EPSRC Reference: 
EP/W024500/1 
Title: 
Multidimensional moment problems and the quantumclassic divide 
Principal Investigator: 
Kimsey, Dr D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Maths, Statistics and Physics 
Organisation: 
Newcastle University 
Scheme: 
Standard Research  NR1 
Starts: 
01 October 2022 
Ends: 
30 September 2023 
Value (£): 
98,551

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Mathematical Analysis 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Imagine you are given a graph that gives you the distribution of child heights in a school. From it you could easily calculate the average height of the children. With a bit more work you could calculate the typical variation of heights about that average. These two quantities  the mean and the variance  are the first two in a sequence of numbers, known as the moments of the probability distribution. They give important information about the shape of the probability distribution graph (e.g., where its "middle" is in the the case of the mean).
Whilst going from a probability distribution to the collection of moments is trivial, going in the other direction is not so simple. If one has access to the complete (in principle infinite) set of moments, then reconstructing the distribution is no problem. However, what about the case when we are only given a finite set of moments? What can we say about the underlying distribution? Is there any guarantee that, given a set of moments, that such a distribution even exists?
These questions come under the title of the univariate truncated moment problem (TMP). And this is well understood from several different points of view (e.g., matrix theory, operator theory, probability theory and optimisation theory). However, multidimensional analogues of the TMP, where the given finite list is multiply indexed and we are worried about the probability of several different things happening, have proven to be much more elusive. Standard approaches that are known to work for in the univariate setting are inadequate in the multidimensional setting (insofar, as the discovery of concrete necessary and sufficient conditions for a solution are only understood in a few limited settings). The progress so far made with multidimensional TMPs has stimulated advances in function theory, operator theory and real algebraic geometry. Applications for multidimensional TMPs are in full abundance, e.g., probability theory, signal processing and theoretical physics.
With applications in mind, we have identified an important and asyet unexplored connection between multidimensional TMPs and foundational questions in quantum theory. In particular, we assert that the question of the existence of solutions to the TMP where the measure is supported on a given finite set is fundamental to understanding whether a set of observations is consistent with quantum or classical mechanics. Utilising recent advances in operator theory and real algebraic geometry, this project will develop new tools for solving these moment problems which will enable us to create a novel, unified and formallyrigorous framework for investigating the divide between classical and quantum theories.

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Organisation Website: 
http://www.ncl.ac.uk 