Grothendieck Inequalities, Nonlocal Games and Optimization
Jop BriĆ«t
Abstract:
Motivated by applications in quantum information theory and
optimization we introduce new variants of a celebrated inequality
known as Grothendieck's Inequality. In quantum information theory we
apply these mathematical tools to study of one of the most surprising
and counter-intuitive predictions of Quantum Mechanics:
entanglement. In optimization we use them to determine the precision
of efficient approximation algorithms for geometric problems that
arise naturally from the study of entanglement and from models of
interacting particles considered in classical statistical physics.
In this thesis we study entanglement by using nonlocal games. A
nonlocal game involves two or more players who are not allowed to
communicate with each other, but do interact with an extra party
usually referred to as the referee. At the start of the game the
referee asks each of the players a question, upon which they each
reply to him with some answer. Then, the referee decides if the
players win or lose based only on the questions he asked and the
answers he received. The players know in advance what set of answers
would cause them to win, which of course is their objective. The catch
is that they only know the question that was aimed directly at them
and not any of the other players' questions. The players thus don't
play against each other, but should somehow coordinate their
strategies to win.
The best course of action for players who live in a world described by
Classical Mechanics is the simplest kind imaginable: just fix in
advance what to answer to each question. In a Quantum Mechanical
world, more sophisticated strategies sometimes give better
results. Each player can base their answer on the outcome of an
experiment done on some private physical system. The key feature of
such strategies is that they can cause the players to produce answers
that are correlated in ways that are impossible in a classical
world. In this case the players are said to be entangled.
The fact that Quantum Mechanics predicts such phenomena was used by
Einstein, Podolski and Rosen in 1935 to argue that this theory must be
incomplete, as surely entanglement could not be part of a reasonable
description of Nature. Surprisingly, experiments done by Aspect et
al. in the 1980's gave convincing evidence that the world we live does
in fact allow for this!
Entanglement is usually mathematically described by a vector in a
Hilbert space. Such a vector is referred to as a state. We prove that
for a large class of states the advantage gained by using them over
classical strategies in the simplest nonlocal games involving three or
more players is severely limited. As a bonus, the proof of this result
can also be used to resolve a 35-year-old open problem posed by
Varopoulos in an area of mathematics called Banach Space Theory.
Optimization means searching over a huge collection to find some
element with the best characteristics. One example of such a problem
is finding a strategy for a nonlocal game that maximizes the players'
winning probability. An second example is to optimize the directions
of the magnetic fields of interacting particles so as to minimize the
total energy of the system.
The most-studied optimization problems usually have a combinatorial
nature. For example, finding an optimal classical strategy for a
nonlocal game or minimizing the energy of interacting particles
described by the celebrated Ising model amounts to searching over a
discrete set of possibilities. In this thesis we consider problems
with a more geometric flavor. To picture this, imagine searching for
some optimal configuration of a finite number of points on a
three-dimensional sphere. Such problems arise naturally from the study
of entanglement when one restricts the amount of entanglement players
are allowed to use in nonlocal games, and from the Heisenberg model of
interacting particles in classical statistical physics.
Unfortunately, most problems like the ones described above likely
can't be solved exactly by any computer in a reasonable amount of
time. If time is of the essence, then the next-best thing is to search
for any solution that is near-optimal, but can be found in a
reasonable amount of time. We will use new variants of Grothendieck's
Inequality to analyze algorithms that offer exactly such an
alternative for the geometric optimization problems mentioned above.