Topological R\'enyi entropy after a quantum quench

We present an analytical study on the resilience of topological order after a quantum quench. The system is initially prepared in the ground state of the toric-code model, and then quenched by switching on an external magnetic field. During the subsequent time evolution, the variation in topological order is detected via the topological Renyi entropy of order 2. We consider two different quenches: the first one has an exact solution, while the second one requires perturbation theory. In both cases, we find that the long-term time average of the topological Renyi entropy in the thermodynamic limit is the same as its initial value. Based on our results, we argue that topological order is resilient against a wide range of quenches.Comment: 5 pages, 4 figures, published version with structural changes, see supplemental material at http://link.aps.org/supplemental/10.1103/PhysRevLett.110.17060

Quantum geometric tensor away from Equilibrium

The manifold of ground states of a family of quantum Hamiltonians can be endowed with a quantum geometric tensor whose singularities signal quantum phase transitions and give a general way to define quantum phases. In this paper, we show that the same information-theoretic and geometrical approach can be used to describe the geometry of quantum states away from equilibrium. We construct the quantum geometric tensor $Q_{\mu\nu}$ for ensembles of states that evolve in time and study its phase diagram and equilibration properties. If the initial ensemble is the manifold of ground states, we show that the phase diagram is conserved, that the geometric tensor equilibrates after a quantum quench, and that its time behavior is governed by out-of-time-order commutators (OTOCs). We finally demonstrate our results in the exactly solvable Cluster-XY model

Thermalization of Topological Entropy after a Quantum Quench

In two spatial dimensions, topological order is robust for static deformations at zero temperature, while it is fragile at any finite temperature. How robust is topological order after a quantum quench? In this paper we show that topological order thermalizes under the unitary evolution after a quantum quench. If the quench preserves gauge symmetry, there is a residual topological entropy exactly like in the finite temperature case. We obtain this result by studying the time evolution of the topological 2-R\'enyi entropy in a fully analytical, exact way. These techniques can be then applied to systems with strong disorder to show whether a many-body localization phenomenon appears in topologically ordered systems.Comment: new plot with finite time results added; typos fixe

Bipartite entanglement and entropic boundary law in lattice spin systems

We investigate bipartite entanglement in spin-1/2 systems on a generic lattice. For states that are an equal superposition of elements of a group $G$ of spin flips acting on the fully polarized state $\ket{0}^{\otimes n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems $A$ and $B$. These states are stabilized by the group $G$. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus $\mathfrak{g}$. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem $A$ and is equal to the perimeter (up to an additive constant) when $A$ is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.Comment: 10 pages, figs, RevTeX

Adiabatic Preparation of Topological Order

Topological order characterizes those phases of matter that defy a description in terms of symmetry and cannot be distinguished in terms local order parameters. This type of order plays a key role in the theory of the fractional quantum Hall effect, as well as in topological quantum information processing. Here we show that a system of n spins forming a lattice on a Riemann surface can undergo a second order quantum phase transition between a spin-polarized phase and a string-net condensed phase. This is an example of a phase transition between magnetic and topological order. We furthermore show how to prepare the topologically ordered phase through adiabatic evolution in a time that is upper bounded by O(\sqrt{n}). This provides a physically plausible method for constructing a topological quantum memory. We discuss applications to topological and adiabatic quantum computing.Comment: 4 pages, one figure. v4: includes new error estimates for the adiabatic evolutio

Transitions in Entanglement Complexity in Random Circuits

Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple pattern of entanglement to a universal, complex pattern can be driven by doping a random Clifford circuit with $T$ gates. This work shows that quantum complexity and complex entanglement stem from the conjunction of entanglement and non-stabilizer resources, also known as magic

Entanglement of Random hypergraph states

Random quantum states and operations are of fundamental and practical interests. In this paper, we investigate the entanglement properties of random hypergraph states, which generalize the notion of graph states by applying generalized controlled-phase gates on an initial reference product state. In particular, we study the two ensembles generated by random controlled-Z (CZ) and controlled-controlled-Z (CCZ) gates, respectively. By applying tensor network representation and combinational counting, we analytically show that the average subsystem purity and entanglement entropy for the two ensembles feature the same volume law but greatly differ in typicality, namely, the purity fluctuation is small and universal for the CCZ ensemble while it is large for the CZ ensemble. We discuss the implications of these results for the onset of entanglement complexity and quantum chaos

Universality and robustness of revivals in the transverse field XY model

We study the structure of the revivals in an integrable quantum many-body system, the transverse field XY spin chain, after a quantum quench. The time evolutions of the Loschmidt echo, the magnetization, and the single spin entanglement entropy are calculated. We find that the revival times for all of these observables are given by integer multiples of T_rev \sim L / v_max where L is the linear size of the system and v_max is the maximal group velocity of quasiparticles. This revival structure is universal in the sense that it does not depend on the initial state and the size of the quench. Applying non-integrable perturbations to the XY model, we observe that the revivals are robust against such perturbations: they are still visible at time scales much larger than the quasiparticle lifetime. We therefore propose a generic connection between the revival structure and the locality of the dynamics, where the quasiparticle speed v_max generalizes into the Lieb$-$Robinson speed v_LR.Comment: completely overhauled version including results on non integrable model