Jensen-Shannon divergence ͑JD͒ is a symmetrized and smoothed version of the most important divergence measure of information theory, Kullback divergence. As opposed to Kullback divergence it determines in a very direct way a metric; indeed, it is the square of a metric. We consider a family of divergence measures ͑JD ␣ for ␣ Ͼ 0͒, the Jensen divergences of order ␣, which generalize JD as JD 1 = JD. Using a result of Schoenberg, we prove that JD ␣ is the square of a metric for ␣ ͑0,2͔, and that the resulting metric space of probability distributions can be isometrically embedded in a real Hilbert space. Quantum Jensen-Shannon divergence ͑QJD͒ is a symmetrized and smoothed version of quantum relative entropy and can be extended to a family of quantum Jensen divergences of order ␣ ͑QJD ␣ ͒. We strengthen results by Lamberti and co-workers by proving that for qubits and pure states, QJD ␣ 1/2 is a metric space which can be isometrically embedded in a real Hilbert space when ␣ ͑0,2͔. In analogy with Burbea and Rao's generalization of JD, we also define general QJD by associating a Jensen-type quantity to any weighted family of states. Appropriate interpretations of quantities introduced are discussed and bounds are derived in terms of the total variation and trace distance.

Suppose that Alice and Bob make local two-outcome measurements on a shared entangled quantum state. We show that, for all positive integers d, there exist correlations that can only be reproduced if the local Hilbert-space dimension is at least d. This establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing a lower bound on a new generalization of Grothendieck's constant.

Given a positive integer n and a positive semidefinite matrix A = (Aij) ∈ R m×m , the positive semidefinite Grothendieck problem with rank-n-constraint (SDPn) iswhere x1, . . . , xm ∈ S n−1 .In this paper we design a randomized polynomial-time approximation algorithm for SDPn achieving an approximation ratio of γ(n) = 2 n Γ ((n + 1)/2) Γ (n/2) 2 = 1 − Θ(1/n).We show that under the assumption of the unique games conjecture the achieved approximation ratio is optimal: There is no polynomial-time algorithm which approximates SDPn with a ratio greater than γ(n). We improve the approximation ratio of the best known polynomial-time algorithm for SDP1 from 2/π to 2/(πγ(m)) = 2/π + Θ(1/m), and we show a tighter approximation ratio for SDPn when A is the Laplacian matrix of a graph with nonnegative edge weights.

XOR games are the simplest model in which the nonlocal properties of entanglement manifest themselves. When there are two players, it is well known that the bias -the maximum advantage over random play -of entangled players can be at most a constant times greater than that of classical players. Recently, Pérez-García et al. [Comm. Math. Phys. 279 (2), 2008] showed that no such bound holds when there are three or more players: the advantage of entangled players over classical players can become unbounded, and scale with the number of questions in the game. Their proof relies on non-trivial results from operator space theory, and gives a non-explicit existence proof, leading to a game with a very large number of questions and only a loose control over the local dimension of the players' shared entanglement.We give a new, simple and explicit (though still probabilistic) construction of a family of three-player XOR games which achieve a large quantum-classical gap (QC-gap). This QCgap is exponentially larger than the one given by Pérez-García et. al. in terms of the size of the game, achieving a QC-gap of order √ N with N 2 questions per player. In terms of the dimension of the entangled state required, we achieve the same (optimal) QC-gap of √ N for a state of local dimension N per player. Moreover, the optimal entangled strategy is very simple, involving observables defined by tensor products of the Pauli matrices.Additionally, we give the first upper bound on the maximal QC-gap in terms of the number of questions per player, showing that our construction is only quadratically off in that respect. Our results rely on probabilistic estimates on the norm of random matrices and higher-order tensors which may be of independent interest.