TY - GEN

T1 - A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs

AU - Ben-Aroya, Avraham

AU - Regev, Oded

AU - De Wolf, Ronald

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2008

Y1 - 2008

N2 - The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m < 0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first "non-quantum" proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.

AB - The Bonami-Beckner hypercontractive inequality is a powerful tool in Fourier analysis of real-valued functions on the Boolean cube. In this paper we present a version of this inequality for matrix-valued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m < 0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak's quantum random access code bound. It in turn implies strong direct product theorems for the one-way quantum communication complexity of Disjointness and other problems. Second, we prove that error-correcting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first "non-quantum" proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.

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U2 - 10.1109/FOCS.2008.45

DO - 10.1109/FOCS.2008.45

M3 - Conference contribution

AN - SCOPUS:57949116863

SN - 9780769534367

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 477

EP - 486

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -