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arXiv:1410.1928

Gapped spin Hamiltonian motivated by quantum teleportation

Ari Mizel

Published Oct 7, 2014• 0 citations
arXivHTMLPDF

Abstract

We construct a Hamiltonian whose ground state encodes a time-independent emulation of quan- tum teleportation. We calculate properties of the Hamiltonian, using exact diagonalization and a mean-field theory, and argue that it has a gap. The system exhibits an illuminating relationship to the well-known AKLT (Affleck, Lieb, Kennedy and Tasaki) model.

DOI
10.48550/arxiv.1410.1928

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We construct a Hamiltonian whose ground state encodes a time-independent emulation of quan- tum teleportation. We calculate properties of the Hamiltonian, using exact diagonalization and a mean-field theory, and argue that it has a gap. The system exhibits an illuminating relationship to the well-known AKLT (Affleck, Lieb, Kennedy and Tasaki) model.
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'.
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.
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.