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Applications of Sum of Squares in Quantum Information

Xiaodi Wu
University of Michigan


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.

Tuesday, April 23, 2013
11:00AM AP&M 2402