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arXiv:2103.08929 · Journal of Physics A Mathematical and Theoretical

Link representation of the entanglement entropies for all bipartitions

Sudipto Singha Roy, Silvia N. Santalla, Germán Sierra, Javier Rodríguez-Laguna

Published Jun 10, 2021• 13 citations
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Abstract

Abstract We have recently shown that the entanglement entropy of any bipartition of a quantum state can be approximated as the sum of certain link strengths connecting internal and external sites. The representation is useful to unveil the geometry associated with the entanglement structure of a quantum many-body state which may occasionally differ from the one suggested by the Hamiltonian of the system. Yet, the obtention of these entanglement links is a complex mathematical problem. In this work, we address this issue and propose several approximation techniques for matrix product states, free fermionic states, or in cases in which contiguous blocks are specially relevant. Along with this, we discuss the accuracy of the approximation for different types of states and partitions. Finally, we employ the link representation to discuss two different physical systems: the spin-1/2 long-range XXZ chain and the spin-1 bilinear biquadratic chain.

DOI
10.1088/1751-8121/ac0a30

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Abstract We have recently shown that the entanglement entropy of any bipartition of a quantum state can be approximated as the sum of certain link strengths connecting internal and external sites. The representation is useful to unveil the geometry associated with the entanglement structure of a quantum many-body state which may occasionally differ from the one suggested by the Hamiltonian of the system. Yet, the obtention of these entanglement links is a complex mathematical problem. In this work, we address this issue and propose several approximation techniques for matrix product states, free fermionic states, or in cases in which contiguous blocks are specially relevant. Along with this, we discuss the accuracy of the approximation for different types of states and partitions. Finally, we employ the link representation to discuss two different physical systems: the spin-1/2 long-range XXZ chain and the spin-1 bilinear biquadratic chain.
The entanglement entropy (EE) of any bipartition of a pure state can be approximately expressed as a sum of entanglement links (ELs). In this work, we introduce their exact extension, i.e. the entanglement hyperlinks (EHLs), a type of generalized mutual informations defined through the inclusion-exclusion principle, each of which captures contributions to the multipartite entanglement that are not reducible to lower-order terms. We show that any EHL crossing a factorized partition must vanish, and that the EHLs between any set of blocks can be expressed as a sum of all the EHLs that join all of them. This last result allows us to provide an exact representation of the EE of any block of a pure state, from the sum of the EHLs which cross its boundary. In order to illustrate their rich structure, we discuss some explicit numerical examples using ground states of local Hamiltonians. The EHLs thus provide a remarkable tool to characterize multipartite entanglement in quantum information theory and quantum many-body physics.
Abstract We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv: 1104.1004 , for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.
Abstract Entanglement entropy under a particle bipartition generates complementary information to mode entanglement because it is sensitive to interactions and particle statistics at leading order and does not depend on any externally imposed length scale. In this paper, we investigate the particle entanglement entropy in a system of N interacting spinless lattice fermions in one spatial dimension by combining bosonization techniques with exact and approximate numerical methods. We introduce a general scaling form for the fermionic particle entanglement entropy captured by a shape function that enters as an extensive interaction-induced correction to a known free fermion result. A general asymptotic expansion in the total number of particles demonstrates that its form is robust for different values of the Rényi index and highlights how quantum correlations are encoded in the n-particle density matrix of a pure many-body quantum state.
The time evolution of a quantum state with short-range correlations after a quench to a one-dimensional critical Hamiltonian can be understood using the quasiparticle picture, which states that local entanglement spreads as if it was carried by quasiparticles which separate at a fixed speed. We extend the quasiparticle picture using the recently introduced link representation of entanglement, allowing us to apply it to initial states presenting long-range correlations. The entanglement links are current correlators, and therefore follow a wave equation on the appropriate configurational space which allows us to predict the time evolution of the entanglement entropies. Our results are checked numerically for free fermionic chains with different initial entanglement patterns.
Motivated by recent advances in quantum technologies, we investigate entanglement spreading and excitation transport within quantum complex networks. For this purpose, we adopt a description in which network nodes represent sites of the physical system (e.g. qubits), and links represent pairs of sites coupled by the Hamiltonian [Bentsen et al., Phys Rev Lett (2019)]. We consider a random tunable-range model [Gori et al., Phys Rev E (2017)] which, in terms of only two parameters, encompasses a large variety of graphs and reduces, in limiting cases, to well-studied benchmarks. Then, it is equipped with a paradigmatic Hamiltonian consisting of hopping terms on the network links. Building on known results [Gori et al., Phys Rev E (2017)], we first study the classical case, providing new semi-analytical estimates. We then address quantum dynamics: For the single-particle case, we analyse the asymptotic entanglement entropy between two halves of the system, computed by Exact Diagonalization. Surprisingly, we observe a new localization region for intermediate ranges of the network links. To understand this phenomenon, we introduce an approximate one-dimensional model and try to relate dynamics to network measures. The analysis is extended to half-filling using fermionic Gaussian states, obtaining results close to the single-particle ones that raise new interesting questions for future work. It is concluded that this model exhibits non-trivial behaviour, especially in the region where unexpected localization occurs. We infer that this is determined by the interplay between network features and quantum effects, possibly induced by structural disorder.