Security of Quantum Key Distribution with Entangled Qutrits

The study of quantum cryptography and quantum non-locality have traditionnally been based on two-level quantum systems (qubits). In this paper we consider a generalisation of Ekert's cryptographic protocol [Ekert] where qubits are replaced by qutrits. The security of this protocol is related to non-locality, in analogy with Ekert's protocol. In order to study its robustness against the optimal individual attacks, we derive the information gained by a potential eavesdropper applying a cloning-based attack.Comment: 9 pages original version: july 2002, replaced in january 2003 (reason: minor changes

Parameter estimation of ODE's via nonparametric estimators

Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional sets of coupled nonlinear differential equations. In this setting, we propose a general method for estimating the parameters indexing ODE's from times series. Our method is able to alleviate the computational difficulties encountered by the classical parametric methods. These difficulties are due to the implicit definition of the model. We propose the use of a nonparametric estimator of regression functions as a first-step in the construction of an M-estimator, and we show the consistency of the derived estimator under general conditions. In the case of spline estimators, we prove asymptotic normality, and that the rate of convergence is the usual $\sqrt{n}$-rate for parametric estimators. Some perspectives of refinements of this new family of parametric estimators are given.Comment: Published in at http://dx.doi.org/10.1214/07-EJS132 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adiabatic quantum search algorithm for structured problems

The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speed-up over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.Comment: 7 pages, 0 figur

Exploring pure quantum states with maximally mixed reductions

We investigate multipartite entanglement for composite quantum systems in a pure state. Using the generalized Bloch representation for n-qubit states, we express the condition that all k-qubit reductions of the whole system are maximally mixed, reflecting maximum bipartite entanglement across all k vs. n-k bipartitions. As a special case, we examine the class of balanced pure states, which are constructed from a subset of the Pauli group P_n that is isomorphic to Z_2^n. This makes a connection with the theory of quantum error-correcting codes and provides bounds on the largest allowed k for fixed n. In particular, the ratio k/n can be lower and upper bounded in the asymptotic regime, implying that there must exist multipartite entangled states with at least k=0.189 n when $n\to \infty$. We also analyze symmetric states as another natural class of states with high multipartite entanglement and prove that, surprisingly, they cannot have all maximally mixed k-qubit reductions with k>1. Thus, measured through bipartite entanglement across all bipartitions, symmetric states cannot exhibit large entanglement. However, we show that the permutation symmetry only constrains some components of the generalized Bloch vector, so that very specific patterns in this vector may be allowed even though k>1 is forbidden. This is illustrated numerically for a few symmetric states that maximize geometric entanglement, revealing some interesting structures.Comment: 10 pages, 2 figure

Pathwise stochastic integrals for model free finance

We present two different approaches to stochastic integration in frictionless model free financial mathematics. The first one is in the spirit of It\^o's integral and based on a certain topology which is induced by the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every "typical price path" has a naturally associated It\^o rough path, and justify the application of the controlled rough path integral in finance by showing that it is the limit of non-anticipating Riemann sums, a new result in itself. Compared to the first approach, rough paths have the disadvantage of severely restricting the space of integrands, but the advantage of being a Banach space theory. Both approaches are based entirely on financial arguments and do not require any probabilistic structure.Comment: Published at http://dx.doi.org/10.3150/15-BEJ735 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Continuous-variable entropic uncertainty relations

Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position $x$ and momentum $p$. This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take $x$-$p$ correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to $x$ and $p$, but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that take correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.Comment: Review paper, 42 pages, 1 figure. We corrected some minor errors in V

Cloning a Qutrit

We investigate several classes of state-dependent quantum cloners for three-level systems. These cloners optimally duplicate some of the four maximally-conjugate bases with an equal fidelity, thereby extending the phase-covariant qubit cloner to qutrits. Three distinct classes of qutrit cloners can be distinguished, depending on two, three, or four maximally-conjugate bases are cloned as well (the latter case simply corresponds to the universal qutrit cloner). These results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can also be analyzed.Comment: 14 pages LaTex. To appear in the Journal of Modern Optics for the special issue on "Quantum Information: Theory, Experiment and Perspectives". Proceedings of the ESF Conference, Gdansk, July 10-18, 200

Posterior propriety in Bayesian extreme value analyses using reference priors

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.Comment: 20 pages, 2 figures; typo corrected on page 5 (line -2, Euler's constant corrected to approx. 0.57722). The final publication is available at http://www3.stat.sinica.edu.tw/preprint/SS-14-034_preprint.pdf or http://dx.doi.org/10.5705/ss.2014.03