Single-Copy Entanglement in a Gapped Quantum Spin Chain

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.

Entanglement and Density Matrix of a Block of Spins in AKLT Model

“…In the limit L → ∞, that is when the size of the block becomes large, we learned from Fan, et al (2004), Katsura, et al (2007a), Vidal, et al (2003), Hadley (2008) that the von Neumann entropy reaches the saturated value S v.N = ln (S + 1) 2 . Then the density matrix (denoted by ρ ∞ in the limit) can only take the form (see Nielsen & Chuang 2000 for a general proof)…”

Block spin density matrix of the inhomogeneous AKLT model

“…In Theorem 1 we constructed eigenvectors of the density matrix with non-zero eigenvalues. These eigenvectors |G; J,Ω defined in (22), or |VBS L (J, M ) defined in (29) equivalently, are proved to be the (M 01 + 1)(M L,L+1 + 1) zero-energy ground states of the block Hamiltonian (20). Using orthogonal basis {|VBS L (J, M ) }, in Theorem 2 an explicit expression (43) for corresponding eigenvalues in terms of Wigner 3j-symbols is derived.…”

“…then these (M 01 + 1)(M L,L+1 + 1) states (29) are not only linearly independent but also mutually orthogonal (see Appendix A). Furthermore, any ground state |G; J,Ω can be written as a linear superposition over these degenerate VBS states as…”

“…The set of eigenvectors also spans the subspace that the density matrix projects onto. We prove the theorem by re-writing the density matrix ρ L (17) as a projector in diagonal form onto the orthogonal degenerate VBS states {|VBS L (J, M ) } introduced in (29).…”

“…In the limit L → ∞ (that is when the size of the block becomes large), we learned from [16], [29], [38], [66] that the von Neumann entropy reaches the saturated value S v.N = ln (S + 1) 2 . It was also proved in [71] that the density matrix (denoted by ρ∞ in the limit) takes the form…”