Quantum Mechanics as a Framework for Dealing with Uncertainty

Quantum uncertainty is described here in two guises: indeterminacy with its concomitant indeterminism of measurement outcomes, and fuzziness, or unsharpness. Both features were long seen as obstructions of experimental possibilities that were available in the realm of classical physics. The birth of quantum information science was due to the realization that such obstructions can be turned into powerful resources. Here we review how the utilization of quantum fuzziness makes room for a notion of approximate joint measurement of noncommuting observables. We also show how from a classical perspective quantum uncertainty is due to a limitation of measurability reflected in a fuzzy event structure -- all quantum events are fundamentally unsharp.Comment: Plenary Lecture, Central European Workshop on Quantum Optics, Turku 2009

The Diffusion of Cattle Ranching and Deforestation – Prospects for a Hollow Frontier in Mexico’s Yucatán

This article investigates the behavioral drivers of pasture creation and associated implications for deforestation in a 22,000 km2 agricultural frontier spanning the base of Mexico‘s southern Yucatán. After developing a theoretical model that highlights the role of social networks and information spillovers with respect to the decision to begin cattle ranching, we use household data to estimate an econometric duration model of the determinants of pasture creation. Although pasture fi ts well with the typical household‘s resource constraints, its continued expansion contributes to a hollow frontier dynamic in which the spread of low-value cattle ranching coincides with decreasing population.Pasture creation; information spillovers; duration analysis; farm households; Mexico

Non-disturbing quantum measurements

We consider pairs of quantum observables (POVMs) and analyze the relation between the notions of non-disturbance, joint measurability and commutativity. We specify conditions under which these properties coincide or differ---depending for instance on the interplay between the number of outcomes and the Hilbert space dimension or on algebraic properties of the effect operators. We also show that (non-)disturbance is in general not a symmetric relation and that it can be decided and quantified by means of a semidefinite program.Comment: Minor corrections in v

Characterization of the Sequential Product on Quantum Effects

We present a characterization of the standard sequential product of quantum effects. The characterization is in term of algebraic, continuity and duality conditions that can be physically motivated.Comment: 11 pages. Accepted for publication in the Journal of Mathematical Physic

Approximating incompatible von Neumann measurements simultaneously

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the Luders instrument of the optimal joint observable.Comment: 9 pages, 4 figures; v2 has a more extensive introduction + other minor correction

Uncertainty reconciles complementarity with joint measurability

The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A new joint measurement scheme for complementary observables is proposed. The measured observables are represented as positive operator valued measures (POVMs), whose intrinsic fuzziness parameters are found to satisfy an intriguing pay-off relation reflecting the complementarity. At the same time, this relation represents an instance of a Heisenberg uncertainty relation for measurement imprecisions. A model-independent consideration show that this uncertainty relation is logically connected with the joint measurability of the POVMs in question.Comment: 4 pages, RevTeX. Title of previous version: "Complementarity and uncertainty - entangled in joint path-interference measurements". This new version focuses on the "measurement uncertainty relation" and its role, disentangling this issue from the special context of path interference duality. See also http://www.vjquantuminfo.org (October 2003

The Standard Model of Quantum Measurement Theory: History and Applications

The standard model of the quantum theory of measurement is based on an interaction Hamiltonian in which the observable-to-be-measured is multiplied with some observable of a probe system. This simple Ansatz has proved extremely fruitful in the development of the foundations of quantum mechanics. While the ensuing type of models has often been argued to be rather artificial, recent advances in quantum optics have demonstrated their prinicpal and practical feasibility. A brief historical review of the standard model together with an outline of its virtues and limitations are presented as an illustration of the mutual inspiration that has always taken place between foundational and experimental research in quantum physics.Comment: 22 pages, to appear in Found. Phys. 199

New spin squeezing and other entanglement tests for two mode systems of identical bosons

For any quantum state representing a physical system of identical particles, the density operator must satisfy the symmetrization principle (SP) and conform to super-selection rules (SSR) that prohibit coherences between differing total particle numbers. Here we consider bi-partitite states for massive bosons, where both the system and sub-systems are modes (or sets of modes) and particle numbers for quantum states are determined from the mode occupancies. Defining non-entangled or separable states as those prepared via local operations (on the sub-systems) and classical communication processes, the sub-system density operators are also required to satisfy the SP and conform to the SSR, in contrast to some other approaches. Whilst in the presence of this additional constraint the previously obtained sufficiency criteria for entanglement, such as the sum of the ˆSx and ˆSy variances for the Schwinger spin components being less than half the mean boson number, and the strong correlation test of |haˆm (bˆ†)ni|2 being greater than h(aˆ†)maˆm (bˆ†)nbˆni(m, n = 1, 2, . . .) are still valid, new tests are obtained in our work. We show that the presence of spin squeezing in at least one of the spin components ˆSx , ˆSy and ˆSz is a sufficient criterion for the presence of entanglement and a simple correlation test can be constructed of |haˆm (bˆ†)ni|2 merely being greater than zero.We show that for the case of relative phase eigenstates, the new spin squeezing test for entanglement is satisfied (for the principle spin operators), whilst the test involving the sum of the ˆSx and ˆSy variances is not. However, another spin squeezing entanglement test for Bose–Einstein condensates involving the variance in ˆSz being less than the sum of the squared mean values for ˆSx and ˆSy divided by the boson number was based on a concept of entanglement inconsistent with the SP, and here we present a revised treatment which again leads to spin squeezing as an entanglement test

The structure of classical extensions of quantum probability theory

On the basis of a suggestive definition of a classical extension of quantum mechanics in terms of statistical models, we prove that every such classical extension is essentially given by the so-called Misra–Bugajski reduction map. We consider how this map enables one to understand quantum mechanics as a reduced classical statistical theory on the projective Hilbert space as phase space and discuss features of the induced hidden-variable model. Moreover, some relevant technical results on the topology and Borel structure of the projective Hilbert space are reviewed