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arXiv:0911.4670 · Physical Review A

Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Ching-Yu Huang, Feng-Li Lin

Published Mar 5, 2010• 45 citations
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Abstract

We compute the multipartite entanglement measures such as the global entanglement of various one- and two-dimensional quantum systems to probe the quantum criticality based on the matrix and tensor product states (MPSs and TPSs). We use the infinite time-evolving block decimation (iTEBD) method to find the ground states numerically in the form of MPSs and TPSs, and then evaluate their entanglement measures by the method of tensor renormalization group (TRG). We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points in all models considered here. We also comment on the scaling behaviors of the entanglement measures by the ideas of quantum-state renormalization group transformations.

DOI
10.1103/physreva.81.032304

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We compute the multipartite entanglement measures such as the global entanglement of various one- and two-dimensional quantum systems to probe the quantum criticality based on the matrix and tensor product states (MPSs and TPSs). We use the infinite time-evolving block decimation (iTEBD) method to find the ground states numerically in the form of MPSs and TPSs, and then evaluate their entanglement measures by the method of tensor renormalization group (TRG). We find that these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points in all models considered here. We also comment on the scaling behaviors of the entanglement measures by the ideas of quantum-state renormalization group transformations.
The matrix product state (MPS) is utilized to investigate the ground state properties and quantum phase transitions (QPTs) of the dimerized antiferromagnetic Heisenberg (DAH) model. The ground state MPS wavefunctions determined by the infinite time-evolving block decimation (iTEBD) algorithm are shown to be very efficient descriptions of DAH model. In the thermodynamic limit, the quantum entanglement, the bond energy, and the nearest-neighbor correlations are calculated. It is revealed that the singular behavior of the bipartite entanglement can detect the QPTs directly. The critical point J c 2 = 1.0 is determined evidently, and the quantum phase transition is argued to belong to the second-order category. At the critical point, logarithmic divergent character of the block entanglement is observed, and the system can be described by a free bosonic field theory.
Tensor networks (TNs) have become one of the most essential building blocks for various fields of theoretical physics such as condensed matter theory, statistical mechanics, quantum information, and quantum gravity. This review provides a unified description of a series of developments in the TN from the statistical mechanics side. In particular, we begin with the variational principle for the transfer matrix of the 2D Ising model, which naturally leads us to the matrix product state (MPS) and the corner transfer matrix (CTM). We then explain how the CTM can be evolved to such MPS-based approaches as density matrix renormalization group (DMRG) and infinite time-evolved block decimation. We also elucidate that the finite-size DMRG played an intrinsic role for incorporating various quantum information concepts in subsequent developments in the TN. After surveying higher-dimensional generalizations like tensor product states or projected entangled pair states, we describe tensor renormalization groups (TRGs), which are a fusion of TNs and Kadanoff-Wilson type real-space renormalization groups, focusing on their fixed point structures. We then discuss how the difficulty in TRGs for critical systems can be overcome in the tensor network renormalization and the multi-scale entanglement renormalization ansatz.
We investigate multipartite entanglement and a quantum phase transition for spin chain systems by making use of the density-matrix renormalization-group method in the form of matrix-product states. It is found that multipartite entanglement changes dramatically near the critical region and the residual-to-global entanglement ratio reaches its maximum at the critical point of the quantum phase transition. These results indicate that multipartite entanglement as well as the residual-to-global entanglement ratio can serve as good indicators to detect a quantum phase transition. We also demonstrate a linear relation between the next-nearest-neighbor interaction strength and the critical magnetic field for an XYZXYZXYZ spin chain by using multipartite entanglement and quantum coherence.
Topological order in two-dimensional (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, additionally, 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 us 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.
In this paper we consider the quantum phase transition in the Ising model in the presence of a transverse field in one, two, and three dimensions from a multipartite entanglement point of view. Using exact numerical solutions, we are able to study such systems up to 25 qubits. The Meyer-Wallach measure of global entanglement is used to study the critical behavior of this model. The transition we consider is between a symmetric Greenberger-Horne-Zeilinger--like state to a paramagnetic product state. We find that global entanglement serves as a good indicator of quantum phase transition with interesting scaling behavior. We use finite-size scaling to extract the critical point as well as some critical exponents for the one- and two-dimensional models. Our results indicate that such multipartite measure of global entanglement shows universal features regardless of dimension ddd. Our results also provide evidence that multipartite entanglement is better suited for the study of quantum phase transitions than the much-studied bipartite measures.
Multipartite entanglement offers a powerful framework for understanding the complex collective phenomena in quantum many-body systems that are often beyond the description of conventional bipartite entanglement measures. Here, we propose a class of multipartite entanglement measures that incorporate the matrix product state (MPS) representation, enabling the characterization of the optimality of quantum circuits for state preparation. These measures are defined as the minimal distances from a target state to the manifolds of MPSs with specified virtual bond dimensions χ, and thus are dubbed as χ-specified matrix product entanglement (χ-MPE). We demonstrate superlinear, linear, and sublinear scaling behaviors of χ-MPE with respect to the negative logarithmic fidelity F in state preparation, which correspond to excessive, optimal, and insufficient circuit depth D for preparing χ-virtual-dimensional MPSs, respectively. Specifically, a linearly growing χ-MPE with F suggests H_{χ}≃H_{D}, where H_{χ} denotes the manifold of the χ-virtual-dimensional MPSs and H_{D} denotes that of the states accessible by the D-layer circuits. We provide an exact proof that H_{χ=2}≡H_{D=1}. Our results establish tensor networks as a powerful and general tool for developing parametrized measures of multipartite entanglement. The matrix product form adopted in χ-MPE can be readily extended to other tensor network Ansätze, whose scaling behaviors are expected to assess the optimality of quantum circuit in preparing the corresponding tensor network states.