Topological Geometric Entanglement (original) (raw)

Topological Geometric Entanglement of Blocks

2011

Here we establish the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. As happens for the entanglement entropy, we find that the geometric entanglement is the sum of two terms: a non-universal one obeying a boundary law times the number of blocks, and a universal one quantifying the underlying long-range entanglement of a topologically-ordered state. For simplicity we focus on the case of Kitaev's toric code model.

Topological minimally entangled states via geometric measure

Journal of Statistical Mechanics: Theory and Experiment, 2014

Here we show how the Minimally Entangled States (MES) of a 2d system with topological order can be identified using the geometric measure of entanglement. We show this by minimizing this measure for the doubled semion, doubled Fibonacci and toric code models on a torus with nontrivial topological partitions. Our calculations are done either quasi-exactly for small system sizes, or using the tensor network approach in [R. Orús, T.-C. Wei, O. Buerschaper, A. García-Saez, arXiv:1406.0585] for large sizes. As a byproduct of our methods, we see that the minimisation of the geometric entanglement can also determine the number of Abelian quasiparticle excitations in a given model. The results in this paper provide a very efficient and accurate way of extracting the full topological information of a 2d quantum lattice model from the multipartite entanglement structure of its ground states.

Topological order, entanglement, and quantum memory at finite temperature

Annals of Physics, 2012

We compute the topological entropy of the toric code models in arbitrary dimension at finite temperature. We find that the critical temperatures for the existence of full quantum (classical) topological entropy correspond to the confinement-deconfinement transitions in the corresponding Z 2 gauge theories. This implies that the thermal stability of topological entropy corresponds to the stability of quantum (classical) memory. The implications for the understanding of ergodicity breaking in topological phases are discussed.

Identifying non-Abelian topological order through minimal entangled states

Physical review letters, 2014

The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.

Unveiling topological order through multipartite entanglement

2021

It is well known that the topological entanglement entropy (Stopo) of a topologically ordered ground state in 2 spatial dimensions can be captured efficiently by measuring the tripartite quantum information (I) of a specific annular arrangement of three subsystems. However, the nature of the general N-partite information (I ) and quantum correlation of a topologically ordered ground state remains unknown. In this work, we study such I measure and its nontrivial dependence on the arrangement of N subsystems. For the collection of subsystems (CSS) forming a closed annular structure, the I measure (N ≥ 3) is a topological invariant equal to the product of Stopo and the Euler characteristic of the CSS embedded on a planar manifold, |I | = χStopo. Importantly, we establish that I is robust against several deformations of the annular CSS, such as the addition of holes within individual subsystems and handles between nearest-neighbour subsystems. While the addition of a handle between furt...

Detecting topological entanglement entropy in a lattice of quantum harmonic oscillators

New Journal of Physics, 2014

The Kitaev surface-code model is the most studied example of a topologically ordered phase and typically involves four-spin interactions on a two-dimensional surface. A universal signature of this phase is topological entanglement entropy (TEE), but due to low signal to noise, it is extremely difficult to observe in these systems, and one usually resorts to measuring anyonic statistics of excitations or non-local string operators to reveal the order. We describe a continuous-variable analog to the surface code using quantum harmonic oscillators on a two-dimensional lattice, which has the distinctive property of needing only two-body nearest-neighbor interactions for its creation. Though such a model is gapless, it satisfies an area law and the ground state can be simply prepared by measurements on a finitely squeezed and gapped two-dimensional cluster state without topological order. Asymptotically, the continuous variable surface code TEE grows linearly with the squeezing parameter, and we show that its mixed-state generalization, the topological mutual information, is robust to some forms of state preparation error and can be detected simply using single-mode quadrature measurements. Finally, we discuss scalable implementation of these methods using optical and circuit-QED technology.

Entanglement entropy for (3+1)-dimensional topological order with excitations

Physical Review B, 2018

Excitations in (3+1)D topologically ordered phases have very rich structures. (3+1)D topological phases support both point-like and string-like excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the question how different types of topological excitations contribute to the entanglement entropy, or alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological orders? We are mainly interested in (3+1)D topological orders that can be realized in Dijkgraaf-Witten gauge theories, which are labeled by a finite group G and its group 4-cocycle ω ∈ H 4 [G; U (1)] up to group automorphisms. We find that each topological excitation contributes a universal constant ln di to the entanglement entropy, where di is the quantum dimension that depends on both the structure of the excitation and the data (G, ω). The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory (G, ω). In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles ω from the others.

Quantum entanglement in topological phases on a torus

Physical Review B, 2016

In this paper we study the effect of non-trivial spatial topology on quantum entanglement by examining the degenerate ground states of a topologically ordered system on torus. Using the stringnet (fixed-point) wave-function, we propose a general formula of the reduced density matrix when the system is partitioned into two cylinders. The cylindrical topology of the subsystems makes a significant difference in regard to entanglement: a global quantum number for the many-body states comes into play, together with a decomposition matrix M which describes how topological charges of the ground states decompose into boundary degrees of freedom. We obtain a general formula for entanglement entropy and generalize the concept of minimally entangled states to minimally entangled sectors. Concrete examples are demonstrated with data from both finite groups and modular tensor categories (i.e., Fibonacci, Ising, etc.), supported by numerical verification.

Identifying symmetry-protected topological order by entanglement entropy

Physical Review B, 2013

According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations, and to readily distinguish the SPT phase from the trivial one by evaluating the multiplet entanglement spectrum. In particular, using the entropies S I (S H) of integer (half-integer) representations, we can define an entanglement gap G = S I − S H , which equals 1 in the SPT phase, and −1 in the trivial phase. As application of our proposal, we study the spin-1 models on various 1D and quasi-1D lattices, including the bilinear-biquadratic model on the single chain, and the Heisenberg model on a two-leg ladder and a three-leg tube. Among others, we confirm the existence of an SPT phase in the spin-1 tube model, and reveal that the phase transition between the SPT and the trivial phase is a continuous one. The transition point is found to be critical, with conformal central charge c = 3 determined by fits to the block entanglement entropy.