Optimal distinction between non-orthogonal quantum states

Given a finite set of linearly independent quantum states, an observer who examines a single quantum system may sometimes identify its state with certainty. However, unless these quantum states are orthogonal, there is a finite probability of failure. A complete solution is given to the problem of optimal distinction of three states, having arbitrary prior probabilities and arbitrary detection values. A generalization to more than three states is outlined.Comment: 9 pages LaTeX, one PostScript figure on separate pag

Card shuffling and diophantine approximation

The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th card or the $(n-k)$th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of $k$ and $n$, has surprising behavior. For example, suppose $k$ is the closest integer to $\alpha n$ for a fixed real $\alpha\in(0,1)$. Then for rational $\alpha$ the spectral gap is $\Theta(n^{-2})$, while for poorly approximable irrational numbers $\alpha$, such as the reciprocal of the golden ratio, the spectral gap is $\Theta(n^{-3/2})$.Comment: Published in at http://dx.doi.org/10.1214/07-AAP484 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

KATRIN Sensitivity to Sterile Neutrino Mass in the Shadow of Lightest Neutrino Mass

The presence of light sterile neutrinos would strongly modify the energy spectrum of the Tritium \beta-electrons. We perform an analysis of the KATRIN experiment's sensitivity by scanning almost all the allowed region of neutrino mass-squared difference and mixing angles of the 3+1 scenario. We consider the effect of the unknown absolute mass scale of active neutrinos on the sensitivity of KATRIN to the sterile neutrino mass. We show that after 3 years of data-taking, the KATRIN experiment can be sensitive to mixing angles as small as sin^2 (2\theta_s) ~ 10^-2. Particularly we show that for small mixing angles, sin^2 (2\theta_s) < 0.1, the KATRIN experiment can gives the strongest limit on active-sterile mass-squared difference.Comment: 4 pages, 2 figures, matches the published versio

Nonlinear quantum state transformation of spin-1/2

A non-linear quantum state transformation is presented. The transformation, which operates on pairs of spin-1/2, can be used to distinguish optimally between two non-orthogonal states. Similar transformations applied locally on each component of an entangled pair of spin-1/2 can be used to transform a mixed nonlocal state into a quasi-pure maximally entangled singlet state. In both cases the transformation makes use of the basic building block of the quantum computer, namely the quantum-XOR gate.Comment: 12 pages, LaTeX, amssym, epsfig (2 figures included

Convex probability domain of generalized quantum measurements

Generalized quantum measurements with N distinct outcomes are used for determining the density matrix, of order d, of an ensemble of quantum systems. The resulting probabilities are represented by a point in an N-dimensional space. It is shown that this point lies in a convex domain having at most d^2-1 dimensions.Comment: 7 pages LaTeX, one PostScript figure on separate pag

Tug-of-war and the infinity Laplacian

We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do not intersect Y. This was previously known only for bounded domains R^n, in which case u is infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also prove the first general uniqueness results for Delta_infty u = g on bounded subsets of R^n (when g is uniformly continuous and bounded away from zero), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be the value of the following two-player zero-sum game, called tug-of-war: fix x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y, and player one's payoff is F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i) We show that the u^\epsilon converge uniformly to u as epsilon tends to zero. Even for bounded domains in R^n, the game theoretic description of infinity-harmonic functions yields new intuition and estimates; for instance, we prove power law bounds for infinity-harmonic functions in the unit disk with boundary values supported in a delta-neighborhood of a Cantor set on the unit circle.Comment: 44 pages, 4 figure

No directed fractal percolation in zero area

We show that fractal (or "Mandelbrot") percolation in two dimensions produces a set containing no directed paths, when the set produced has zero area. This improves a similar result by the first author in the case of constant retention probabilities to the case of retention probabilities approaching 1