficulty
of determining consistency of local density matrices).
Dr. Hastings has a body of work based on Lieb-Robinson bounds for
the dynamics of quantum spin systems and lattice fermions that has
been transformative in the sense that it has opened up new approaches
to make progress on important problems in condensed matter physics.
His work provides new insights in the (classical) computability
of ground states and equilibrium states of standard Hamiltonians
of condensed matter physics. His ideas of using Lieb-Robinson bounds
to mathematically replace locality in QFT and to use it to prove
exponential decay of correlations and a higher dimensional Lieb-Schultz-Mattis
theorem are also examples of his ability to bring important new
insights to bear on old quesions.
His proof of what is known as the area law for the local entropy
in the ground state of gapped spin chains is another piece of work
in the interface between quantum information, computation, and basic
questions in many-body physics.
In September 2008, he literally stunned the quantum information
community with a very different piece of work that solved a long
standing question about additivity of certain quantities under tensor
products of quantum channels. This question (or, more properly,
a group of conjectures shown to be equivalent by Shor in 2003) was
open despite concerted efforts by such luminaries as Shor, Holevo,
King, and many others during the past 10-15 years. Hastings showed
that the conjectures are false not by constructing explicit counter-examples,
but by showing the existence of counter-examples by a very delicate
probabilistic argument. Rather than ending the question, Hastings
results show that there are pairs of quantum channels for which
entangled inputs can improve the transmission of classical information,
raising interesting new questions.
Speakers in the Distinguished Lecture Series (DLS) have made outstanding
contributions to their field of mathematics. The DLS consists of
a series of three one-hour lectures.
Index of Fields
Distinguished and Coxeter Lectures