All Publications

We study the entanglement Hamiltonian of an interval for the free massless Dirac field in an inhomogeneous background on a finite segment and in the ground state. We consider a class of metrics that are Weyl equivalent to the flat metric through a Weyl factor that depends only on the spatial coordinate, with the same boundary condition imposed at both endpoints of the segment. The explicit form of the entanglement Hamiltonian is written as the sum of a local and a bilocal term. The weight function of the local term allows us to study a contour function for the entanglement entropies. For the model obtained from the continuum limit of the rainbow chain, the analytic expressions are compared with exact numerical results from the lattice, showing an excellent agreement.

In quantum resource theory, one is often interested in identifying which states serve as the best resources for particular quantum tasks. If a relative comparison between quantum states can be made, this gives rise to a partial order, where states are ordered according to their suitability to act as a resource. In the literature, various different partial orders for a variety of quantum resources have been proposed. In discrete variable systems, vector majorization ofWigner functions in discrete phase space provides a natural partial order between quantum states. In the continuous variable case, a natural counterpart would be continuous majorization of Wigner functions in quantum phase space. Indeed, this concept was recently proposed and explored (mostly restricting to the single-mode case) by Van Herstraeten et al. [Quantum 7, 1021 (2023)]. In this work, we develop the theory of continuous majorization in the general N-mode case. In addition, we propose extensions to include states with finite Wigner negativity. For the special case of the convex hull of N-mode Gaussian states, we prove a conjecture made by Van Herstraeten, Jabbour, and Cerf.We also prove a phase space counterpart of Uhlmann's theorem of majorization.

We study the quantum transport generated by the bipartite entanglement in two-dimensional conformal field theory at finite density with the U(1) × U(1) symmetry associated to the conservation of the electric charge and of the helicity. The bipartition given by an interval is considered, either on the line or on the circle. The continuity equations and the corresponding conserved quantities for the modular flows of the currents and of the energy-momentum tensor are derived. We investigate the mean values of the associated currents and their quantum fluctuations in the finite density representation, which describe the properties of the modular quantum transport. The modular analogues of the Johnson- Nyquist law and of the fluctuation-dissipation relation are found, which encode the thermal nature of the modular evolution.

We study the entanglement Hamiltonian of two disjoint blocks in the harmonic chain on the line and in its ground state. In the regime of large mass, the only non-vanishing terms are the on-site and nearest-neighbour ones. Analytic expressions are obtained for their profiles, which are written in terms of piecewise linear functions that can be discontinuous and display sharp transitions as the separation between the blocks changes. In the regime of vanishing mass, where the matrices characterizing the entanglement Hamiltonian contain couplings at all distances, we explore the location of the subdominant terms and some combinations of matrix elements that are useful for the continuum limit, comparing the results with the corresponding ones for a free chiral current. The single-particle entanglement spectrum is also investigated.

In two-dimensional conformal field theories (CFT) in Minkowski spacetime, we study the spacetime distance between two events along two distinct modular trajectories. When the spatial line is bipartite by a single interval, we consider both the ground state and the state at finite different temperatures for the left and right moving excitations. For the free massless Dirac field in the ground state, the bipartition of the line given by the union of two disjoint intervals is also investigated. The modular flows corresponding to connected subsystems preserve relativistic causality. Locality along the modular flows of some fields is explored by evaluating their (anti-)commutators. In particular, the bilocal nature of the modular Hamiltonian of two disjoint intervals for the massless Dirac field provide multiple trajectories leading to Dirac delta contributions in the (anti-)commutators even when the initial points belong to different intervals, thus being spacelike separated.

The holographic bit threads are an insightful tool to investigate the holographic entanglement entropy and other quantities related to the bipartite entanglement in AdS/CFT. We mainly explore the geodesic bit threads in various static backgrounds, for the bipartitions characterized by either a sphere or an infinite strip. In pure AdS and for the sphere, the geodesic bit threads provide a gravitational dual of the map implementing the geometric action of the modular conjugation in the dual CFT. In Schwarzschild AdS black brane and for the sphere, our numerical analysis shows that the flux of the geodesic bit threads through the horizon gives the holographic thermal entropy of the sphere. This feature is not observed when the subsystem is an infinite strip, whenever we can construct the corresponding bit threads. The bit threads are also determined by the global structure of the gravitational background; indeed, for instance, we show that the geodesic bit threads of an arc in the BTZ black hole cannot be constructed.

We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.

We investigate the complexity of states and operators evolved with the modular Hamiltonian by using the Krylov basis. In the first part, we formulate the problem for states and analyze different examples, including quantum mechanics, two-dimensional conformal field theories and random modular Hamiltonians, focusing on relations with the entanglement spectrum. We find that the modular Lanczos spectrum provides a different approach to quantum entanglement, opening new avenues in many-body systems and holography. In the second part, we focus on the modular evolution of operators and states excited by local operators in two-dimensional conformal field theories. We find that, at late modular time, the spread complexity is universally governed by the modular Lyapunov exponent λLmod=2π and is proportional to the local temperature of the modular Hamiltonian. Our analysis provides explicit examples where entanglement entropy is indeed not enough; however the entanglement spectrum is, and encodes the same information as complexity.

We introduce two infinite sequences of entanglement monotones, which are constructed from expectation values of polynomials in the modular Hamiltonian. These monotones yield infinite sequences of inequalities that must be satisfied in majorizing state transitions. We demonstrate this for information erasure, deriving an infinite sequence of "Landauer inequalities"for the work cost, bounded by linear combinations of expectation values of powers of the modular Hamiltonian. These inequalities give improved lower bounds for the work cost in finite-dimensional systems, and depend on more details of the erased state than just on its entropy and variance of modular Hamiltonian. Similarly one can derive lower bounds for marginal entropy production for a system coupled to an environment. These infinite sequences of entanglement monotones also give rise to relative quantifiers that are monotonic in more general processes, namely those involving so-called σ majorization with respect to a fixed point full rank state σ; such quantifiers are called resource monotones. As an application to thermodynamics, one can use them to derive finite-dimension corrections to the Clausius inequality. Finally, in order to gain some intuition for what (if anything) plays the role of majorization in field theory, we compare pairs of states in discretized theories at criticality and study how majorization depends on the size of the bipartition with respect to the size of the entire chain.

We compare the capacity of entanglement with the entanglement entropy by considering various aspects of these quantities for free bosonic and fermionic models in one spatial dimension, both in the continuum and on the lattice. Substantial differences are observed in the subleading terms of these entanglement quantifiers when the subsystem is made by two disjoint intervals, in the massive scalar field and in the fermionic chain. We define c-functions based on the capacity of entanglement similar to the one based on the entanglement entropy, showing through a numerical analysis that they display a monotonic behaviour under the renormalisation group flow generated by the mass. The capacity of entanglement and its related quantities are employed to explore the symmetry resolution. The temporal evolutions of the capacity of entanglement and of the corresponding contour function after a global quench are also discussed.

We study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisation of entanglement of subsystems in mixed states. We find that the negativity Hamiltonian inherits the structure of the corresponding entanglement Hamiltonian. We finally show how the continuum expressions for both these operators can be recovered from exact numerical computations in free-fermion chains.

We study the geometric action of some modular conjugations in two dimensional (2D) conformal field theories. We investigate the bipartition given by an interval when the system is in the ground state, either on the line or on the circle, and in the thermal Gibbs state on the line. We find that the restriction of the corresponding inversion maps to a spatial slice is obtained also in the gauge/gravity correspondence through the geodesic bit threads in a constant time slice of the dual static asymptotically AdS background. For a conformal field theory in the thermal state on the line, the modular conjugation suggests the occurrence of a second world which can be related through the geodesic bit threads to the horizon of the BTZ black brane background. An inversion map is constructed also for the massless Dirac fermion in the ground state and on the line bipartite by the union of two disjoint intervals.

We study the entanglement entropies of an interval adjacent to the boundary of the half line for the free fermionic spinless Schrödinger field theory at finite density and zero temperature, with either Neumann or Dirichlet boundary conditions. They are finite functions of the dimensionless parameter given by the product of the Fermi momentum and the length of the interval. The entanglement entropy displays an oscillatory behaviour, differently from the case of the interval on the whole line. This behaviour is related to the Friedel oscillations of the mean particle density on the half line at the entangling point. We find analytic expressions for the expansions of the entanglement entropies in the regimes of small and large values of the dimensionless parameter. They display a remarkable agreement with the curves obtained numerically. The analysis is extended to a family of free fermionic Lifshitz models labelled by their integer Lifshitz exponent, whose parity determines the properties of the entanglement entropies. The cumulants of the local charge operator and the Schatten norms of the underlying kernels are also explored.

We consider free-fermion chains in the ground state and the entanglement Hamiltonian for a subsystem consisting of two separated intervals. In this case, one has a peculiar long-range hopping between the intervals in addition to the well-known and dominant short-range hopping. We show how the continuum expressions can be recovered from the lattice results for general filling and arbitrary intervals. We also discuss the closely related case of a single interval located at a certain distance from the end of a semi-infinite chain and the continuum limit for this problem. Finally, we show that for the double interval in the continuum a commuting operator exists which can be used to find the eigenstates.

We study the entanglement entropies of an interval on the infinite line in the free fermionic spinless Schrödinger field theory at finite density and zero temperature, which is a non-relativistic model with Lifshitz exponent z = 2. We prove that the entanglement entropies are finite functions of one dimensionless parameter proportional to the area of a rectangular region in the phase space determined by the Fermi momentum and the length of the interval. The entanglement entropy is a monotonically increasing function. By employing the properties of the prolate spheroidal wave functions of order zero or the asymptotic expansions of the tau function of the sine kernel, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular region in the phase space. These expansions lead to prove that the analogue of the relativistic entropic C function is not monotonous. Extending our analyses to a class of free fermionic Lifshitz models labelled by their integer dynamical exponent z, we find that the parity of this exponent determines the properties of the bipartite entanglement for an interval on the line.

We study the continuum limit of the entanglement Hamiltonian of a sphere for the massless scalar field in its ground state by employing the lattice model defined through the discretisation of the radial direction. In two and three spatial dimensions and for small values of the total angular momentum, we find numerical results in agreement with the corresponding ones derived from the entanglement Hamiltonian predicted by conformal field theory. When the mass parameter in the lattice model is large enough, the dominant contributions come from the on-site and the nearest-neighbour terms, whose weight functions are straight lines.

In the conformal field theories given by the Ising and Dirac models, when the system is in the ground state, the moments of the reduced density matrix of two disjoint intervals and of its partial transpose have been written as partition functions on higher genus Riemann surfaces with symmetry. We show that these partition functions can be expressed as the grand canonical partition functions of the two-dimensional two component classical Coulomb gas on certain circular lattices at specific values of the coupling constant.

We study the temporal evolution of the circuit complexity after the local quench where two harmonic chains are suddenly joined, choosing the initial state as the reference state. We discuss numerical results for the complexity for the entire chain and the subsystem complexity for a block of consecutive sites, obtained by exploiting the Fisher information geometry of the covariance matrices. The qualitative behaviour of the temporal evolutions of the subsystem complexity depends on whether the joining point is inside the subsystem. The revivals and a logarithmic growth observed during these temporal evolutions are discussed. When the joining point is outside the subsystem, the temporal evolutions of the subsystem complexity and of the corresponding entanglement entropy are qualitatively similar.

We study the temporal evolution of the circuit complexity for a subsystem in harmonic lattices after a global quantum quench of the mass parameter, choosing the initial reduced density matrix as the reference state. Upper and lower bounds are derived for the temporal evolution of the complexity for the entire system. The subsystem complexity is evaluated by employing the Fisher information geometry for the covariance matrices. We discuss numerical results for the temporal evolutions of the subsystem complexity for a block of consecutive sites in harmonic chains with either periodic or Dirichlet boundary conditions, comparing them with the temporal evolutions of the entanglement entropy. For infinite harmonic chains, the asymptotic value of the subsystem complexity is studied through the generalised Gibbs ensemble.

We study the massless Dirac field on the line in the presence of a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. Considering this system in its ground state, we derive the modular Hamiltonians of the subregion given by the union of two disjoint equal intervals at the same distance from the defect. The absence of energy dissipation at the defect implies the existence of two phases, where either the vector or the axial symmetry is preserved. Besides a local term, the densities of the modular Hamiltonians contain also a sum of scattering dependent bi-local terms, which involve two conjugate points generated by the reflection and the transmission. The modular flows of each component of the Dirac field mix the trajectory passing through a given initial point with the ones passing through its reflected and transmitted conjugate points. We derive the two-point correlation functions along the modular flows in both phases and show that they satisfy the Kubo-Martin-Schwinger condition. The entanglement entropies are also computed, finding that they do not depend on the scattering matrix.