My research is focused on the foundations of quantum mechanics and its overlap with quantum computation and quantum information. Some big questions in these fields that I try to address are: how to interpret the nature of the reality described by quantum mechanics? Which features have to be considered as inherently non-classical? Which is the source of the quantum computational speed-up?
I am happy to work with students in any stage of their studies, even if students with a background of linear algebra and basic quantum mechanics are more suitable. Some knowledge of quantum computing and quantum information would also be useful, but not essential.
The projects can be adjusted according to the level of the students, and they usually involve a literature review, discussions of what to get from the literature and consequent development of new ideas, analytical calculations and possibly numerical calculations.
I can work both with individuals and with multiple students simultaneously. My aim is to feed the interest of the student in doing research, obtain novel insights in the proposed topic and produce original results. Hopefully this will also lead to works that can be published in peer reviewed journals.
A field of research in quantum foundations aims at understanding what are the features of quantum theory that have to be considered as inherently non-classical, meaning that they are not present in any classical theory, even when the classical theory is powered with some intuitive extra structures.
General Probabilistic Theories (GPTs) provides a general framework to distinguish classical from quantum theory, and quantum theory from more general theories in terms of natural axioms. Just to give an example of an axiom that holds both for quantum and classical theory, I recall the causality axiom, saying that the probability of an outcome of a measurement does not depend on later measurements.
Looking at quantum theory ‘’from outside’’ through this framework has allowed people to understand what is special about it. In this project we want to understand what is special about quantum theory ‘’from inside’’, instead.
I propose to explore and study the axioms that can distinguish subtheories within quantum theory and quantum theory. In particular, I propose to study popular quantum subtheories that are efficiently simulatable by classical computers, like the so called stabilizer theory. This may shed light on which principles underlie the onset of the quantum computational power. The project will consist of reviewing the main results on GPTs (2-3 papers) and analyze which of the axioms therein considered apply to the stabilizer theory (or other quantum subtheories that we may decide to study). It will involve the study of the geometry of the state spaces for these theories (we will probably first focus on the qubit case) and analytic calculations.
It is possible to rephrase quantum mechanics in the formalism of classical mechanics – based on the phase space – at the price of accepting that probabilities in the phase space can sometimes be negative. The negativity has been usually associated with a signature of quantumness and a popular theorem shows that negativity and contextuality are equivalent notions. The latter is one of the main features of quantum mechanics that is considered to be inherently nonclassical and it is phrased in the framework of ontological models (which is the mathematical way we use to define reality).
Recent results have put into question this theorem, in particular when transformations are included in the experimental scenario.
The project will focus on reviewing the notions of quasiprobability representation and contextuality and on studying the mentioned theorem (pin down all its assumptions and unveil the hidden ones) to reconcile it with the results that are in conflict with it when considering transformations.
The features of quantum mechanics that are considered to be inherently nonclassical so far all come from no go theorems. These theorems are about comparing some formal notion of classicality in the framework of ontological models (which is the mathematical way we use to define reality) with quantum theory and prove an inconsistency between the two.
The most popular no go theorem is Bell’s theorem, which seems to suggest that the world is nonlocal.
The scenario studied in Bell’s theorem can be also studied to address the question of whether the invariance of the physical description under Lorentz transformations (so for two reference frames in relative motion) – one of the cornerstone of relativity – is preserved at the ontological level. It this is not the case, this would prove that there is a preferred reference frame at the ontological level. Definitely a stunning result.
The project will focus on critically reviewing the literature on previous attempts to do that and also on examples of ontological models for certain interpretations of quantum mechanics (de Broglie-Bohm theory and collapse theories) where the breaking of Lorentz invariance is manifested. A good summary of the issues arising from the literature on the topic and possible ideas on how to proceed would be enough for this challenging project. However we will also try to come up with ideas to prove the no go theorem in general.
A popular field of research in quantum information is the one of the so called ‘’non-local games’’. It allows to show the quantum computational supremacy over classical computation for some particular tasks and to study the physical origin of this supremacy.
Probably the most popular of such games is the CHSH game, where a referee asks two binary (either 0 or 1) questions to two players that have to provide a binary answer each. They are very distant and cannot communicate after they receive the questions. They win the game if the sum of their answers is equal to the product of the questions (arithmetic modulo 2). It results that by using strategies based on quantum mechanics the probability of success is greater than if they use classical strategies.
The CHSH game is of great importance because the sensitivity of its optimal success probability depending on the underlying physical model gives us a tool to distinguish different types of theories experimentally, and allows us to test nature. It also reveals insights into a non-classical feature of quantum mechanics, known colloquially as ‘’non-locality’’.
The CHSH game is stated in arithmetic modulo 2, and shows a bound on the performances of classical strategies known as the Bell bound and a (higher) bound on the performances of quantum strategies known as Tsirelson’s bound. When the game is extended to arithmetic modulo d, for d an arbitrary number, it is still unclear what the exact value of the Tsirelson’s bound is.
Recently, a game related to the CHSH game has been developed. A mapping from strategies in this game and strategies in the CHSH game has been proven. Moreover, the search for the Tsirelson bound in this game for arithmetic modulo d looks easier. This could lead to the exact value of the Tsirelson bound for the CHSH game in arithmetic modulo d, which is a long standing question.
The project will consist of reviewing a couple of papers on the CHSH game and the related one, and develop ideas on how to obtain the Tsirelson bound when considering arithmetic modulo d (starting with d=3). This will probably consist of studying in details the alleged optimal quantum strategies. The math is simple and the games are very easy to understand (they also have a clear geometric interpretation in d=2). The project will probably involve some simple numerics to compute the probability of success for the different strategies.