A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs

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 $m<0.7 n$, 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, and answers a question by Trevisan.Comment: This is the full version of a paper that will appear in the proceedings of the IEEE FOCS 08 conferenc

Multipartite entanglement in XOR games

We study multipartite entanglement in the context of XOR games. In particular, we study the ratio of the entangled and classical biases, which measure the maximum advantage of a quantum or classical strategy over a uniformly random strategy. For the case of two-player XOR games, Tsirelson proved that this ratio is upper bounded by the celebrated Grothendieck constant. In contrast, Pérez-García et al. proved the existence of entangled states that give quantum players an unbounded advantage over classical players in a three-player XOR game. We show that the multipartite entangled states that are most often seen in today’s literature can only lead to a bias that is a constant factor larger than the classical bias. These states include GHZ states, any state local-unitarily equivalent to combinations of GHZ and maximally entangled states shared between different subsets of the players (e.g., stabilizer states), as well as generalizations of GHZ states of the form ∑iɑi|i〉...|i〉 for arbitrary amplitudes ɑi. Our results have the following surprising consequence: classical three-player XOR games do not follow an XOR parallel repetition theorem, even a very weak one. Besides this, we discuss implications of our results for communication complexity and hardness of approximation. Our proofs are based on novel applications of extensions of Grothendieck’s inequality, due to Blei and Tonge, and Carne, generalizing Tsirelson’s use of Grothendieck’s inequality to bound the bias of two-player XOR games

N-particle nonclassicality without N-particle correlations

Most of known multipartite Bell inequalities involve correlation functions for all subsystems. They are useless for entangled states without such correlations. We give a method of derivation of families of Bell inequalities for N parties, which involve, e.g., only (N-1)-partite correlations, but still are able to detect proper N-partite entanglement. We present an inequality which reveals five-partite entanglement despite only four-partite correlations. Classes of inequalities introduced here can be put into a handy form of a single non-linear inequality. An example is given of an N qubit state, which strongly violates such an inequality, despite having no N-qubit correlations. This surprising property might be of potential value for quantum information tasks.Comment: 5 page