Yuming Zhao, Department of Pure Mathematics, University of Waterloo
"Positivity and sum of squares in quantum information"
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. 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 quantum information, we are interested in noncommutative polynomials in *-algebras. A well-known theorem of Helton states that an element of a free *-algebra is positive in all *-representations if and only if it is a sum of squares. The theorem provides an effective way to determine if a given element is positive, by searching through sums of squares decompositions. In this talk, I'll present joint work with Arthur Mehta and William Slofstra in which we show that no such procedure exists for the tensor product of two free *-algebras: determining whether an element of such an algebra is positive is coRE-hard. Consequently, tensor products of free *-algebras contain elements which are positive but not sums of squares. I will also discuss the connections to quantum information theory.
This seminar will be held both online and in person:
- Room: MC 5479
- Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09