The norm-1-property of a quantum observable

A normalized positive operator measure $X\mapsto E(X)$ has the norm-1-property if \no{E(X)}=1 whenever $E(X)\ne O$. This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrary close to one with suitable state preparations. Some general implications of the norm-1-property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.Comment: 14 page

Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds

This paper proposes to generalize the histories included in Euclidean functional integrals from manifolds to a more general set of compact topological spaces. This new set of spaces, called conifolds, includes nonmanifold stationary points that arise naturally in a semiclasssical evaluation of such integrals; additionally, it can be proven that sequences of approximately Einstein manifolds and sequences of approximately Einstein conifolds both converge to Einstein conifolds. Consequently, generalized Euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary point of the Einstein action. Therefore sums over conifold histories provide a useful and self-consistent starting point for further study of topological effects in quantum gravity. Postscript figures available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.

The intuitionistic temporal logic of dynamical systems

A dynamical system is a pair $(X,f)$, where $X$ is a topological space and $f\colon X\to X$ is continuous. Kremer observed that the language of propositional linear temporal logic can be interpreted over the class of dynamical systems, giving rise to a natural intuitionistic temporal logic. We introduce a variant of Kremer's logic, which we denote ${\sf ITL^c}$, and show that it is decidable. We also show that minimality and Poincar\'e recurrence are both expressible in the language of ${\sf ITL^c}$, thus providing a decidable logic expressive enough to reason about non-trivial asymptotic behavior in dynamical systems

Approximated Symbolic Computations over Hybrid Automata

Hybrid automata are a natural framework for modeling and analyzing systems which exhibit a mixed discrete continuous behaviour. However, the standard operational semantics defined over such models implicitly assume perfect knowledge of the real systems and infinite precision measurements. Such assumptions are not only unrealistic, but often lead to the construction of misleading models. For these reasons we believe that it is necessary to introduce more flexible semantics able to manage with noise, partial information, and finite precision instruments. In particular, in this paper we integrate in a single framework based on approximated semantics different over and under-approximation techniques for hybrid automata. Our framework allows to both compare, mix, and generalize such techniques obtaining different approximated reachability algorithms.Comment: In Proceedings HAS 2013, arXiv:1308.490

A doubly exponential upper bound on noisy EPR states for binary games

This paper initiates the study of a class of entangled games, mono-state games, denoted by $(G,\psi)$, where $G$ is a two-player one-round game and $\psi$ is a bipartite state independent of the game $G$. In the mono-state game $(G,\psi)$, the players are only allowed to share arbitrary copies of $\psi$. This paper provides a doubly exponential upper bound on the copies of $\psi$ for the players to approximate the value of the game to an arbitrarily small constant precision for any mono-state binary game $(G,\psi)$, if $\psi$ is a noisy EPR state, which is a two-qubit state with completely mixed states as marginals and maximal correlation less than $1$. In particular, it includes $(1-\epsilon)|\Psi\rangle\langle\Psi|+\epsilon\frac{I_2}{2}\otimes\frac{I_2}{2}$, an EPR state with an arbitrary depolarizing noise $\epsilon>0$.The structure of the proofs is built the recent framework about the decidability of the non-interactive simulation of joint distributions, which is completely different from all previous optimization-based approaches or "Tsirelson's problem"-based approaches. This paper develops a series of new techniques about the Fourier analysis on matrix spaces and proves a quantum invariance principle and a hypercontractive inequality of random operators. This novel approach provides a new angle to study the decidability of the complexity class MIP$^*$, a longstanding open problem in quantum complexity theory.Comment: The proof of Lemma C.9 is corrected. The presentation is improved. Some typos are correcte

Knowability as continuity: a topological account of informational dependence

We study knowable informational dependence between empirical questions, modeled as continuous functional dependence between variables in a topological setting. We also investigate epistemic independence in topological terms and show that it is compatible with functional (but non-continuous) dependence. We then proceed to study a stronger notion of knowability based on uniformly continuous dependence. On the technical logical side, we determine the complete logics of languages that combine general functional dependence, continuous dependence, and uniformly continuous dependence.Comment: 65 page