# Program

TUTORIALS:

An information theoretic approach to the second law of thermodynamics
Matteo Lostaglio
Basic ideas and tools of the resource theoretic approach to the second law of thermodynamics and its connection with quantum information theory. I will present a simple model in which thermodynamic accessibility is characterised by majorisation ⨪ the same relation characterising bipartite pure state entanglement. This will allow to discuss the meaning of entropy and what is single-shot thermodynamics about. Time permitting, I will hint at the role of quantum coherence in these models.

Geometrical approach to quantum statistical inference
Jan Kolodynski
We consider the statistical data inference tasks such as hypothesis testing, parameter estimation and their canonical variations, in order to review the relations between their corresponding figures of merit—measures—and demonstrate the crucial differences which arise in the quantum setting as compared to the classical one. In our analysis, we primarily focus on the geometrical approach to data inference problems, in which the corresponding measures can be neatly interpreted as particular forms of distances, divergences or metrics in the space of probability distributions and, when dealing with quantum systems, density matrices. As a result, we naturally generalise and establish novel relations between statistical inference techniques that have been previously developed when considering problems of: quantum state discrimination, quantum parameter estimation, limiting the speed of quantum evolution, and the interpretation of renormalisation group in quantum theory.

TALKS:

No-Go Theorem for the Characterisation of Work Fluctuations in Coherent Quantum Systems
Marti Perarnau Llobet
An open question of fundamental importance in thermodynamics is how to describe the fluctuations of work for quantum coherent processes. In the standard approach, based on a projective energy measurement both at the beginning and at the end of the process, the first measurement destroys any initial coherence in the energy basis. Here we seek for extensions of this approach which can possibly account for initially coherent states. We consider all measurement schemes to estimate work and require that (i) the difference of average energy corresponds to average work for closed quantum systems, and that (ii) the work statistics agree with the standard two-measurement scheme for states with no coherence in the energy basis. We first show that such a scheme cannot exist. Next, we consider the possibility of performing collective measurements on several copies of the state and prove that it is still impossible to satisfy simultaneously requirements (i) and (ii). Nevertheless, improvements do appear, and in particular we develop a measurement scheme which acts simultaneously on two copies of the state and allows to describe a whole class of coherent transformations.

Detecting the degree of nonlocality with two-body correlators
Flavio Baccari
It has already been shown in [1] that it is possible to construct valid Bell inequalities expressed in terms of up to two-body correlators, for an arbitrarily large number of parties. More importantly, these inequalities can be violated by physically relevant many-body quantum states. In this work we address the question of whether this kind of inequalities is able to capture more detailed information about the correlations in the system. We focus on the so-called degree of nonlocality, which is defined as follows: a set of correlations displays nonlocality of degree k if it can be described by a model in which groups of at most k parties are sharing nonlocal correlations. We are able to derive two-body inequalities that distinguish between different degrees of nonlocality, for the cases k <= 5. Our results prove that the knowledge of two-body correlations is enough to certify the number of particles that are sharing nonlocal correlation in a multipartite system. [1] J. Tura, R. Augusiak, A.B. Sainz, T. Vértesi, M. Lewenstein and A. Acín, Science 344, 1256-1258 (2014).

Time-ordered no-signalling randomness generation
Boris Bourdoncle
Non-local correlations contain some intrinsic private randomness. Quantitative relations between these two notions can be obtained in various framework, for different levels of trust, and under diverse assumptions on the abilities of the agents. There is a trade-off between the strength of the hypotheses that one assumes and the amount of randomness that one certifies. In particular, any information tasks, such as randomness generation, or more generally, privacy amplification or key distribution, is based on a protocol, i.e., successive choices of inputs to generate successive outputs. In this work, we address the problem of generating randomness when the correlations at each round are constrained by the natural assumption of time-ordered no-signalling, that is, when information can propagate forward in time but not backward. We give adversarial strategies that makes use of this information flow, and thus gives a higher predictability on the outputs, compared with the case where no information at all propagate between the rounds of a protocol.

Extended moment matrices for characterizing nonconvex sets
Alejandro Pozas-Kerstjens
Bell inequalities define the boundary between local and nonlocal correlations. Given that the set of correlations admitting local hidden variable models is a polytope, the enumeration of the Bell inequalities is easy after the vertices of the polytope are known and the resulting inequalities are linear in the probability distribution. There are, however, sets of correlations that are either nonconvex or have nonlinear facets, such as the bilocal set or the quantum set, the boundary of which will in general be described by nonlinear Bell-like inequalities. In this talk we propose a modification of the Navascués-Pironio-Acín hierarchy that is able to characterize nonconvex sets with possibly nonlinear facets. This modification allows for the inclusion of nonlinear constraints at the level of the probability distribution while maintaining solvability through semidefinite programming. We additionally report on recent findings on the bilocality scenario and the triangle scenario using this technique.

Quantum light propagation through atomic ensembles using matrix product states
Marco Manzoni
Systems consisting of optical fields propagating through and interacting with atomic ensembles constitute a major platform for the interface of quantum light and matter. Recently, a number of such systems have emerged, including Rydberg ensembles or atoms coupled to nanophotonic systems, in which strong nonlinear interactions between individual photons can be attained. An interesting and largely open problem is whether these systems can produce exotic many-body states of light, as the number of input photons is increased. To gain insight into this problem, it would be highly desirable to find approaches to numerically simulate light propagation in the many-body limit, which have remained elusive thus far. Here, we describe an approach to this problem by using a “spin model” that maps a quasi one-dimensional light propagation problem to the dynamics of an open 1D interacting spin system, where all of the photon correlations are obtained from those of the spins. This approach allows us to use the powerful toolbox of matrix product states to solve the spin problem and study multi photon effects. As a specific example, we apply this formalism to investigate vacuum induced transparency, wherein a pulse propagates through an atomic medium with a photon number-dependent group velocity, thereby enabling separation of different photon number components at the output.

Bose polaron as an instance of quantum Brownian motion
Aniello Lampo
Quantum Brownian motion represents a paradigmatic model of open quantum system. It describes the dynamics of a particle coupled to a huge environment. In our work we employ such a model to investigate the physical behaviour of an impurity embedded in an ultra colds gas (Polaron Problem). This approach allows to calculate measurable observables and predict new effects, such as superdiffusion and squeezing.

Topology of a dissipative spin
Loic Henriet
Topological quantum computation benefits from the protection of the encoded information from dissipation due to the existence of global topological invariants. In this talk, we analyse the effect of dissipation on topological properties. More precisely, we study the topological and geometrical deformations in a simple spin-1/2 model induced by an ohmic quantum dissipative environment at zero temperature. This model is known to display a Kosterlitz-Thouless quantum phase transition from a delocalized to a localized phase when increasing the coupling with the environment. From Bethe Ansatz results and a variational approach, we confirm that the Chern number is preserved in the delocalized phase (low coupling) and report a divergence of the Berry curvature at the equator at the transition. Recent experiments in quantum circuits have engineered non-equilibrium protocols in time to access topological properties at equilibrium from the measure of the quasi-adiabatic nonequilibrium spin expectation values. Applying a numerically exact stochastic Schrodinger equation we find that, for a fixed sweep velocity, the bath induces a crossover from quasi-adiabatic to non-adiabatic dynamical behaviour when the spin bath coupling increases.

Schur complements and matrix means in quantum optics
Ludovico Lami
Gaussian states and Gaussian operations are of primary interest in quantum optics, due to the ease of their practical implementations. Remarkably, the quadratic nature of the correlations displayed by these states makes matrix analysis tools suitable for their analysis. This work exploits this connection to prove novel results in quantum optics. – We derive fundamental constraints for the Schur complement of positive matrices, which provide an operator strengthening to recently established information inequalities for quantum covariance matrices, including strong subadditivity. As an application, we establish fundamental properties of a recently proposed Gaussian steerability measure [Phys. Rev. Lett. 114, 060403 (2015)]. – Since the correlations exhibited by Gaussian states are quadratic by definition, natural quantifiers of those are based on Rényi-2 entropies. We show that the Rényi-2 mutual information of a bipartite Gaussian state is lower bounded by twice its Rényi-2 Gaussian entanglement of formation (R2 EoF), an inequality that fails to hold for conventional von Neumann entropies. An important consequence of this inequality is the monogamy of the R2 EoF, here established for the first time. Perhaps surprisingly, this entanglement measure has been recently conjectured to be connected to cryptographic quantifiers [Phys. Rev. Lett. 117, 240505 (2016)].

Decay of correlations in systems of fermions with long-range interactions at non-zero temperature
Senaida Hernández
We study correlations in fermionic systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anti-commuting operators based on long-range Lieb-Robinson type bounds. Our result shows that correlations between such operators in fermionic long-range systems of spatial dimension $D$ with at most two-site interactions decaying algebraically with the distance with an exponent $\alpha \geq 2\,D$, decay at least algebraically with an exponent arbitrarily close to $\alpha$. Our bound is asymptotically tight, which we demonstrate by numerically analyzing density-density correlations in a 1D quadratic (free, exactly solvable) model, the Kitaev chain with long-range interactions. Away from the quantum critical point correlations in this model are found to decay asymptotically as slowly as our bound permits.

Using Random Boundary Conditions to simulate disordered quantum spin models in 2D-systems
Abel Yuste

Disordered quantum antiferromagnets in two-dimensional compounds have been a focus of interest in the last years due to their exotic properties. However, with very few exceptions, the ground states of the corresponding Hamiltonians are notoriously difficult to simulate making their characterization and detection very elusive, both, theoretically and experimentally. Here we propose a method to signal quantum disordered antiferromagnets by doing exact diagonalization in small lattices using random boundary conditions and averaging the observables of interest over the different disorder realizations. We apply our method to study a Heisenberg spin-1/2 model in an anisotropic triangular lattice. In this model, the competition between frustration and quantum fluctuations might lead to some spin liquid phases as predicted from different methods ranging from spin wave mean field theory to 2D-DMRG or PEPS. Our method accurately reproduces the ordered phases expected of the model and signals disordered phases by the presence of a large number of quasi degenerate ground states together with the absence of a local order parameter. The method presents a weak dependence on finite size effects.

Self-testing protocols based on the chained Bell inequalities
Alexia Salavrakos
Self testing is a device-independent technique based on non-local correlations whose aim is to certify the effective uniqueness of the quantum state and measurements needed to produce these correlations. It is known that the maximal violation of some Bell inequalities suffices for this purpose. However, most of the existing self-testing protocols for two devices exploit the well-known CHSH Bell inequality or modifications of it, and always with two measurements per party. In our work, we construct self-testing protocols based on the chained Bell inequalities, which are defined for two devices implementing an arbitrary number of two-output measurements. Our results have several implications : on the one hand, they imply that the quantum state and measurements leading to the maximal violation of the chained Bell inequality are unique. On the other hand, if we consider the limit of a large number of measurements, our approach allows the self-testing of the entire plane of measurements spanned by the Pauli matrices X and Z. Our results also imply that the chained Bell inequalities can be used to certify two bits of perfect randomness.

Device independent detection of entanglement depth with 2-body correlation functions
Albert Aloy
We construct device independent witnesses that involve at most 2-body correlation functions and that are able to detect how strongly entangled a quantum many-body system is. We construct the entanglement depth witness from Bell Inequalities that are constrained by symmetry and involving only one-and two-body correlation functions. Thus, in contrast with the usual constructions of Bell inequalities for many-body systems that quickly become computationally intractable and hard to test experimentally, we provide a tool to compute and measure experimentally how strongly entangled a quantum system is.

The large dimensional limit of multipartite entanglement
Sara Di Martino
Since its early origins entanglement has been considered as one of the most basic and intriguing features of quantum mechanics. However, the characterization and quantification of quantum correlations is not a simple task. On one side bipartite entanglement, i.e. the entanglement of two subsystems, is well understood and can be completely characterized, but on the other hand, multipartite entanglement is less understood and more elusive even if widely investigated. Our aim is to study the properties of multipartite entanglement of a system composed by n d- level particles (qudits). Focussing our attention on pure states we want to tackle the problem of the maximization of the entanglement for such system. In particular we consider the problem of minimizing the local purities of the system. It has been shown that in general not for all subsystems this function can reach its lower bound [1]. However it can be proved that for all values of n a d can always be found such that the lower bound can be reached. Adopting the concepts and tools of classical statistical mechanics, we introduce a Hamiltonian representing the average bipartite purity over all balanced bipartition [2] and examine its high- temperature expansion. In particular we make use of techniques that are based on the analysis of diagrams that naturally arise when one considers this expansion of the distribution function. We prove that the sum of these diagrams converges and we analyze its behavior as d goes to infinite. References 1. A. J. Scott. Phys. Rev. A, 69:052330 (2004). 2. Paolo Facchi, Giuseppe Florio, Ugo Marzolino, Giorgio Parisi, and Saverio Pascazio. Journal of Physics A: Mathematical and Theoretical, 42(5):055304 (2009).

Absolutely maximally entangled states in optimal quantum error correcting codes
Zahra Raissi
Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT correspondence. There is a direct correspondence between closed form expression for minimal support AME states, and classical maximal distance separable (MDS) error correcting codes of n parties with local dimension q (a power of prime). Further, from the single AME state, we develop orthonormal basis of AME states and stabilizers operators. We also show that, from these AME states, how to construct QECC for all n up to n=8 and q>=n-1, by finding several suitable incompressible operators. In these QECCs, a logical qudit is encoded in a q-dimensional subspace spanned by AME states of n parties.