# Research

Our research focuses on the study of quantum chaos and out-of-equilibrium dynamics of quantum many-body systems viewed through the lens of quantum information theory and statistical mechanics. The recent experimental and technological progress in quantum simulation and computation has stimulated enormous theoretical interest in the coherent dynamics of complex quantum systems, motivating a renaissance for the topic, which has become a broad multidisciplinary field, with concepts and approaches stemming from condensed matter, high energy, atomic physics and quantum optics and information theory. The group interests are guided by two main questions: How does chaos emerge from the dynamics of many-particle quantum systems? Can we use quantum correlations to challenge classical dynamics?

Below you can a selection of the research highlights in the group.

## The eigenstate thermalization hypothesis

Understanding how isolated quantum systems thermalise under unitary evolution is a theme as old as quantum mechanics itself. In isolated classical systems, thermalisation relies on the emergence of chaos and ergodicity, which together lead phase-space trajectories starting from the same energy to become indistinguishable when averaged over time. In the quantum domain, relaxation to thermal equilibrium is nowadays well understood by the Eigenstate Thermalization Hypothesis (ETH). According to ETH, local observables in the energy eigenbasis are pseudorandom matrices, whose statistical properties are smooth thermodynamic functions. This simple assumption on the structure of matrix elements has proved to be extremely successful in describing the dynamical properties of physical Hamiltonian systems: from their thermalization to the entanglement of chaotic eigenstates and to two-point dynamical correlations at equilibrium.

We are interested in understanding the effects of ETH on quantum entanglement and multi-time correlation functions. The latter are the relevant quantities beyond linear response, to characterize higher-order hydrodynamics, quantum chaos and scrambling. For instance, very recently we have shown that general ETH, which accounts for higher-order correlations among matrix elements, can be rationalized theoretically using the language of Free Probability. Free Probability is a branch of math which generalizes probability to non-commuting variables. (Check out the Free probability blog here.) This mathematical framework seems to be very promising in describing higher-order correlations in physical lattice systems and beyond...

Selected publications:

ETH and multipartite entanglement:

M. Brenes, S. Pappalardi, J. Goold, A. Silva - Phys.Rev.Lett. (2020).ETH and Free Probability:

S. Pappalardi, L.Foini and J.Kurchan - Phys.Rev.Lett. (2022).

S. Pappalardi, F Fritzsch, T. Prosen - Arxiv 2303.00713 (2023).

M. Fava, L. Foini and S. Pappalardi - Arxiv 2308.06200 (2023).

## Multipartite entanglement in many-body systems

There are many facets to entanglement theory and, in particular, many-body systems offer the perfect playground to explore multipartite entanglement. In our group, we are interested in the Quantum Fisher Information (QFI), which characterizes the usefulness of quantum states for phase-estimation purposes. It is also a witness of multipartite entanglement: it gives the size of the biggest entangled block of a composite system. This quantity has recently attracted growing attention in the community of many-body systems, mostly due to its remarkable relation to thermal susceptibilities, which makes the QFI experimental accessible in condensed matter physics.

We are interested in establishing the QFI as a complementary quantity to characterise quantum many-body states, thus identifying their usefulness for metrological purposes. So far, besides their connection in the semi-classical limit with the entanglement entropy and chaos quantifiers 1., we have shown that the QFI allows distinguishing thermal density matrices from ETH eigenstates at the corresponding energy 2.. Another convincing example is the one of many-body quantum scars (athermal eigenstates in the middle of the spectrum) which possesses a QFI that scales extensively with system size, in contrast to generic thermal states, hence signalling their significance as a resource in quantum-enhanced metrology 3..

Selected publications

QFI in semi-classical dynamics where QFI=EE (entanglement entropy)

A. Lerose and S. Pappalardi - Phys. Rev. A (2020).QFI for ETH eigenstates

M. Brenes, S. Pappalardi, J. Goold and A. Silva - Phys. Rev. Lett. (2019).QFI for many-body quantum scars

J.Y. Desaules, F. Pietracaprina, Z. Papić, J. Goold and S. Pappalardi - Phys. Rev. Lett. (2022)

S. Dooley, S. Pappalardi and J. Goold - Phys.Rev.B. (2023). .

## "Planckian" bounds

There exist a set of bounds imposed by quantum effects on the physical properties of many-body systems, such as viscosity, conductivity or Lyapunov exponent. They are characterized by a time-scale which only depends on temperature and the Planck constant and for this reason they have been dubbed ``"Planckian bounds". Their remarkable feature is that they are all saturated by models of black holes. The physical mechanism underlying these bounds is still unknown. We are interested in understanding the quantum bounds as a problem of basic quantum and statistical mechanics.

Selected publications:

bounds from dynamics of free particles on curved manifolds

S. Pappalardi and J. Kurchan - SciPost (2022)

bounds from the Fluctuation-Dissipation theorem

## Long-range interacting quantum systems

In recent years, there has been growing attention towards the study of the quantum dynamics of long-range (LR) interacting particles, motivated by the fact that they effectively describe most atomic molecular and optical systems, such as Bose-Einstein condensates, cavity QED and trapped ions. The prominent collective character of these systems gives rise to novel dynamical phenomena, metastable phases of matter, and forms of dynamical criticality, which do not have a counterpart in traditional quantum systems with local interactions. We are interested in the effects of LRI on the standard paradigm of quantum thermalization, with a quantum information perspective. For instance, we have illustrated how the standard quasi-particle contribution to the EE dynamics gets suppressed, leading to a time-dependent logarithmic scaling of the EE that is governed instead by a collective (semi-classical) contribution 2.. This provides a very transparent and quantitative relationship between entanglement propagation measures (such as entropy, quantum Fisher information, spin squeezing) and chaos quantifiers (such as Lyapunov exponents and out-of-time-order correlations) in the semiclassical regime 2.. This phenomenology is consistent with a many-body spectrum characterised by athermal scarred eigenstates...

Selected publications:

entanglement entropy in for ground states of the Dyson LRI model

S. Pappalardi, P. Calabrese and G. Parisi - J. Stat. Mech. (2019)the origin of the slow entanglement dynamics in LRI spin systems

A. Lerose and S.Pappalardi - Phys. Rev. Research (R) (2020)

A. Lerose and S. Pappalardi - Phys. Rev. A, (2020)

Review on out-of-equilibrium dynamics

N. Defenu, A. Lerose and S.Pappalardi - Arxiv 2307.04802 (2023).

## Online talks

2024, Chaos and Designs by Free Probability, Talk at the workshop “Novel Emergent Phenomena in Quantum Many-Body Dynamics”, Les Houches - France.

2024, Low-temperature quantum bounds on curved manifolds, Seminar the course “Spectres en géométrie hyperbolique”, College de France - France.

2022, Entanglement enhanced metrology via many-body quantum scar, QPequi Talks, Goiânia - Brazil.

2022, Quantum Bounds and the Fluctuation-Dissipation-Theorem, Conformal field theory and quantum many-body physics, CRM, Montreal, Canada.

2022, Quantum Bounds on simple models, Leeds Loughborough Nottingham NonEqulibrium Seminar.

2022, Quantum Bounds and the Fluctuation-Dissipation-Theorem, Tsung-Dao Lee Institute Youth Forum for Quantum Physics.

2021, Dynamics, entanglement and chaos in systems with collective and long-range interactions, PCS Institute for Basic Science.

2020, Entanglement dynamics and chaos in systems with collective and long-range interactions, Qchaos2020 seminars YouTube series,