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DTSTART:20230312T070000
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URL:https://uwaterloo.ca/pure-mathematics/events/analysis-seminar-171
SUMMARY:Analysis Seminar
CLASS:PUBLIC
DESCRIPTION:YUMING ZHAO\, DEPARTMENT OF PURE MATHEMATICS\, UNIVERSITY OF WA
 TERLOO\n\n\"POSITIVITY AND SUM OF SQUARES IN QUANTUM INFORMATION\"\n\nA mu
 ltivariate polynomial is said to be positive if it takes only\nnon-negativ
 e values over reals. Hilbert's 17th problem concerns\nwhether every positi
 ve polynomial can be expressed as a sum of squares\nof other polynomials. 
 Many problems in math and computer science are\nclosely connected to decid
 ing whether a given polynomial is positive\nand finding certificates (e.g.
 \, sum-of-squares) of positivity. In\nquantum information\, we are interes
 ted in noncommutative polynomials\nin *-algebras. A well-known theorem of 
 Helton states that an element\nof a free *-algebra is positive in all *-re
 presentations if and only\nif it is a sum of squares. The theorem provides
  an effective way to\ndetermine if a given element is positive\, by search
 ing through sums of\nsquares decompositions. In this talk\, I'll present j
 oint work with\nArthur Mehta and William Slofstra in which we show that no
  such\nprocedure exists for the tensor product of two free *-algebras:\nde
 termining whether an element of such an algebra is positive is\ncoRE-hard.
  Consequently\, tensor products of free *-algebras contain\nelements which
  are positive but not sums of squares. I will also\ndiscuss the connetions
  to quantum information theory.\n\nThis seminar will be held both online a
 nd in person:\n\n* Room: MC 5479\n * Zoom link:\nhttps://uwaterloo.zoom.us
 /j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09\n[https://uwaterloo.zo
 om.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09]
DTSTAMP:20260505T204720Z
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