Entanglement in Quantum Systems

A superconducting vortex lattice in proximity to a topological insulator is predicted to have a Majorana mode at each vortex core. The Majorana modes can hop between lattice sites and interact with each other. I will present an analysis of a model of such interacting Majorana modes based on a combination of mean field theory, renormalization group and a study of ladder models. At strong coupling, the Majorana modes tend to entangle in pairs.

In this talk I will review the results of recent work in collaboration with Cecilia De Fazio, Benjamin Doyon and István M. Szécsényi. We studied the entanglement of excited states consisting of a finite number of particle excitations. More precisely, we studied the difference between the entanglement entropy of such states and that of the ground state in a simple bi-partition of a quantum system, where both the size of the system and of the bi-partition are infinite, but their ratio is finite. We originally studied this problem in massive 1+1 dimensional QFTs where analytic computations were possible. We have found the results to apply more widely, including to higher dimensional free theories. In all cases we find that the increment of entanglement is a simple function of the ratio between region's and system's size only. Such function, turns out to be exactly the entanglement of a qubit state where the coefficients of the state are simply associated with the probabilities of particles being localised in one or the other part of the bi-partition. In this talk I will describe the results in some detail and discuss their domain of applicability. I will also highlight the main QFT techniques that we have used in order to obtain them analytically and present some numerical data.

Purification is a powerful technique in which the entropy of a mixed quantum state is represented as the entanglement between a pure state and a larger system. This representation is not unique, but in many-body systems one natural choice is the purification with minimal entanglement. I will discuss the entropy of the minimally entangled purification, called the “entanglement of purification,” in three model systems: an Ising spin chain, holographic theories, and random stabilizer tensor networks. I will conjecture a holographic interpretation of entanglement of purification, and compare this predictions with one-dimensional tensor network calculations.

In this talk, I will explain how the study of entanglement in conformal field theories leads directly to nonlinear gravitational physics, without any assumptions from string theory or the AdS/CFT correspondence. In particular, the entanglement entropies for ball-shaped regions (up to second order in perturbations about the vacuum state) in a certain class of CFT states match precisely with extremal surface areas in associated asymptotically AdS geometries, and constraints on entanglement entropies imply that these geometries satisfy Einstein's equations to second order in perturbation theory around empty AdS space. I will also show that other constraints from quantum information theory suggest new gravitational positive energy theorems.

One approach to obtain a structure in the exponentially large Hilbert space corresponding to composite systems is to study local transformations of multipartite states. In this talk I will focus on systems which are composed of n d-level subsystems. I will first show that non-trivial LOCC (Local operations assisted by classical communication) transformations among generic fully entangled pure states are almost never possible. This implies that the maximally entangled set, which can be viewed as the generalization of the maximally entangled bipartite state, is of full measure. The consequences of these findings in the context of entanglement theory will be discussed. Moreover, I will present some recent results on probabilistic transformations from a higher dimensional Hilbert space to a lower dimensional one.

I will describe how to define the Hilbert space of an interval in a 2D CFT. For CFTs with a bulk dual, this leads to a theory of membranes which determines the leading order entanglement entropy. They give a concrete realization of the infinite-dimensional BMS symmetry in AdS/CFT and allow a new understanding of spacetime entropy. The construction resolves a seeming paradox about assigning an entropy to a collection of marginal density matrices for subregions of a CFT.

Tensor Network States are ansatzes for the efficient description of the state of a quantum many-body system. They can be used to study static and dynamic properties of strongly correlated states. In this talk I will present some recent work on the application of these techniques to study Lattice Gauge Theories. In particular, using the Schwinger model as a testbench, we have shown that these ansatzes are suitable to approximate low energy states precisely enough to allow for accurate finite size and continuum limit extrapolations of ground state properties, mass gaps and temperature dependent quantities. Beyond this case the feasibility of the method has already been tested also for non-Abelian models, out-of-equilibrium scenarios, and non-vanishing chemical potential, the latter cases being specially relevant, as they offer difficulties to standard Montecarlo techniques.

The subtle interplay of infrared singularities in quantum electrodynamics and perturbative quantum gravity and information theoretic issues such as quantum entanglement between soft and hard degrees of freedom will be discussed. It will be argued that the inevitable loss of soft photons and gravitons in a scattering experiment leads to decoherence of the out-going state, that this decoherence persists even when the incoming states are infrared-safe coherent states, and that mathematical issues in scattering theory with normalizable incoming wave packets distinguishes between competing methods and requires infrared-safe in-states.

I will discuss several aspects of entanglement in non-hermitian spin chains, in particular those having quantum group or super-group symmetry. Applications to non-unitary CFTs and non-unitary RG flows will then be considered. I may also have time for some remarks about CFTs with non-compact target spaces.

In this talk, I will introduce a novel class of phase transitions separating quantum states with different entanglement features. An example of such an “entanglement phase transition” is provided by the many-body localization transition in disordered quantum systems, as it separates highly entangled thermal states at weak disorder from many-body localized states with low entanglement at strong disorder. In the spirit of random matrix theory, I will describe a simple model for such transitions where a physical quantum many-body system lives at the “holographic” boundary of a bulk random tensor network.

I give an overview of work with Aasen and Mong on topologically invariant defects in two-dimensional classical lattice models, quantum spin chains and tensor networks. We show how to find defects that satisfy commutation relations guaranteeing the partition function depends only on their topological properties. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. These lattice topological defects have a variety of useful applications. In the Ising model, the fusion of duality defects allows Kramers-Wannier duality to be enacted on the torus and higher genus surfaces easily, implementing modular invariance directly on the lattice. These results can be extended to a very wide class of models, giving generalised dualities previously unknown in the statistical-mechanical literature. A consequence is an explicit definition of twisted boundary conditions that yield the precise shift in momentum quantization and thus the spin of the associated conformal field. Other universal quantities we compute exactly on the lattice are the ratios of g-factors for conformal boundary conditions.

Bit threads offer a novel perspective on holographic entanglement entropy. Using tools from network theory, specifically the concept of multicommodity flows, we will use bit threads to prove the “monogamy of mutual information” property of holographic entanglement entropies. The proof will lead to a conjecture about the general entanglement structure of holographic states.

Gauge theories are fundamental to our understanding of interactions between the elementary constituents of matter as mediated by gauge bosons. However, computing the real-time dynamics in gauge theories is a notorious challenge for classical computational methods. In the spirit of Feynman's vision of a quantum simulator, this has recently stimulated theoretical effort to devise schemes for simulating such theories on engineered quantum-mechanical devices, with the difficulty that gauge invariance and the associated local conservation laws (Gauss laws) need to be implemented. Here we report on a proposal and an experimental demonstration of a digital quantum simulation of a lattice gauge theory, by realising 1+1-dimensional quantum electrodynamics (Schwinger model) on a few-qubit trapped-ion quantum computer. We are interested in the real-time evolution of the Schwinger mechanism, describing the instability of the bare vacuum due to quantum fluctuations, which manifests itself in the spontaneous creation of electron-positron pairs. To make efficient use of our quantum resources, we map the original problem to a spin model by eliminating the gauge fields in favour of exotic long-range interactions, which have a direct and efficient implementation on an ion trap architecture. We explore the Schwinger mechanism of particle-antiparticle generation by monitoring the mass production and the vacuum persistence amplitude. Moreover, we track the real-time evolution of entanglement in the system, which illustrates how particle creation and entanglement generation are directly related. Our work represents a first step towards quantum simulating high-energy theories with atomic physics experiments, the long-term vision being the extension to real-time quantum simulations of non-Abelian lattice gauge theories.

The entanglement negativity -- a quantum entanglement measure that can be applied to mixed states -- is defined in terms of the partial transpose of density matrices. In this talk, we discuss the partial transpose and the entanglement negativity in fermionic many-body systems. In addition to quantity quantum entanglement, the partial transpose is also useful to extract topological invariants of symmetry-protected topological phases protected by an antiunitary symmetry.

We consider the computation of volumes contained in a spatial slice of AdS3 in terms of observables in a dual CFT. Our main tool is kinematic space, defined either from the bulk perspective as the space of oriented bulk geodesics, or from the CFT perspective as the space of entangling intervals. We give an explicit formula for the volume of a general region in the spatial slice as an integral over kinematic space. For the region lying below a geodesic, we show how to write this volume purely in terms of entangling entropies in the dual CFT. This expression is perhaps most interesting in light of the complexity=volume proposal, which posits that complexity of holographic quantum states is computed by bulk volumes. An extension of this idea proposes that the holographic subregion complexity of an interval, defined as the volume under its Ryu-Takayanagi surface, is a measure of the complexity of the corresponding reduced density matrix. If this is true, our results give an explicit relationship between entanglement and subregion complexity in CFT, at least in the vacuum. Talk based on arXiv:1805.10298.

I will discuss the procedures of gauging and ungauging, reveal their operational meaning and propose their generalization within the framework of quantum error-correcting codes. I will describe ungauging of two subsystem codes with unusual symmetries: the Bacon-Shor code and the gauge color code (GCC). In case of the GCC, three different stabilizer Hamiltonians correspond to distinct thermal symmetry-protected topological (SPT) phases. Lastly, I will explain how to find a 2D fracton SPT phase protected by fractal-like symmetries by exploiting a relation between gapped domain walls and transversal gates. Joint work with B. Yoshida; arXiv:1805.01836.

One can encode a qubit in a bosonic mode and the Gottesman-Preskill-Kitaev (GKP) code is an interesting example of such bosonic code, of experimental interest in the circuit-QED community. We show how the decoding problem of repeated noisy error correction on a single GKP qubit represents the evaluation of a 1D Euclidean path-integral of a particle moving in a random cosine potential. We demonstrate the efficiency of a minimal-action decoding strategy numerically. One may concatenate these GKP qubits with Kitaev's surface or toric code to obtain a surface-GKP code for scalable quantum error correction. We sketch how this code can have a threshold by arguing about its relation to a quenched-randomness version of 3D compact-QED coupled to a matter field and compare this to the usual Kitaev toric code picture. We discuss our new result showing that continuous-variable (topological) codes are useless for the protection of quantum information, confirming that a sufficient departure from Gaussianity is necessary to preserve a discrete amount of quantum information and justify the usual discrete error models. Joint work with Christophe Vuillot, Hamed Asasi, Yang Wang, Leonid Pryadko.

I will report on recent experimental work performed at University of Florence with degenerate gases of ultracold 173Yb fermions. These two-electron atoms are a powerful resource for engineering synthetic many-body quantum systems, as they exhibit distinct internal degrees of freedom – nuclear spin and electronic state – that can be both manipulated with high levels of quantum control. By coupling different internal states we have demonstrated the possibility of engineering “synthetic dimensions”, in which effective lattice dynamics are encoded in the internal Hilbert space of single atoms: by using this approach, we have demonstrated new techniques for the production of synthetic gauge fields and studied the emergence of edge currents in fermionic ladders with large tunable flux. I will also show that the nuclear and electronic degrees of freedom are naturally entangled by strong spin-exchange interactions, that allowed us to demonstrate new possibilities of interaction tuning and open new perspectives for the investigation of strongly correlated states of matter.

Tensor networks impose a notion of geometry on the entanglement of a quantum system. In some cases, this geometry is found to reproduce key properties of holographic dualities, and subsequently much work has focused on using tensor networks as tractable models for holographic dualities. Conventionally, the structure of the network - and hence the geometry - is largely fixed a priori by the choice of tensor network ansatz. Here, we evade this restriction and describe an unbiased approach that allows us to extract the appropriate geometry from a given quantum state. We develop an algorithm that iteratively finds a unitary circuit that transforms a given quantum state into an unentangled product state. We then analyze the structure of the resulting unitary circuits. In the case of non-interacting, critical systems in one dimension, we recover signatures of scale invariance in the unitary network, and we show that appropriately defined geodesic paths between physical degrees of freedom exhibit known properties of a hyperbolic geometry.

There exists a rich set of relations between entanglement entropies of a multi-partite quantum system, some applying universally to all quantum systems (such as strong sub-additivity), some pertaining to some limited but interesting subclass of systems (such as monogamy of mutual information in holography). We focus on QFT systems with a holographic dual describing a classical geometry in the bulk, and discuss work in progress on obtaining new entanglement relations for such systems.

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the post-quench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for non-interacting systems, and it was believed to be unfeasible for genuinely interacting models. Conversely, here we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the space-time scaling limit. Our framework requires only knowledge about the steady state, and the velocities of the low-lying excitations around it

Operator entanglement is the analog of the standard entanglement entropy, but for operators: it tells us how far an operator acting on $\mathcal{H}_A \otimes \mathcal{H}_B$ is from a product $O_A \otimes O_B$. I will review some of the motivations for looking at that quantity, and explain how it can be computed in 2d CFT. I will focus on a few situations that are interesting for questions about numerical simulations of time-evolution in 1d with Matrix Product Operators.

In this talk, I'll review definitions of entanglement entropy in gauge theory and explain why an analogy between EE in emergent gauge theories and AdS/CFT suggests what the Ryu-Takayanagi formula might be counting from a bulk point of view. I will report on ongoing attempts to make this analogy precise in low dimensional theories of gravity.

In this talk, I will show how to design new examples of AdS₃/CFT₂ via the study of Siegel paramodular forms. In other words, I will attempt to bootstrap holography via number theory. The talk is based on 1611.04588 and 1805.09336.

I will discuss new phases of matter that arise in the presence of periodic (Floquet) driving. In contrast to ground states, these can be realized in the absence of cooling and may provide novel ways to manipulate quantum information. Examples include a Floquet generalization of the Kitaev honeycomb lattice model, which combines bulk topological order, boundary modes and breaking of time translation symmetry. Time permitting I will also discuss operator scrambling induced by random Hamiltonians and random unitary circuits.