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Journal of Physics Condensed Matter

Entanglement versus gap, quantum teleportation, and the AKLT model

Ari Mizel

Published Jun 14, 2021• 4 citations
Publisher

Abstract

Quantum-mechanical entanglement is notoriously volatile because of its susceptibility to external disturbances. However, entanglement can be stabilized if it is present in the non-degenerate ground state of a gapped, time-independent Hamiltonian. In this paper, we devise a spin-chain Hamiltonian whose ground state contains a Bell pair, with one member of the pair at each end of the chain. We study the Hamiltonian numerically, using full numerical diagonalization and a carefully tailored mean-field theory, to show that it is gapped. Whenever the Hamiltonian is tuned to increase its gap, the fidelity of its Bell pair decreases, manifesting a fundamental contention. The form of the Hamiltonian is motivated by quantum teleportation. Comparing it to the canonical Affleck, Kennedy, Lieb, and Tasaki (AKLT) model, we find that the AKLT model exhibits a sort of 'failed quantum teleportation'.

DOI
10.1088/1361-648x/ac0384

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Quantum-mechanical entanglement is notoriously volatile because of its susceptibility to external disturbances. However, entanglement can be stabilized if it is present in the non-degenerate ground state of a gapped, time-independent Hamiltonian. In this paper, we devise a spin-chain Hamiltonian whose ground state contains a Bell pair, with one member of the pair at each end of the chain. We study the Hamiltonian numerically, using full numerical diagonalization and a carefully tailored mean-field theory, to show that it is gapped. Whenever the Hamiltonian is tuned to increase its gap, the fidelity of its Bell pair decreases, manifesting a fundamental contention. The form of the Hamiltonian is motivated by quantum teleportation. Comparing it to the canonical Affleck, Kennedy, Lieb, and Tasaki (AKLT) model, we find that the AKLT model exhibits a sort of 'failed quantum teleportation'.
In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages
Key properties of a physical system depend on whether it is gapped, i.e., whether its spectral gap has a positive lower bound that is independent of system size. Here, we provide a prescription for renormalizing a spin chain Hamiltonian in such a way that the renormalized Hamiltonian is gapped if and only if the original Hamiltonian is gapped. Then, we articulate a set of conditions that guarantees the renormalized Hamiltonian is gapped. These conditions are built on a certain strong notion of decaying correlations involving an operator norm of non-commuting terms in the renormalized Hamiltonian. We apply the method to show that two interesting models, with forms motivated by quantum circuits, are gapped. We also confirm the generality of the method by successfully applying it to a somewhat different case, the well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) model.
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of SU(3)\text{SU}(3)SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating SU(3)\text{SU}(3)SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its SU(2)\text{SU}(2)SU(2) analog. We discuss the SPT feature of the state.
Generating high-quality multiparticle entanglement between communicating parties is the primary resource in quantum teleportation protocols. To this aim, we show that the natural dynamics of a single spin chain is able to sustain the generation of two pairs of Bell states----possibly shared between a sender and a distant receiver---which can in turn enable two-qubit teleportation. In particular, we address a spin-12\frac{1}{2}21​ chain with XXXXXX interactions, connecting two pairs of spins located at its boundaries, playing the roles of sender and receiver. In the regime where both end pairs are weakly coupled to the spin chain, it is possible to generate at predefinite times a state that has vanishing infidelity with the product state of two Bell pairs, thereby providing nearly unit fidelity of teleportation. We also derive an effective Hamiltonian via a second-order perturbation approach that faithfully reproduces the dynamics of the full system.
Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e0<0{e}_{0}<0e0​<0, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.