I started working on this project when I joined the Theory Division of the Max Planck Institute of Quantum Optics. There is a major worldwide effort and a fierce competition among companies such as IBM, Google, Microsoft or Nokia Bell Labs to build a scalable quantum computer: Larger and larger computers are even placed online. Although the **First Generation** of **Quantum Computers** may look like stone-age quantum computers compared to universal, fault-tolerant ones (i.e., they will not be able to implement quantum error correction just yet), it is still open whether they can already outperform classical computers for specific problems.

I started to work in this field in 2013 during my Ph.D. studies at ICFO - The Institute of Photonic Sciences. This was the central topic of my Ph.D. thesis. Bell correlations are those that do not admit a **local hidden variable model** (LHVM) and constitute a resource for device-independent quantum information processing. Such correlations are strictly stronger than entanglement.

We started by developing **multipartite Bell inequalities** built from **few-body correlators**, typically one- and two-body. We focused on **Translationally Invariant**^{1 } (TI) and **Permutationally Invariant**^{2 } (PI) Bell inequalities. In 2014 we proposed an experimentally-friendly method to detect the existence of Bell correlations in many-body systems requiring onyl access to **total-spin components.**^{3 } This enabled the first experimental detection of Bell correlations in a **Bose-Einstein condensate** of 480 ^{87}Rb atoms in 2016^{4 } and in a **thermal ensemble** of 5·10^{5} atoms in 2017.^{5 }

In 2015, during my research stay at the Theory Division of the Max Planck Institute of Quantum Optics we showed that Bell correlations could be revealed in many-body systems simply by measuring their **energy**.^{6 } This method is specially tailored to spin systems in one spatial dimension, where the limit within LHVM theories can be efficiently computed via **dynamic programming**. In some special cases, the Hamiltonian can even be exactly diagonalized by mapping it to a **free fermion system** via the **Jordan-Wigner transformation**, thus even yielding analytically closed formulas.

In 2017, we proposed a way of efficiently approximating the set of correlations of a many-body system admitting an LHVM from the outside.^{7 } This consists on a hierarchy of **Semidefinite Programming** (SdP) tests that approximate **convex hulls of semialgebraic sets**,^{8 } based on **Lasserre's method of moments**.^{9 } This enables one to check e.g. experimental data against all PI few-body Bell inequalities with a simple test.

Currently, we are working on several research directions. First, to understand the structure behind operator-sums-of-squares for PI Bell inequalities that one obtains with the **NPA** (Navascués-Pironio-Acín) hierarchy. Second, to quantify how much **entanglement** and **nonlocality depth** can be certified by measuring such inequalities. Third, to understand the role of **temperature** as a detector of nonlocality and the connection between the Bell operator and the **quantum harmonic oscillator** via the **Holstein-Primakoff** approximate transformation. Fourth, we are exploring the role of energy as a detector of nonlocality in 2D systems, building upon **chordal extensions** of graphs.

I started this project at at the Theory Division of the Max Planck Institute of Quantum Optics in 2017.

With the advent of the **Device-Independent** (DI) paradigm to quantum information processing, **Bell inequalities** have gained an important role as certificates of a series quantum properties. I started working in that direction in 2013, during my Ph.D. studies, in the context of elemental monogamies of correlations, applied to DI **Randomness Amplification**.^{10 }

In 2015 I started working on DI **Self-Testing** for **Maximally Entangled** (MaxEnt) states. We proposed the **SATWAP** (Salavrakos-Augusiak-Tura-Wittek-Acín-Pironio) Bell inequality, which has the property that is maximally violated by MaxEnt states.^{11 }

In 2017 we started an experimental collaboration with the University of Bristol. We probed the SATWAP inequality on a **Large-scale Integrated Optics** platform embedding more than 550 optical elements, where programmable bipartite quantum states up to local dimension 15 can be prepared.^{12 } Our goal is now to extend this certification to multipartite **GHZ** quantum states.

I started working on this project in 2010 during my summer internship at ICFO - The Institute of Photonic Sciences, finding a numerical example proving the existence of **PPT** (positivity under partial transposition) symmetric entangled states of four qubits,^{13 } hence solving a long-standing open problem. Symmetric quantum states share a deep connection with mixtures of **Dicke states**. As a warm-up project in my Ph.D. studies, we delved further into the study of entanglement in PPT symmetric states of several qubits.^{14 }

In 2016, I have revisited this problem on various occasions: First, with my former B.Sc. student, in the context of Device-Independent **Tsirelson's bounds** for PPT states. Second, with my current co-supervised Ph.D. student, in the context **Diagonal-Symmetric** (DS) PPT states, where we have unveiled a connection with the field of **Quadratic Conic Optimization**, which enables one to show, in a natural way, that deciding membership in the set of DS separable states is, even in this very simplified case, **NP-hard**.^{15 }

I started working on this project in 2010, also during my summer internship at ICFO - The Institute of Photonic Sciences, which resulted in my first publication.^{16 } There, we investigate the connection between (decomposable) **Entanglement Witnesses** (EWs) and their optimality properties in terms of their action on **Completely Entangled Subspaces** (subspaces orthogonal to **Unextendible Product Basis**).

Also, during my Ph.D. studies we rediscovered a method to check for **optimality** of EWs,^{17 } which found application in the study of the now disproven **Structural Physical Approximations** conjecture.