An improved 1D area law for frustration-free systems

We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings' 1D area law, and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\epsilon>0$ and interaction strength of at most $J$, our entropy bound is S_{1D}\le \orderof{1}X^3\log^8 X where X\EqDef(J\log d)/\epsilon. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. Incorporating locality into the proof when applied to the 2D case gives an entanglement bound that is at the cusp of being non-trivial in the sense that any further improvement would yield a sub-volume law.Comment: 15 pages, 6 figures. Some small style corrections and updated ref

How Powerful is Adiabatic Quantum Computation?

We analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'.Comment: 12 pages, LaTeX2e, requires fullpage, times, amssymb, and amsmath packages. This article appeared in the proceedings of FOCS'01; original submission date: April 27, 200

Robust device independent quantum key distribution

Quantum cryptography is based on the discovery that the laws of quantum mechanics allow levels of security that are impossible to replicate in a classical world. Can such levels of security be guaranteed even when the quantum devices on which the protocol relies are untrusted? This fundamental question in quantum cryptography dates back to the early nineties when the challenge of achieving device independent quantum key distribution, or DIQKD, was first formulated. We answer this challenge affirmatively by exhibiting a robust protocol for DIQKD and rigorously proving its security. The protocol achieves a linear key rate while tolerating a constant noise rate in the devices. The security proof assumes only that the devices can be modeled by the laws of quantum mechanics and are spatially isolated from each other and any adversary's laboratory. In particular, we emphasize that the devices may have quantum memory. All previous proofs of security relied either on the use of many independent pairs of devices, or on the absence of noise. To prove security for a DIQKD protocol it is necessary to establish at least that the generated key is truly random even in the presence of a quantum adversary. This is already a challenge, one that was recently resolved. DIQKD is substantially harder, since now the protocol must also guarantee that the key is completely secret from the quantum adversary's point of view, and the entire protocol is robust against noise; this in spite of the substantial amounts of classical information leaked to the adversary throughout the protocol, as part of the error estimation and information reconciliation procedures. Our proof of security builds upon a number of techniques, including randomness extractors that are secure against quantum storage as well as ideas originating in the coding strategy used in the proof of the Holevo-Schumacher-Westmoreland theorem which we apply to bound correlations across multiple rounds in a way not unrelated to information-theoretic proofs of the parallel repetition property for multiplayer games. Our main result can be understood as a new bound on monogamy of entanglement in the type of complex scenario that arises in a key distribution protocol