Applications of Sum of Squares in Quantum Information
Xiaodi Wu
University of Michigan
Abstract:
Quantum information is an emerging research area that investigates the
power and the limitations of quantum systems performing informational and
computational tasks. In this talk, I will introduce two examples about how the
sum of squares relaxation and related techniques could be instrumental to study
the properties of certain quantum systems.
The first example is about the description of the set of quantum separable
states, an important class of quantum states that is hard to characterize. It
is related to the sum of squares relaxation for polynomial optimizations of
commutative variables, e.g., the Lasserre hierarchy. I will talk about how a
specific optimization problem, called "Bi-Quadratic Optimization over Unit
Spheres", is related to the quantum Merlin-Arthur games with unentangled two
provers, which refers to a genuine quantum phenomenon comparing to its
classical counterpart. I will also talk about how a quantum technique, called
the quantum de Finetti theorem, could be utilized to analyze the converging
rate of the Lasserre hierarchy for "Bi-Quadratic Optimization over Unit
Spheres".
The second example is about calculating the quantum value of a non-local game,
which is related to the non-commutative sum of squares relaxation. In a
non-local game, two physically separated players are given randomly sampled
questions (they, however, do not know the question given to each other) and then
required to output their answers. A certain game value will then be assigned to
this question-answer combination. Two quantum players, even though physically
separated, can nevertheless share an entangled quantum state and then generate
better correlated answers than classical players to obtain higher game values.
This phenomenon is called non-locality. I will talk about how a non-commutative
sum of squares technique can help calculate how much better the quantum players
are for any given non-local game, although maybe in infinite time.