Research

I am currently a visiting assistant professor at Bucknell University. I previously held a postdoctoral position at Louisiana State University where I studied the mathematics of quantum information theory with Mark Wilde and the rest of the Quantum Science and Technology Group. I am interested in how ideas from functional analysis can be applied to problems in quantum information. I am also interested in operator algebras and how ideas from harmonic analysis on groups can be extended to locally compact quantum groups.

Before coming to Baton Rouge, I was a postdoc in the Mathematics Department of Universidad Complutense de Madrid, working with the Mathematics and Quantum Information research group lead by David Perez Garcia.

I received my Ph.D. in Mathematics from the University of Illinois at Urbana-Champaign. My thesis advisor was Professor Zhong-Jin Ruan and my thesis title was "Noncommutative Lp-spaces Associated with Locally Compact Quantum Groups". This involved studying the interaction between the quantum group and the structure of the noncommutative Lp-space constructed using the Haar weight.


Research Interests:
  • Functional analysis,
  • Operator algebras,
  • Operator spaces,
  • Locally compact quantum groups,
  • Noncommutative Lp spaces,
  • Quantum information theory,
  • Quantum game theory,
  • Quantum many-body systems.


Publications






  • Rank-One Quantum Games (with Marius Junge, Carlos Palazuelos Cabezon, and David Perez Garcia) (computational complexity, 24 (1), 133-196, arXiv:1112.3563)





Abstract: Bipartite correlations generated by non-signalling physical systems that admit a finite-dimensional local quantum description cannot exceed the quantum limits, i.e., they can always be interpreted as distant measurements of a bipartite quantum state. Here we consider the effect of dropping the assumption of finite dimensionality. Remarkably, we find that the same result holds provided that we relax the tensor structure of space-like separated measurements to mere commutativity. We argue why an extension of this result to tensor representations seems unlikely.




Abstract: Tsirelson's problem deals with how to model separate measurements in quantum mechanics. In addition to its theoretical importance, the resolution of Tsirelson's problem could have great consequences for device independent quantum key distribution and certified randomness. Unfortunately, understanding present literature on the subject requires a heavy mathematical background. In this paper, we introduce quansality, a new theoretical concept that allows to reinterpret Tsirelson's problem from a foundational point of view. Using quansality as a guide, we recover all known results on Tsirelson's problem in a clear and intuitive way.


Abstract: Let G be a locally compact abelian group with dual group ^G. The Hausdorff-Young theorem states that if $f \in L_p(G)$, where $1 \leq p \leq 2$, then its Fourier transform $F_p(f)$ belongs to $L_q(^G)$ (where 1/p + 1/q = 1) and $ || F_p(f) ||_q \leq || f ||_p $. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G by defining a Fourier transform $F_p : L_p(G) \to L_q(^G)$ and showing that this Fourier transform satisfies the Hausdorff-Young inequality.

  • Noncommutative L_p spaces associated with locally compact quantum groups (Ph.D. thesis) 

Abstract:
Results from abstract harmonic analysis are extended to locally compact quantum groups by considering the noncommutative L_p-spaces associated with the locally compact quantum groups.

Let
G be a locally compact abelian group with dual group Gˆ. The Hausdorff–Young theorem states that if f L_p(G), where 1 p 2, then its Fourier transform F_p(f) belongs to Lq(Gˆ) (where 1/p+1/q= 1) and || F_p(f) ||_q ≤  || f ||_p. Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G by defining a Fourier transform F_p : L_p(G) Lq(Gˆ) and showing that this Fourier transform satisfies the Hausdorff–Young inequality.

Let G be a locally compact group. Then L_1(G) acts on L_p(G) by convolution. We extend this result to Kac algebras and also discuss an operator space version of this result. Ruan and Junge showed that if G is a discrete group with the approximation property, then L_p(VN(G)) has the operator space approximation property. Let G be a discrete Kac algebra with the approximation property. The aforementioned action of L_1(G) is used to show that L_p(Gˆ) has the operator space approximation property. Similarly, if G is a weakly amenable discrete Kac algebra, then L_p(Gˆ) has the completely bounded approximation property. Available at:  http://www.ideals.illinois.edu/handle/2142/16895

ċ
CooneyHausdorffYoung.pdf
(212k)
Tom Cooney,
Dec 16, 2011, 6:24 AM
Comments