Positivity and Sum-of-squares in Quantum Information

Title: Positivity and Sum-of-squares in Quantum Information

Speaker: Yuming Zhao

Institute for Quantum Computing and the Department of Pure Mathematics, University of Waterloo, Canada

Time: November 28, 2023,  10.00-11.30

Location: Dingding Group： 2023-Autumn Quantum Theory Group

Abstract:A multivariate polynomial is said to be positive if it takes only non-negative values over reals. Hilbert's 17th problem concerns whether every positive polynomial can be expressed as a sum of squares of other polynomials. In general, we say a noncommutative polynomial is positive (resp. matrix positive) if plugging operators (resp. matrices) always yields a positive operator. Many problems in math and computer science are closely connected to deciding whether a given polynomial is positive and finding certificates (e.g., sum-of-squares) of positivity.

In the study of nonlocal games in quantum information, we are interested in tensor product of free algebras. Such an algebra models a physical system with two spatially separated subsystems, where in each subsystem we can make different quantum measurements. The recent and remarkable MIP*=RE result shows that it is undecidable to determine whether a polynomial in a tensor product of free algebras is matrix positive. In this talk, I'll present joint work with Arthur Mehta and William Slofstra, in which we show that it is undecidable to determine positivity in tensor product of free algebras. As a consequence, there is no sum-of-square certificate for positivity in such algebras. I will also discuss relevant topics including self-testing of quantum systems and delegation of quantum computation.

Bio: Yuming Zhao is a Ph.D. candidate in the Institute for Quantum Computing and the Department of Pure Mathematics at the University of Waterloo, supervised by William Slofstra. Previously, he received his bachelor's degree in Mathematics and Applied Mathematics from Zhejiang University in 2019, under the supervision of Junde Wu. He is broadly interested in the mathematics of quantum information and computation. Recent areas of study include quantum self-testing, delegation of quantum computation, complexity theory in operator algebras, and approximate representation theory.

Contact Person: Junde Wu(wjd@zju.edu.cn)