Temporal quantum correlations and Leggett-Garg inequalities in multi-level systems

We show that the quantum bound for temporal correlations in a Leggett-Garg test, analogous to the Tsirelson bound for spatial correlations in a Bell test, strongly depends on the number of levels $N$ that can be accessed by the measurement apparatus via projective measurements. We provide exact bounds for small $N$, that exceed the known bound for the Leggett-Garg inequality, and show that in the limit $N\rightarrow \infty$ the Leggett-Garg inequality can be violated up to its algebraic maximum.Comment: 6 pages, 2 figure

Theoretical research without projects

We propose a funding scheme for theoretical research that does not rely on project proposals, but on recent past scientific productivity. Given a quantitative figure of merit on the latter and the total research budget, we introduce a number of policies to decide the allocation of funds in each grant call. Under some assumptions on scientific productivity, some of such policies are shown to converge, in the limit of many grant calls, to a funding configuration that is close to the maximum total productivity of the whole scientific community. We present numerical simulations showing evidence that these schemes would also perform well in the presence of statistical noise in the scientific productivity and/or its evaluation. Finally, we prove that one of our policies cannot be cheated by individual research units. Our work must be understood as a first step towards a mathematical theory of the research activity.Comment: Some edits to the published versio

Leggett-Garg Macrorealism and temporal correlations

Leggett and Garg formulated macrorealist models encoding our intuition on classical systems, i.e., physical quantities have a definite value that can be measured with minimal disturbance, and with the goal of testing macroscopic quantum coherence effects. The associated inequalities, involving the statistics of sequential measurements on the system, are violated by quantum mechanical predictions and experimental observations. Such tests, however, are subject to loopholes: a classical explanation can be recovered assuming specific models of measurement disturbance. We review recent theoretical and experimental progress in characterizing macrorealist and quantum temporal correlations, and in closing loopholes associated with Leggett-Garg tests. Finally, we review recent definitions of nonclassical temporal correlations, which go beyond macrorealist models by relaxing the assumption on the measurement disturbance, and their applications in sequential quantum information processing.Comment: 21 pages, 7 figures. Comments are welcome

Logica e probabilita' in meccanica quantistica

E' noto che i valori di aspettazione calcolati tramite il formalismo della meccanica quantistica sono interpretabili in termini di probabilita' classiche se ci si restringe a misure "compatibili", cioe' descritte da operatori commutanti. La collezione di teorie classiche che ne risulta ha assunto nel tempo un ruolo sempre piu' centrale nell'analisi delle interpretazioni della meccanica quantistica. Partendo da questa osservazione ci proponiamo di analizzare la struttura logica e probabilistica della meccanica quantistica all'interno di una classe di teorie della probabilita' che generalizzano quella classica, in cui la struttura logica degli osservabili (si/no) e' quella di algebra di Boole parziale. Tali teorie sono definite da una collezione di algebre booleane, in generale non disgiunte, in cui le operazioni sono definite in modo indipendente dall'algebra considerata e la struttura probabilistica e' data da una collezione di probabilita' classiche, cioe' una collezione di misure normalizzate definite sulle algebre che formano l'algebra parziale. Mostriamo che tale struttura e' definibile sulla base di richieste espresse in termini empirici soddisfatte in particolare dalla meccanica quantistica. Successivamente studiamo le proprieta' algebriche di queste teorie della probabilita' generalizzata e in particolare la possibilita' di "completarle" o "estenderle" ad una teoria probabilistica classica; cioe' la possibilita' di estendere l'algebra di Boole parziale ad un'algebra di Boole e la collezione di misure ad un'unica misura definita sull'algebra ottenuta, utilizzando due criteri, uno basato sostanzialmente sul politopo di correlazione di Pitowsky e l'altro basato sulla nozione di misura parziale introdotta da Tarski e Horn. Poiche' tali "completamenti" sono in realta' impliciti in ogni tentativo di interpretazione classica della meccanica quantistica, siamo in grado di discutere, sulla base dei risultati ottenuti, i problemi che nascono da tali tentativi, in particolare discutiamo le interpretazioni delle disuguaglianze alla Bell distinguendo il ruolo delle pure ipotesi probabilistiche rispetto alle nozioni usuali di "causalita'" e "localita'"

Temporal quantum correlations and hidden variable models

This thesis is devoted to the investigation of the differences between the predictions of classical and quantum theory. More precisely, we shall analyze such differences starting from their consequences on quantities with a clear empirical meaning, such as probabilities, or relative frequencies, that can be directly observed in experiments. Different kind of classical probability theories, or hidden variable theories, corresponding to different physical constraints imposed on the measurement scenario are discussed, namely, locality,noncontextuality and macroscopic realism. Each of these theories predicts bounds on the strength of correlations among different variables, and quantum mechanical predictions violate such bounds, thus revealing a stark contrast with our classical intuition. Our work starts with the investigation of the set of classical probabilities by means of the correlation polytope approach, which provides a minimal and optimal set of bounds for classical correlations. In order to overcome some of the computational difficulties associated with it, we develop an alternative method that avoid the direct computation of the polytope and we apply it to Bell and noncontextuality scenarios showing its advantages both for analytical and numerical computations. A different notion of optimality is then discussed for noncontextuality scenarios that provide a state-independent violation: Optimal expression are those maximizing the ratio between the quantum and the classical value. We show that this problem can be formulated as a linear program and solved with standard numerical techniques. Moreover, optimal inequalities for the cases analyzed are also proven to be part of the minimal set described above. Subsequently, we provide a general method to analyze quantum correlations in the sequential measurement scenario, which allows us to compute the maximal correlations. Such a method has a direct application for computation of maximal quantum violations of Leggett-Garg inequalities, i.e., the bounds for correlation in a macroscopic realist theories, and it is relevant in the analysis of noncontextuality tests, where sequential measurements are usually employed. Finally, we discuss a possible application of the above results for the construction of dimension witnesses, i.e., as a certification of the minimal dimension of the Hilbert spaces needed to explain the arising of certain quantum correlations.to Bell and noncontextuality scenarios showing its advantages both for analytical and numerical computations.Diese Doktorarbeit befasst sich mit der Untersuchung der unterschiedlichen Vorhersagen von klassischen Theorien und Quantenmechanik. Es werden verschiedene klassische Wahrscheinlichkeitstheorien oder Theorien, die auf der Existenz versteckter Variablen basieren, diskutiert und besonders auf ihre Vorhersagen bezüglich der möglichen Stärke der Korrelationen zwischen verschiedenen Variablen eingegangen. Die klassischen Theorien machen dabei unterschiedliche physikalischen Annahmen wie Lokalität, Nichtkontextualität oder makroskopischer Realismus. Für jede dieser Theorien sagt die Quantenmeachnik stärkere Korrelationen voraus, die die klassischen Schranken verletzen und damit im Widerspruch zu unserer klassisch geprägten Intuition stehen. Unsere Arbeit beginnt mit der Untersuchung der Menge von klassischen Wahrscheinlichkeiten mittels des Korrelations-Polytop-Verfahrens, welches einen minimalen und optimalen Satz an Grenzen für klassische Korrelationen liefert. Um einige der mit diesem Verfahren verbundenen rechnerischen Schwierigkeiten zu überwinden, entwickeln wir eine alternative Methode, die die direkte Berechnung des Polytops umgeht. Angewendet auf Bell- und Kontextualitätsszenarien zeigen wir die Vorteile unserer Methode, sowohl bezüglich analytischer, als auch numerischer Berechnungen. Danach wird eine andere Möglichkeit betrachtet, Optimalität für Nichtkontextualitätsungleichungen zu definieren, die eine zustandsunabhängige Verletzung aufweisen: Optimale Ungleichungen sind solche, die das Verhältnis zwischen quantenmachanischem und klassischem Wert maximieren. Wir zeigen, dass dieses Problem als lineares Programm formuliert und mit standardmäßigen, numerischen Methoden gelöst werden kann. Darüber hinaus beweisen wir, dass die optimalen Ungleichungen für die betrachteten Fälle jene sind, die Teil des oben beschriebenen minimalen Satzes von Grenzflächen sind. Anschließend stellen wir eine allgemeine Methode vor mit der man Quantenkorrelationen bei sequentiellen Messungen analysieren kann und die maximalen Korrelationen berechnen kann. Ein solches Verfahren hat als direkte Anwendung die Berechnung maximaler Quantenverletzung von Leggett-Garg Ungleichungen, d.h. der Grenzen für Korrelationen in Theorien, die auf der Annahme des makroskopischem Realismus basieren. Zudem ist diese Methode relevant in der analytischen Betrachtung von Kontextualitätstests, in denen üblicherweise sequentielle Messungen verwendet werden. Abschließend diskutieren wir für die obigen Resultate Anwendungen bei der Konstruktion von Zeugenoperatoren für die Dimension von Quantensystemen. Damit ist es möglich, die minimale Dimension des Hilbertraums zu zertifizieren, die nötig ist, um das Auftreten von gegebenen Quantenkorrelationen zu erklären