38 research outputs found
Device-Independent Quantum Key Distribution
Cryptographic key exchange protocols traditionally rely on computational
conjectures such as the hardness of prime factorisation to provide security
against eavesdropping attacks. Remarkably, quantum key distribution protocols
like the one proposed by Bennett and Brassard provide information-theoretic
security against such attacks, a much stronger form of security unreachable by
classical means. However, quantum protocols realised so far are subject to a
new class of attacks exploiting implementation defects in the physical devices
involved, as demonstrated in numerous ingenious experiments. Following the
pioneering work of Ekert proposing the use of entanglement to bound an
adversary's information from Bell's theorem, we present here the experimental
realisation of a complete quantum key distribution protocol immune to these
vulnerabilities. We achieve this by combining theoretical developments on
finite-statistics analysis, error correction, and privacy amplification, with
an event-ready scheme enabling the rapid generation of high-fidelity
entanglement between two trapped-ion qubits connected by an optical fibre link.
The secrecy of our key is guaranteed device-independently: it is based on the
validity of quantum theory, and certified by measurement statistics observed
during the experiment. Our result shows that provably secure cryptography with
real-world devices is possible, and paves the way for further quantum
information applications based on the device-independence principle.Comment: 5+1 pages in main text and methods with 4 figures and 1 table; 37
pages of supplementary materia
Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems
We consider chains of random constraint satisfaction models that are
spatially coupled across a finite window along the chain direction. We
investigate their phase diagram at zero temperature using the survey
propagation formalism and the interpolation method. We prove that the SAT-UNSAT
phase transition threshold of an infinite chain is identical to the one of the
individual standard model, and is therefore not affected by spatial coupling.
We compute the survey propagation complexity using population dynamics as well
as large degree approximations, and determine the survey propagation threshold.
We find that a clustering phase survives coupling. However, as one increases
the range of the coupling window, the survey propagation threshold increases
and saturates towards the phase transition threshold. We also briefly discuss
other aspects of the problem. Namely, the condensation threshold is not
affected by coupling, but the dynamic threshold displays saturation towards the
condensation one. All these features may provide a new avenue for obtaining
better provable algorithmic lower bounds on phase transition thresholds of the
individual standard model
Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices
Compressed sensing is a signal processing method that acquires data directly
in a compressed form. This allows one to make less measurements than what was
considered necessary to record a signal, enabling faster or more precise
measurement protocols in a wide range of applications. Using an
interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a
strategy that allows compressed sensing to be performed at acquisition rates
approaching to the theoretical optimal limits. In this paper, we give a more
thorough presentation of our approach, and introduce many new results. We
present the probabilistic approach to reconstruction and discuss its optimality
and robustness. We detail the derivation of the message passing algorithm for
reconstruction and expectation max- imization learning of signal-model
parameters. We further develop the asymptotic analysis of the corresponding
phase diagrams with and without measurement noise, for different distribution
of signals, and discuss the best possible reconstruction performances
regardless of the algorithm. We also present new efficient seeding matrices,
test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe
Design of capacity-approaching irregular low-density parity-check codes
We design low-density parity-check (LDPC) codes that perform at rates extremely close to the Shannon capacity. The codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides. Our theoretical analysis of the codes is based on the work of Richardson and Urbanke (see ibid., vol.47, no.2, p.599-618, 2000). Assuming that the underlying communication channel is symmetric, we prove that the probability densities at the message nodes of the graph possess a certain symmetry. Using this symmetry property we then show that, under the assumption of no cycles, the message densities always converge as the number of iterations tends to infinity. Furthermore, we prove a stability condition which implies an upper bound on the fraction of errors that a belief-propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution. Our codes are found by optimizing the degree structure of the underlying graphs. We develop several strategies to perform this optimization. We also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bound