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arXiv:0801.3681 · Physical Review Letters

Single-Copy Entanglement in a Gapped Quantum Spin Chain

Christopher Hadley

Published Apr 29, 2008• 9 citations
arXivHTMLPDF

Abstract

The single-copy entanglement of a given many-body quantum system is defined [J. Eisert and M. Cramer, Phys. Rev. A 72, 042112 (2005)10.1103/PhysRevA.72.042112] as the maximal entanglement deterministically distillable from a bipartition of a single specimen of that system. For critical (gapless) spin chains, it was recently shown that this is exactly half the von Neumann entropy [R. Orús, J. I. Latorre, J. Eisert, and M. Cramer, Phys. Rev. A 73, 060303(R) (2006)], itself defined as the entanglement distillable in the asymptotic limit-i.e., given an infinite number of copies of the system. It is an open question as to what the equivalent behavior for gapped systems is. In this Letter, I show that for the paradigmatic spin-S Affleck-Kennedy-Lieb-Tasaki chain (the archetypal gapped chain), the single-copy entanglement is equal to the von Neumann entropy; i.e., all the entanglement present may be distilled from a single specimen.

DOI
10.1103/physrevlett.100.177202

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•Jan 1, 2008•0 citations
The single-copy entanglement of a given many-body quantum system is defined [J. Eisert and M. Cramer, Phys. Rev. A 72, 042112 (2005)10.1103/PhysRevA.72.042112] as the maximal entanglement deterministically distillable from a bipartition of a single specimen of that system. For critical (gapless) spin chains, it was recently shown that this is exactly half the von Neumann entropy [R. Orús, J. I. Latorre, J. Eisert, and M. Cramer, Phys. Rev. A 73, 060303(R) (2006)], itself defined as the entanglement distillable in the asymptotic limit-i.e., given an infinite number of copies of the system. It is an open question as to what the equivalent behavior for gapped systems is. In this Letter, I show that for the paradigmatic spin-S Affleck-Kennedy-Lieb-Tasaki chain (the archetypal gapped chain), the single-copy entanglement is equal to the von Neumann entropy; i.e., all the entanglement present may be distilled from a single specimen.
We establish a quantitative relationship between the entanglement content of a single quantum chain at a critical point and the corresponding entropy of entanglement. We find that, surprisingly, the leading critical scaling of the single-copy entanglement with respect to any bipartitioning is exactly one-half of the entropy of entanglement, in a general setting of conformal field theory and quasifree systems. Conformal symmetry imposes that the single-copy entanglement scales as ${E}_{1}({\ensuremath{\rho}}_{L})=(c∕6)\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L\ensuremath{-}(c∕6)({\ensuremath{\pi}}^{2}∕\mathrm{ln}\phantom{\rule{0.2em}{0ex}}L)+O(1∕L)$, where LLL is the number of constituents in a block of an infinite chain and ccc denotes the central charge. This shows that from a single specimen of a critical chain, already half the entanglement can be distilled compared to the rate that is asymptotically available. The result is substantiated by a quantitative analysis for all translationally invariant quantum spin chains corresponding to all isotropic quasifree fermionic models. An example of the XYXYXY spin chain shows that away from criticality the above relation is maintained only near the quantum phase transition.
We consider the single-copy entanglement as a quantity to assess quantum correlations in the ground state in quantum many-body systems. We show for a large class of models that already on the level of single specimens of spin chains, criticality is accompanied with the possibility of distilling a maximally entangled state of arbitrary dimension from a sufficiently large block deterministically, with local operations and classical communication. These analytical results---which refine previous results on the divergence of block entropy as the rate at which maximally entangled pairs can be distilled from many identically prepared chains---are made quantitative for general isotropic translationally invariant spin chains that can be mapped onto a quasifree fermionic system, and for the anisotropic XYXYXY model. For the XXXXXX model, we provide the asymptotic scaling of $\ensuremath{\sim}(1∕6){\mathrm{log}}_{2}(L)$, and contrast it with the block entropy.
We investigate boundary critical phenomena from a quantum-information perspective. Bipartite entanglement in the ground state of one-dimensional quantum systems is quantified using the R\'enyi entropy ${S}_{\ensuremath{\alpha}}$, which includes the von Neumann entropy $(\ensuremath{\alpha}\ensuremath{\rightarrow}1)$ and the single-copy entanglement $(\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty})$ as special cases. We identify the contribution of the boundaries to the R\'enyi entropy, and show that there is an entanglement loss along boundary renormalization group (RG) flows. This property, which is intimately related to the Affleck-Ludwig ggg theorem, is a consequence of majorization relations between the spectra of the reduced density matrix along the boundary RG flows. We also point out that the bulk contribution to the single-copy entanglement is half of that to the von Neumann entropy, whereas the boundary contribution is the same.
We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φS provides a particular example for this type of entanglement.
We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L . We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l , and the total system size, L , for all four degenerate ground states for both odd and even L . In the large l , L limits the entanglement entropy approaches ln (2) for the S z T = ±1 while it approaches twice that value, 2ln (2), for the S z T = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.
Quantum information theory is an exciting, inter-disciplinary field, combining elements of condensed matter theory, quantum mechanics and information theory. In this thesis, I shall make a modest contribution to this field by examining entanglement in many-body systems with more than two levels. In the first section, I consider the dynamics of a system of qutrits three-level quantum systems which are coupled through an SU(3)-invariant permutation Hamiltonian. Each term in this Hamil- tonian is a nearest-neighbour permutation operator, and thus this Hamiltonian may be considered a generalisation of the standard SU (2)-invariant Heisenberg Hamiltonian, in which every term (up to the addition of the identity operator) is a nearest-neighbour permutation operator for two-level system. The system considered has the topology of a cross, and thus may be considered (to a limited extent) analogous to a beam-splitter. The aim of the study is to establish a Bell singlet state between two distant parties. Building on this work, I shall go on to consider the ground state of a system made up of many-level systems coupled by the same Hamiltonian I shall show that this state is a generalisation of the two-level singlet to many levels and many systems. It thus has a high degree of symmetry. I will consider its application in entanglement distribution through measurements (localisable entanglement), and discuss how it may be physically implemented in systems of ultracold atoms, through the Hubbard model. I shall also show that in the famous valence bond solid (the ground state of the Affleck-Kennedy- Lieb-Tasaki spin chain), all the entanglement present in the state may be extracted from a single copy of the chain this is in contrast to gapless, critical chains, in which only half the total entanglement is extractable from a single copy.