Entanglement distillation, a fundamental building block of quantum networks, enables the purification of noisy entangled states shared among distant nodes by local operations and classical communication. Its practical realization presents several technical challenges, including the storage of quantum states in quantum memory and the execution of coherent quantum operations on multiple copies of states within the quantum memory. In this work, we present an entanglement purification protocol via quantum communication, namely a carrier-assisted entanglement purification protocol, which utilizes two elements only: i) quantum memory for a single-copy entangled state shared by parties and ii) single qubits travelling between parties. We show that the protocol, when single-qubit transmission is noiseless, can purify a noisy entangled state shared by parties. When single-qubit transmission is noisy, the purification relies on types of noisy qubit channels; we characterize Pauli channels such that the protocol works for the purification. We address this limitation by using multiple carrier qubits, and show that for any non-entanglement-breaking Pauli channel, the protocol's fixed-point fidelity approaches unity as the number of carriers increases. Our results significantly reduce the experimental overhead required for distilling entanglement: the practical advantage is demonstrated through parameters directly related to the capability of entanglement purification, such as noise in quantum memory, local measurements, channel use, and entanglement fidelity. We envisage that the protocol would make long-distance pure entanglement closer to a practical realization.

We consider estimation of multiple high-dimensional Gaussian graphical models corresponding to a single set of nodes under several distinct conditions. We assume that most aspects of the networks are shared, but that there are some structured differences between them. Specifically, the network differences are generated from node perturbations: a few nodes are perturbed across networks, and most or all edges stemming from such nodes differ between networks. This corresponds to a simple model for the mechanism underlying many cancers, in which the gene regulatory network is disrupted due to the aberrant activity of a few specific genes. We propose to solve this problem using the structured joint graphical lasso, a convex optimization problem that is based upon the use of a novel symmetric overlap norm penalty, which we solve using an alternating directions method of multipliers algorithm. Our proposal is illustrated on synthetic data and on an application to brain cancer gene expression data.

Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including ℓ1and nuclear norm minimization as well as ℓpminimization with p <; 1. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs for the recovery of matrices have so far been much more involved than in the vector case. In this paper, we show how several classes of recovery conditions can be extended from vectors to matrices in a simple and transparent way, leading to the best known restricted isometry and nullspace conditions for matrix recovery. Our results rely on the ability to “vectorize” matrices through the use of a key singular value inequality.

The problem of recovering a low-rank matrix consistent with noisy linear measurements is a fundamental problem with applications in machine learning, statistics, and control. Reweighted trace minimization, which extends and improves upon the popular nuclear norm heuristic, has been used as an iterative heuristic for this problem. In this paper, we present theoretical guarantees for the reweighted trace heuristic. We quantify its improvement over nuclear norm minimization by proving tighter bounds on the recovery error for low-rank matrices with noisy measurements. Our analysis is based on the Restricted Isometry Property (RIP) and extends some recent results from Compressed Sensing. As a second contribution, we improve the existing RIP recovery results for the nuclear norm heuristic, and show that recovery happens under a weaker assumption on the RIP constants.