We study the entanglement between disjoint subregions in quantum critical systems through the lens of the logarithmic negativity. We work with systems in arbitrary dimensions, including conformal field theories and their corresponding lattice Hamiltonians, as well as resonating valence-bond states. At small separations, the logarithmic negativity is big and displays universal behavior, but we show nonperturbatively that it decays faster than any power at large separations. This can already be seen in the minimal setting of single-spin subregions. The corresponding absence of distillable entanglement at large separations generalizes the one-dimensional result, and indicates that quantum critical ground states do not possess long-range bipartite entanglement, at least for bosons. For systems with fermions, a more suitable definition of the logarithmic negativity exists that takes into account fermion parity, and we show that it decays algebraically. Along the way we obtain general results for the moments of the partially transposed density matrix. Published by the American Physical Society 2024
We study the entanglement between disjoint subregions in quantum critical systems through the lens of the logarithmic negativity. We work with systems in arbitrary dimensions, including conformal field theories and their corresponding lattice Hamiltonians, as well as resonating valence-bond states. At small separations, the logarithmic negativity is big and displays universal behavior, but we show nonperturbatively that it decays faster than any power at large separations. This can already be seen in the minimal setting of single-spin subregions. The corresponding absence of distillable entanglement at large separations generalizes the one-dimensional result, and indicates that quantum critical ground states do not possess long-range bipartite entanglement, at least for bosons. For systems with fermions, a more suitable definition of the logarithmic negativity exists that takes into account fermion parity, and we show that it decays algebraically. Along the way we obtain general results for the moments of the partially transposed density matrix. Published by the American Physical Society 2024
In this thesis we study entanglement in one-dimensional critical quantum many-body systems and in particular we will focus on disconnected regions. Given a system in a pure state, to quantify the amount of entanglement between a multicomponent subsystem and the rest of the full system we can use as an entanglement measure the renowned entanglement entropy. However, if we are interested in the entanglement shared among the disconnected regions, the entanglement entropy fails to be a good quantifier. The reason is that the state of the subsystem is in general mixed once the rest is traced out, and the entanglement entropy is a good measure of entanglement only for pure states. A good measure of entanglement in mixed states is the logarithmic negativity, which is the quantitative version of Peres' criterion of separability. The main advantage of the negativity with respect to other entanglement measures is the simplicity of its definition in terms of the density matrix describing the quantum state. Since the definition does not require any variational calculus, it is much more easily computable than any other entanglement measure, and therefore we can obtain some results also in complicated settings such as many-body systems. We will be mostly interested in the configuration where the full system is divided into two parts, A and B. If we do not have access to the degrees of freedom in B, we can describe subsystem A through its reduced density matrix, where the degrees of freedom in B have been traced out. The residual subsystem in A will in general be left in a mixed state. Subsystem A is then divided again into two parts, A_1 and A_2, and we will be interested in the entanglement shared by these two components. Knowing the full density matrix, the logarithmic negativity can be easily computed from the eigenvalues of its partial transpose with respect to the degrees of freedom living in one of the two subsystems A_1 or A_2. However, the full density matrix of a many-body state is in general unaccessible, even numerically, since the size of the matrix grows exponentially with the size of the system. If we concentrate on critical systems whose low-energy physics can be described by a quantum field theory, we can resort to its powerful tools to compute the entanglement properties. In particular the main tool is the replica trick. The entanglement entropy is obtained from the moments of the reduced density matrix, while the negativity can be computed from the moments of the partial transpose. However, even the computation of the integer-order moments is not an easy task, and exact analytical results are known only for the simplest quantum field theories. Hence, throughout the thesis we will usually consider conformal field theories, which have an enhanced set of symmetries and therefore allow for some exact computations. In the Introduction, we will try to frame this work by briefly reviewing some of the main topics in the literature of quantum many-body systems and quantum field theories where entanglement plays a crucial role. We will also describe what are the main features that are requested to a `good' measure of entanglement, and we will review some of the main measures that have been considered so far in the context of quantum information. We will define the entanglement negativity and stress its main advantages, as well as its drawbacks. After reviewing some basic facts of conformal field theories, we will give some technical details on the computation of entanglement entropy and logarithmic negativity in general quantum field theories, which will be needed in the rest of the thesis. In Chap. 2 we will start the study of the entanglement of several disjoint disconnected regions by considering the entanglement that the union of these regions share with the rest of the system. We will compute the integer-order Rényi entropies of the subsystem from which the entanglement entropy can be obtained through the replica limit. Unfortunately, the exact analytic continuation to real order is still out of reach and therefore the entanglement entropy cannot be computed exactly. However numerical extrapolations allow to obtain some accurate estimates. In the remaining chapters we will focus on the entanglement shared between two non complementary disjoint regions, through the computations of the entanglement negativity in different settings. In Chap. 3, we will consider a global quantum quench starting from a conformal boundary state and evolving through a conformal evolution. A general formula for the mutual information and the logarithmic negativity for two adjacent and disjoint intervals is given in the spacetime limit. In Chap. 4 we focus on the XY chain and we recover a formula for the partial transpose as a sum of four fermionic auxiliary Gaussian density matrices, by generalizing some previous results for the reduced density matrix of spin systems and for the partial transpose of pure fermionc Gaussian states. Even if the computation of the negativity is still out of reach, we can obtain formulas for the integer-order moments of the partial transpose in terms of the correlation matrices relative to the two components and to the region in between them. In Chap. 5 we will study the entanglement negativity for a free Dirac fermion field. Starting from the lattice results, we will obtain a path integral representation of the partial transpose which can be easily generalized to the continuum. With this representation of the partial transpose, we can construct all its integer-order moments and compute them for the simple conformal field theory of the free fermion. Again, the computation of the negativity could not be accomplished due to technical difficulties in the analytic continuation to real order. The same computation has been extended also to the modular invariant Dirac fermion and the Ising model, and the obtained formulas coincide with the ones already present in the literature. This analysis draws an interesting connection between some terms appearing in the formulas for the moments of the partial transpose (as well as of the R\'enyi entropies) found for the lattice models and the ones found for the corresponding theories describing their scaling limit. Whenever possible, all the analytical results will be checked against numerical calculations performed on simple free chain models. In Chaps. 2 and 3 we will consider bosonic Gaussian states, specifically the harmonic chain in its ground state and out of equilibrium, while in Chaps. 4 and 5 we will consider fermionic systems, specifically the XX and Ising spin chains, and the tight-binding model. Finally, we will draw some conclusions and discuss some open problems.
The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.
We study two entanglement measures in a large family of isotropic many-body states including incompressible quantum Hall liquids and quantum critical systems: the logarithmic negativity (LN), and mutual information (MI). For pure states, obtained for example from a bipartition at zero temperature, these provide distinct characterizations of the entanglement present between two spatial subregions, while for mixed states (such as at finite temperature) only the LN remains a good entanglement measure. Our focus is on regions that have corners, either adjacent or tip-touching. We first obtain general non-perturbative properties regarding the geometrical dependence of the LN and MI. A close similarity is observed with mutual charge fluctuations, where super-universal angle dependence holds allowing for explicit checks. For the MI, we make stronger statements due to strong subadditivity. We also give ramifications of our general analysis to conformal field theories (CFTs) in two spatial dimensions. We then explicitly verify these properties with integer quantum Hall states. To do so we develop two independent approaches to obtain the fermionic LN, which takes into account Fermi statistics: an overlap-matrix method, and a real-space lattice discretization. At finite temperature, we find a rapid decrease of the LN well inside the cyclotron gap at integer fillings. We further show that the LN decays faster compared to the MI at high temperatures.
The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics.The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve.In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties.We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in 1/L, the inverse of the Hilbert-space dimension.We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity.As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned.This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point.We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement.Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes.We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.
Entanglement exhibits universal behavior near the ground-state critical point where correlations are long ranged and the thermodynamic entropy is vanishing. On the other hand, a quantum quench imparts extensive energy and results in a build up of entropy, hence no critical behavior is expected at long times. In this work, we present a new paradigm in the quench dynamics of integrable spin chains which exhibit a ground-state order-disorder phase transition at a critical line. Specifically, we consider a quench along the critical line which displays a volume-law behavior of the entropy and exponentially decaying correlations; however, we show that quantum criticality is hidden in higher-order correlations and becomes manifest via measures such as the mutual information and logarithmic negativity. Furthermore, we showcase the scale invariance of the Rényi mutual information between disjoint regions as further evidence for genuine critical behavior. We attribute the emergent quantum criticality to the soft mode not getting excited in spite of the quench. Moreover, the results presented here are universal to models whose low-energy or long-wavelength dynamics are well described by a free-fermionic field theory. Our results are amenable to an experimental realization on different quantum simulator platforms, particularly the Rydberg simulators. Published by the American Physical Society 2024
Abstract We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1 + 1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length r and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance r . We show that the former saturates to a finite value, and that the latter tends to zero, as . We show that in both cases, the leading corrections are exponential decays in r (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy (EE), the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the EE, a large- r analysis of the negativity allows for the detection of bound states.