From computation to black holes and space-time foam

We show that quantum mechanics and general relativity limit the speed $\tilde{\nu}$ of a simple computer (such as a black hole) and its memory space $I$ to \tilde{\nu}^2 I^{-1} \lsim t_P^{-2}, where $t_P$ is the Planck time. We also show that the life-time of a simple clock and its precision are similarly limited. These bounds and the holographic bound originate from the same physics that governs the quantum fluctuations of space-time. We further show that these physical bounds are realized for black holes, yielding the correct Hawking black hole lifetime, and that space-time undergoes much larger quantum fluctuations than conventional wisdom claims -- almost within range of detection with modern gravitational-wave interferometers.Comment: A misidentification of computer speeds is corrected. Our results for black hole computation now agree with those given by S. Lloyd. All other conclusions remain unchange

Certainty and Uncertainty in Quantum Information Processing

This survey, aimed at information processing researchers, highlights intriguing but lesser known results, corrects misconceptions, and suggests research areas. Themes include: certainty in quantum algorithms; the "fewer worlds" theory of quantum mechanics; quantum learning; probability theory versus quantum mechanics.Comment: Invited paper accompanying invited talk to AAAI Spring Symposium 2007. Comments, corrections, and suggestions would be most welcom

The thermodynamic meaning of negative entropy

Landauer's erasure principle exposes an intrinsic relation between thermodynamics and information theory: the erasure of information stored in a system, S, requires an amount of work proportional to the entropy of that system. This entropy, H(S|O), depends on the information that a given observer, O, has about S, and the work necessary to erase a system may therefore vary for different observers. Here, we consider a general setting where the information held by the observer may be quantum-mechanical, and show that an amount of work proportional to H(S|O) is still sufficient to erase S. Since the entropy H(S|O) can now become negative, erasing a system can result in a net gain of work (and a corresponding cooling of the environment).Comment: Added clarification on non-cyclic erasure and reversible computation (Appendix E). For a new version of all technical proofs see the Supplementary Information of the journal version (free access

Squeezed coherent states for noncommutative spaces with minimal length uncertainty relations

We provide an explicit construction for Gazeau-Klauder coherent states related to non-Hermitian Hamiltonians with discrete bounded below and nondegenerate eigenspectrum. The underlying spacetime structure is taken to be of a noncommutative type with associated uncertainty relations implying minimal lengths. The uncertainty relations for the constructed states are shown to be saturated in a Hermitian as well as a non-Hermitian setting for a perturbed harmonic oscillator. The computed value of the Mandel parameter dictates that the coherent wavepackets are assembled according to sub-Poissonian statistics. Fractional revival times, indicating the superposition of classical-like sub-wave packets are clearly identified.Comment: 14 pages, 2 figure

How to simulate a quantum computer using negative probabilities

The concept of negative probabilities can be used to decompose the interaction of two qubits mediated by a quantum controlled-NOT into three operations that require only classical interactions (that is, local operations and classical communication) between the qubits. For a single gate, the probabilities of the three operations are 1, 1, and -1. This decomposition can be applied in a probabilistic simulation of quantum computation by randomly choosing one of the three operations for each gate and assigning a negative statistical weight to the outcomes of sequences with an odd number of negative probability operations. The exponential speed-up of a quantum computer can then be evaluated in terms of the increase in the number of sequences needed to simulate a single operation of the quantum circuit.Comment: 11 pages, including one figure and one table. Full paper version for publication in Journal of Physics A. Clarifications of basic concepts and discussions of possible implications have been adde

Quantum Cryptography Beyond Quantum Key Distribution

Quantum cryptography is the art and science of exploiting quantum mechanical effects in order to perform cryptographic tasks. While the most well-known example of this discipline is quantum key distribution (QKD), there exist many other applications such as quantum money, randomness generation, secure two- and multi-party computation and delegated quantum computation. Quantum cryptography also studies the limitations and challenges resulting from quantum adversaries---including the impossibility of quantum bit commitment, the difficulty of quantum rewinding and the definition of quantum security models for classical primitives. In this review article, aimed primarily at cryptographers unfamiliar with the quantum world, we survey the area of theoretical quantum cryptography, with an emphasis on the constructions and limitations beyond the realm of QKD.Comment: 45 pages, over 245 reference