A precise definition of “weak [quantum] measurements” and “weak value” (of a quantum observable) is offered, and simple finite dimensional examples are given showing that weak values are not unique and therefore probably do not correspond to any physical attribute of the system being “weakly” measured, contrary to impressions given by most of the literature on weak measurements. A possible math...

1. Incompleteness of weak duals of reasonable spaces 2. Appendix: locally-convex limits and colimits 3. Appendix: ubiquity of quasi-completeness The point here is to prove that the weak duals of reasonable topological vector spaces, such as infinite-dimensional Hilbert, Banach, or Fréchet spaces, are not complete. That is, in these weak duals there are Cauchy nets which do not converge. Happily...

Hall’s recent derivation of an exact uncertainty relation [Phys. Rev. A64, 052103 (2001)] is revisited. It is found that the Bayes estimator of an observable between preand postselection equals the real part of the weak value. The quadratic loss function equals the expectation of the squared imaginary part of the weak value.

Various quantum measurement procedures are analyzed and it is shown that under certain conditions they yield consistently weak values which might be very different from the eigenvalues, the allowed outcomes according to the standard quantum formalism. The weak value outcomes result from peculiar quantum interference of the pointer variable of the measuring device.

The article [1] presents a theorem which gives sufficient conditions for a “general conditioned average” introduced in [3] to yield the traditional weak value in the limit of arbitrarily weak measurements. The “general conditioned average” corresponds to what would more usually be called a (preselected and) postselected weak measurement. Since there is only one theorem in that article, we will ...

Broadly distributed random variables with a power-law distribution f (m) ∼ m−(1+α) are known to generate condensation effects. This means that, when the exponent α lies in a certain interval, the largest variable in a sum of N (independent and identically distributed) terms is for large N of the same order as the sum itself. In particular, when the distribution has infinite mean (0 < α < 1) one...

Hirschfeldt and Shore have introduced a notion of stability for infinite posets. We define an arguably more natural notion called weak stability, and we study the existence of infinite computable or low chains or antichains, and of infinite Π1 chains and antichains, in infinite computable stable and weakly stable posets. For example, we extend a result of Hirschfeldt and Shore to show that ever...

Let −∆ denote the Dirichlet Laplace operator on a bounded open set in Rd. We study the sum of the negative eigenvalues of the operator −h∆− 1 in the semiclassical limit h → 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on th...

Procedures designed to select alternatives on the basis of the results of pairwise contests between them have received much attention in literature. The particular case of tournaments has been studied in depth. More recently weak tournaments and valued generalizations thereof have been investigated. The purpose of this paper is to investigate to what extent these choice procedures may be meanin...