# Hypercontractivity and Log Sobolev Inequalities in Quantum Information Theory (15w5098)

Arriving in Banff, Alberta Sunday, February 22 and departing Friday February 27, 2015

## Organizers

(Stanford University)

(Northeastern University)

(University of Bristol)

(University of Vermont)

## Objectives

This workshop will provide an opportunity to bring together several communities: mathematicians who work on hypercontractivity and log Sobolev inequalities, scientists (including physicists and engineers) who work in quantum information theory, and computer scientists.Quantum information theory is a highly interdisciplinary field in which important developments have relied on a variety of sophisticated mathematical tools including operator spaces, random matrix theory, and high dimensional convex bodies. Previous workshops at BIRS, beginning with the February 2007 workshop on Operator Structures in Quantum Information Theory played a key role by helping to introduce these mathematical methods to QIT researchers. One example of this interplay was the inter-disciplinary collaboration at the 2007 workshop which led to the solution of an open problem concerning multipartite Bell inequality violations~cite{PWPVJ}, and the reformulation of an 30-year-old open question in Banach algebras, which was then solved by computer scientists cite{BBHV}. This workshop was followed by others at the Fields Institute in July, 2009 and at BIRS in February 2012. In addition, the June 2010 BIRS workshop on Noncommutative $L_p$ Spaces, Operator Spaces and Applications included participants from QIT. These workshops contributed to the resolution (in the negative) of the so-called additivity conjecture'' in QIT and led directly to alternate proofs and improved estimates on the violation of additivity. Other developments to which these workshop contributed include Haagerup and Musat's cite{HM} resolution (in the negative) of the asymptotic Birkhoff conjecture, and several reformulations of the Connes' embedding problem including a connection with Tsirelson's problem cite{JNPPSW}.The success of these workshops was due, in part, to a decision to focus on a particular area of mathematics of potential importance to quantum information and to bring together people with very different backgrounds in mathematics and in the physical sciences. An important feature of these workshops were sessions devoted to the presentation and discussion of open problems, summaries of which were made available in the BIRS reports and on various web pages. We plan to include such discussions in the proposed workshop. Recently, hypercontractivity and log Sobolev inequalities have been used to obtain a variety of results in quantum information theory as described in the overview section. An informal lecture this June at a workshop at the Benasque Centre for Science attracted forty participants, more than ten of whom stayed for intensive discussions afterwords. There is a strong and growing interest in this topic. The fact that Montanaro cite{M} was able to apply these techniques to several different problems in QIT, as well as the work in cite{KT}, indicates that the potential applications to QIT go well beyond complexity theory. The time is ripe for a workshop which will bring together experts from the mathematical community and scientists in QIT. Based on the success of such previously focused workshops we expect an eager audience for this program. Indeed, a number of prominent mathematicians and quantum information scientists have already expressed a strong interest in participating.begin{thebibliography}{99} bibitem{Aud} K.M.R Audenaert, newblock On a norm compression inequality for $2 times N$ partitioned block matrices. newblock {em Linear Algebra Appl.} 428, 781--795 (2008). bibitem{BCL} K.~Ball, E.~A.~Carlen and E.~H.~Lieb, newblock Sharp Uniform Convexity and Smoothness Inequalities for Trace Norms. newblock {em Invent. Math.} 115, 463--482, (1994). bibitem{benaroya08} A.~Ben-Aroya, O.~Regev, and R.~de~Wolf, newblock A hypercontractive inequality for matrix-valued functions with applications to quantum computing and {LDC}s. newblock In {em Proc. 49textsuperscript{th} Annual Symp. Foundations of Computer Science}, pp 477--486, (2008). newblock {arXiv:0705.3806}. bibitem{BBHKSY} Boaz Barak, Fernando G.S.L. Brand{~{a}}o, Aram W. Harrow, Jonathan A. Kelner, David Steurer, Yuan Zhou newblock Hypercontractivity, Sum-of-Squares Proofs, and their Applications newblock {em Proc. STOC} pp. 307--326 (2012) {arXiv:1205.4484} bibitem{Biane} P.~Biane, newblock Free Hypercontractivity. newblock {em Commun. Math. Phys.} 184, 457--474, (1997).bibitem{BBHV} J.~Briet, H.~Buhrman, T.~Lee, T.~Vidick, newblock Multiplayer XOR games and quantum communication complexity with clique-wise entanglement {arXiv:0911.4007}; newblock All Schatten spaces endowed with the Schur product are Q-algebras. newblock {em J. Funct. Anal. } 262 1--9 (2012).bibitem{CL1} E.~A.~Carlen and E.~H.~Lieb, newblock Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities. newblock {em Commun. Math. Phys.} 155, 27--46, (1993).bibitem{friedgut98} E.~Friedgut. newblock Boolean functions with low average sensitivity depend on few coordinates. newblock {em Combinatorica}, 18, 27--35, (1998).bibitem{friedgut96} E.~Friedgut and G.~Kalai. newblock Every monotone graph property has a sharp threshold. newblock {em Proceedings of the American Mathematical Society}, 124, 2993--3002, (1996).bibitem{gavinsky07} D.~Gavinsky, J.~Kempe, I.~Kerenidis, R.~Raz, and R.~de~Wolf. newblock Exponential separations for one-way quantum communication complexity, with applications to cryptography. newblock In {em Proc. 39textsuperscript{th} Annual ACM Symp. Theory of Computing}, pages 516--525, (2007). newblock {quant-ph/0611209}.bibitem{HM} U. Haagerup and M.~Musat, newblock Factorization and dilation problems for completely positive maps on von Neumann algebras. newblock {em Comm. Math. Phys.} 303, 555--594 (2011). bibitem{JNPPSW} M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, R. F. Werner newblock Connes' embedding problem and Tsirelson's problem newblock {em J. Math. Phys.} 52}, 012102 (2011) { arXiv:1008.1142 bibitem{KT} M.~J.~Kastoryano and K.~Temme, newblock Quantum logarithmic Sobolev inequalities and rapid mixing. newblock {em J. Math. Phys} 54}, 052202 (2013). {arXiv:1207.3261.bibitem{king} C. King, newblock Inequalities for trace norms of $2 times 2$ block matrices. newblock {em Commun. Math. Phys.} 242, 531-545, 2003.bibitem{klartag11} B.~Klartag and O.~Regev. newblock Quantum one-way communication can be exponentially stronger than classical communication. newblock In {em Proc. 43textsuperscript{rd} Annual ACM Symp. Theory of Computing}, pages 31--40, (2011). newblock {arXiv:1009.3640}. bibitem{M} A.~Montanaro, newblock Some applications of hypercontractive inequalities in quantum information theory newblock{em J. Math. Phys.} 53, 122206 (2012). newblock {arXiv:1208.0161}bibitem{MO1} A.~Montanaro and T.~Osborne, newblock Quantum boolean functions. newblock {em Chicago Journal of Theoretical Computer Science}, Article 1, (2010).bibitem{mossel10} E.~Mossel, R.~O'Donnell, and K.~Oleszkiewicz. newblock Noise stability of functions with low influences: Invariance and optimality. newblock {em Ann. of Math.}, 171(1), (2010).bibitem{odonnell08} R.~O'Donnell. newblock Some topics in analysis of boolean functions. newblock In {em Proc. 40textsuperscript{th} Annual ACM Symp. Theory of Computing}, pages 569--578, (2008).bibitem{OZ} R.~Olkiewicz and B.~Zegarlinski. newblock Hypercontractivity in noncommutative $L_p$ spaces. newblock {em Journal of Functional Analysis}, 161(1):246-285, (1999).bibitem{PWPVJ} D.~Perez-Garcia, M.~M.~Wolf, C.~Palazuelos, I.~Villanueva and M.~Junge. newblock Unbounded violation of tripartite Bell inequalities. newblock {em Comm. Math. Phys.} 279, 455, (2008).bibitem{dewolf08} R.~de Wolf. newblock A brief introduction to {F}ourier analysis on the boolean cube. newblock {em Theory of Computing Library Graduate Surveys}, 1, 1--20, (2008). end{thebibliography}