Abstract. We show norm estimates for the sum of independent random variables in noncommutative L p spaces for 1 < p < ∞ following previous work by the authors. These estimates generalize Rosenthal's inequalities in the commutative case. Among other applications, we derive a formula for p-norm of the eigenvalues for matrices with independent entries, and characterize those symmetric subspaces and unitary ideal spaces which can be realized as subspaces of noncommutative L p for 2 < p < ∞.

Let 1 ≤ p < ∞ and (x n ) n∈N be a sequence of positive elements in a noncommutative L p space and (E n ) n∈N be an increasing sequence of conditional expectations, then n

This paper is devoted to the study of various maximal ergodic theorems in noncommutative L p L_p -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on L p L_p and the analogue of Stein’s maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

Abstract. The data processing inequality states that the quantum relative entropy between two states ρ and σ can never increase by applying the same quantum channel N to both states. This inequality can be strengthened with a remainder term in the form of a distance between ρ and the closest recovered state (R • N )(ρ), where R is a recovery map with the property that σ = (R • N )(σ). We show the existence of an explicit recovery map that is universal in the sense that it depends only on σ and the quantum channel N to be reversed. This result gives an alternate, information-theoretic characterization of the conditions for approximate quantum error correction.

We propose robust and efficient algorithms for the joint sparse recovery problem in compressed sensing, which simultaneously recover the supports of jointly sparse signals from their multiple measurement vectors obtained through a common sensing matrix.In a favorable situation, the unknown matrix, which consists of the jointly sparse signals, has linearly independent nonzero rows.In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally proposed by Schmidt for the direction of arrival problem in sensor array processing and later proposed and analyzed for joint sparse recovery by Feng and Bresler, provides a guarantee with the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank-defect or ill-conditioning. This situation arises with limited number of measurement vectors, or with highly correlated signal components. In this case MUSIC fails, and in practice none of the existing methods can consistently approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC), which improves on MUSIC so that the support is reliably recovered under such unfavorable conditions. Combined with subspace-based greedy algorithms also proposed and analyzed in this paper, SA-MUSIC provides a computationally efficient algorithm with a performance guarantee. The performance guarantees are given in terms of a version of restricted isometry property. In particular, we also present a non-asymptotic perturbation analysis of the signal subspace estimation that has been missing in the previous study of MUSIC. Index TermsCompressed sensing, joint sparsity, multiple measurement vectors (MMV), subspace estimation, restricted isometry property (RIP), sensor array processing, spectrum-blind sampling.