1. We will study the specific traits differentiating the quantum description of physical systems from the classical one, within the context of the multi-time statistics associated with sequential measurements on an open quantum system, with a particular focus on the role played by the presence of correlations in time in the statistics and non-classical correlations between the open system and its environment. In particular, we will study the continuous-measurement interpretation of the dynamics of open quantum systems via the definition of quantum-jump unravellings associated with the master equation and will investigate the different kinds of memory effects which appear in such a context.
2. We will study the energy exchanges in driven quantum systems. Particular attention will be devoted to the theoretical and operational definition of the work done and the dissipated heat in quantum systems. We will propose an operational and consistent definition of quantum heat that is still missing in the literature. We will extend to the quantum level the classical concept of power and clarify how its measure can give information about the work and heat at the quantum level. We will study the relation between the statistics of quantum work and heat generated during the quantum dynamics.
Mathematics of Quantum Physics
1. We will further develop relativistic quantum theories of a particle according to methodological commitments that prevent from the problems that plagued the early theories. We will determine general quantum transformation relations (KQ) of position with respect to boosts. Whenever successful, an answer about the consistency of Dirac theory with Poincaré invariance will be attained. These relations will be the basis for developing a relativistic quantum theory of an interacting particle, according to the same commitments that prevent from inconsistencies. Contextually, the extension of the approach to zero mass particles will be pursued.
2. We shall study some aspects of interacting quantum field theories on flat and curved spacetime. We shall further analyze the perturbative construction of states at finite temperature in the adiabatic limits. Furthermore, we shall focus on foundational aspects in the quantization of gauge fields. A deeper understanding of the structure of gauge theories has recently been proposed following a geometric interpretation of the well-established BRST-BV approach using higher and derived geometry. We proceed in the refinement of the BV-BRST quantization developed in the framework of perturbative algebraic quantum field theory.
3. Regarding the application of noncommutative geometry to high energy physics, we will study how to implement Lorentz invariance within Connes theory of spectral triple. The goal is to show that the transition from the Euclidean to the Lorentzian case occurs also for the fermionic action of the Standard Model. Hence, we shall develop models beyond the Standard Model and we aim to study the phenomenology of this model.
4. We will study the interplay between a quantization map acting on the classical Curie-Weiss model, giving rise to the quantum Curie-Weiss model (a mean field approximation of the Heisenberg model) viewed as a theory in a Poisson-manifold described as the ball B^3, and a similar description using the boundary S^2 of that Poisson manifold. This latter model is technically easier to handle since the technology of coherent spin states can be exploited. In particular we will study the appearance of a spontaneous breaking of symmetry with respect to a discrete symmetry in a completely analytic approach.
5. We will apply the methods previously developed for the mathematical definition of Feynman path integrals on Riemannian manifolds. In addition, we will investigate the relations between Feynman quantization techniques and "traditional" quantization methods, in particular for non-essentially self-adjoint Hamiltonians.
1. We shall further analyze the interplay of gravity and quantum physics in two semiclassical ways. First, we aim to further analyze semiclassical Einstein equations and to study the form of its solution. Second, we shall study the geometrodynamics in the Hamilton-Jacobi formulation. We aim to obtain models which describe the evolution of our universe at early stage, and models of black hole evaporation in the semi-classical regime.
2. The problem of a global parametrix in a generic globally hyperbolic spacetime will be addressed, in order to generally define, for instance, the T-product for arguments arbitrarily far in the spacetime. The idea is to improve recent results in order to obtain a Hadamard parametrix which is as close as possible to a covariant notion.
3. We will continue our quantum spacetime investigations, with special emphasis on the relativistic aspects. We will continue our investigations on the postulates of quantum mechanics (e.g. the derivation of the tensor product postulate). We will investigate new possibilities for entropic uncertainty relations.
Beyond Quantum Mechanics
1. We will carry on the analysis of theories sharing one particular property, for example simplicial systems, the possibility/impossibility of extracting information without disturbance, bilocal discriminability, and the way in which systems compose. Specific models of such theories will also be developed and studied, such as a theory whose systems are all classical or quantum, but whose composite systems have a larger information capacity than the sum of the capacities of the components. The possibility of consistently developing such theories has important consequences for some mainstream approaches to foundations, exclusively focused on the geometry of convex sets of states of single systems and/or the correlations of bipartite systems.
2. We will tackle the challenge of describing physical phenomena in an alternative universe, whose systems are different form the quantum ones. The study consists in the first place in a general theory of cellular automata (CA), that represent homogeneous and local evolution algorithms, which in turn represent microscopic physical laws. This approach requires also the study of the connections of the local behaviour of CAs with their large-scale behaviour, with techniques reminiscent of renormalization, bridging microscopic theories with large scale effective theories. This program requires an extension of OPTs allowing to define and treat systems made by composing infinitely many elementary ones.
3. In attempting to overcome the difficulties connected with the quantum measurement problem, two types of stochastic processes in Hilbert space have been considered, namely the so-called hitting or discontinuous processes (GRW) and the continuous ones (CSL). All proposed processes aim at describing reduction as a physical process. The relationship between discontinuous and continuous processes will be explored and a detailed study of collapse dynamics in Hilbert space will be developed. Relativistic collapse models presented in the literature will also be studied, in order to assess to which extent they represent fully relativistic (but necessarily nonlocal) theories.
Theoretical input for new experiments
1. We will develop new tests for quantum incompatibility by using communication protocols involving a single qudit rather than multipartite systems. In such tests, incompatibility is revealed when the protocol violates some classical limit to the reconstruction of the transmitted information.
2. We will study novel experimental schemes for non-interferometric tests of spontaneous collapse models. We will exploit quantum-enhanced metrological techniques to assess and optimize the sensitivity of the experiment to the tiny space and energy fluctuations predicted by the models. Different degrees of freedom will be considered (center of mass motion, thinness of the object, internal dynamics…). Parallel to this, an accurate analysis of decoherence effects will be performed.
Towards Quantum Technologies
1. Methods to achieve accessible bounds to the classical capacity of unknown quantum communication channels has been derived. A quantum thermodynamics approach to study energy exchanges between two boson modes has been developed. Now, entanglement properties of hypergraph states in dimension higher than 2 will be analysed. Methods to detect lower bounds to the quantum capacities for correlated two-bit noise and to the classical capacities of single qubit noise will be tested experimentally on quantum optical platforms.
2. We will continue the investigations on quantum metrology. We will explore the practical applications of the general theory of squeezing metrology we recently developed. We will also work on quantum computation, regarding novel algorithms for the calculation of matrix manipulations. We will finalize our project on quantum tomography for imaging, where techniques developed for quantum mechanics are applied to medical tomography.
3. We will study, both from the theoretical and experimental viewpoint, the possibility to construct a quantum random number generator whose quantum nature is certified by the violation of the CHSH inequality and making use of single-particle-entanglement. Already performed experiments indicate that single-particle entangled states of single photons can be produced from attenuated classical sources of light. This fact suggests that low-power entangled photon sources can be used for a range of quantum technology applications.