Topological minimally entangled states via geometric measure (original) (raw)
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Topological Geometric Entanglement
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.
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.
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.
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.
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.
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...
Physical Review Letters, 2014
Topological order in a 2d quantum matter can be determined by the topological contribution to the entanglement Rényi entropies. However, when close to a quantum phase transition, its calculation becomes cumbersome. Here we show how topological phase transitions in 2d systems can be much better assessed by multipartite entanglement, as measured by the topological geometric entanglement of blocks. Specifically, we present an efficient tensor network algorithm based on Projected Entangled Pair States to compute this quantity for a torus partitioned into cylinders, and then use this method to find sharp evidence of topological phase transitions in 2d systems with a string-tension perturbation. When compared to tensor network methods for Rényi entropies, our approach produces almost perfect accuracies close to criticality and, on top, is orders of magnitude faster. The method can be adapted to deal with any topological state of the system, including minimally entangled ground states. It also allows to extract the critical exponent of the correlation length, and shows that there is no continuous entanglement-loss along renormalization group flows in topological phases.
Tensor Network States, Entanglement, and Anomalies of Topological Phases of Matters
2020
This dissertation investigates two aspects of topological phases of matter: 1) the tensor network state (TNS) representations of the ground states as well as their entanglement entropies of gapped Hamiltonians in diverse dimensions; 2) the anomalies and dynamics of strongly coupled quantum field theories. For the first aspect, we first show an efficient method of analytically deriving the translation invariant TNS and matrix product state (MPS) representation for the ground state of translation invariant stabilizer code Hamiltonians in both 1d and higher dimensions. These TNS/MPS states have minimal virtual bond dimension. Using the TNS, we derive the entanglement entropy for a variety of stabilizer codes, including the fracton models the Haah code. We further go beyond the stabilizer codes and study the structure of entanglement entropy for generic 3d gapped Hamiltonians. In particular, an explicit formula for a universal physical observable – topological entanglement entropy (TEE)...
Multipartite entanglement in fermionic systems via a geometric measure
Physical Review A, 2010
We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for N spin-1 2 fermions distributed over 2L modes (single particle states). The measure is defined on the 2L qubit space isomorphic to the Fock space for 2L single particle states. This entanglement measure is defined for a given partition of 2L modes containing m ≥ 2 subsets. Thus this measure applies to m ≤ 2L partite fermionic system where L is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets may have different dimensions. Further, we have defined the local quantum operations with respect to a given partition of modes. This definition is generic and unifies different ways of dividing a fermionic system into subsystems. We have shown, using a representative case, that the geometric measure is invariant under local unitaries corresponding to a given partition. We explicitly demonstrate the use of the measure to calculate multipartite entanglement in some correlated electron systems. To the best of our knowledge, there is no usable entanglement measure of m > 3 partite fermionic systems in the literature, so that this is the first measure of multipartite entanglement for fermionic systems going beyond the bipartite and tripartite cases.