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arXiv:quant-ph/0412167 · Europhysics Letters (EPL)

Multi-distributed entanglement in finitely correlated chains

F Benatti, Beatrix C. Hiesmayr, H Narnhofer

Published Sep 1, 2005• 8 citations

Abstract

The entanglement-sharing properties of an infinite spin-chain are studied when the state of the chain is a pure, translation-invariant state with a matrix-product structure (Klümper A., Schadschneider A. and Zittartz J., J. Phys. A , 24 (1991) L955; Z. Phys. B , 87 (1992) 281; Europhys. Lett. , 24 (1993) 293). We study the entanglement properties of such states by means of their finitely correlated structure (Fannes M., Nachtergaele B. and Werner R. F., Comm. Math. Phys. , 144 (1992) 443; Europhys. Lett. , 10 (1989) 633; J. Phys. A , 24 (1991) L185). These states are recursively constructed by means of an auxiliary density matrix ρ on a matrix algebra ℬ and a completely positive map : ⊗ ℬ → ℬ, where is the spin 2 × 2 matrix algebra. General structural results for the infinite chain are therefore obtained by explicit calculations in (finite) matrix algebras. In particular, we study not only the entanglement shared by nearest-neighbours, but also, differently from previous works (Wootters W. K., Contemp. Math. , 305 (2002) 299) the entanglement shared between connected regions of the spin-chain. This range of possible applications is illustrated and the maximal concurrence = 1/√2 (Coffman V., Kundu J. and Wootters W. K., Phys. Rev. A , 61 (2000) 052306) for the entanglement of connected regions can actually be reached.

DOI: 10.1209/epl/i2005-10204-2

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The entanglement-sharing properties of an infinite spin-chain are studied when the state of the chain is a pure, translation-invariant state with a matrix-product structure (Klümper A., Schadschneider A. and Zittartz J., J. Phys. A , 24 (1991) L955; Z. Phys. B , 87 (1992) 281; Europhys. Lett. , 24 (1993) 293). We study the entanglement properties of such states by means of their finitely correlated structure (Fannes M., Nachtergaele B. and Werner R. F., Comm. Math. Phys. , 144 (1992) 443; Europhys. Lett. , 10 (1989) 633; J. Phys. A , 24 (1991) L185). These states are recursively constructed by means of an auxiliary density matrix ρ on a matrix algebra ℬ and a completely positive map : ⊗ ℬ → ℬ, where is the spin 2 × 2 matrix algebra. General structural results for the infinite chain are therefore obtained by explicit calculations in (finite) matrix algebras. In particular, we study not only the entanglement shared by nearest-neighbours, but also, differently from previous works (Wootters W. K., Contemp. Math. , 305 (2002) 299) the entanglement shared between connected regions of the spin-chain. This range of possible applications is illustrated and the maximal concurrence = 1/√2 (Coffman V., Kundu J. and Wootters W. K., Phys. Rev. A , 61 (2000) 052306) for the entanglement of connected regions can actually be reached.

The entanglement-sharing properties of an infinite qubit-chain are studied when the state of the chain is pure, translation-invariant and with a finitely correlated structure. These states are iteratively constructed by means of auxiliary states and completely positive maps. They allow for a variety of entanglement distributions along the chain that can be studied by means of the entanglement properties of an auxiliary state over an auxiliary bipartite qubit-system. In particular, we constructed an example which reaches the maximal bound of concurrence C=1/Sqrt{2} for the entanglement of connected regions.

We consider translationally invariant states of an infinite one-dimensional chain of qubits or spin-

\frac{1}{2}

particles. We maximize the entanglement shared by nearest neighbors via a variational approach based on finitely correlated states. We find an upper bound of nearest-neighbor concurrence equal to

\mathcal{C}=0.434\phantom{\rule{0.2em}{0ex}}095

which is

0.09%

away from the bound

{\mathcal{C}}_{\mathrm{W}}=0.434\phantom{\rule{0.2em}{0ex}}467

obtained by a completely different procedure. The obtained state-maximizing nearest-neighbor entanglement seems to approximate the maximally entangled mixed states. Further we investigate in detail several other properties of the so-obtained optimal state.

Gaussian matrix-product states are obtained as the outputs of projection operations from an ancillary space of

0.09%

infinitely entangled bonds connecting neighboring sites, applied at each of

{\mathcal{C}}_{\mathrm{W}}=0.434\phantom{\rule{0.2em}{0ex}}467

sites of a harmonic chain. Replacing the projections by associated Gaussian states, the building blocks, we show that the entanglement range in translationally invariant Gaussian matrix-product states depends on how entangled the building blocks are. In particular, infinite entanglement in the building blocks produces fully symmetric Gaussian states with maximum entanglement range. From their peculiar properties of entanglement sharing, a basic difference with spin chains is revealed: Gaussian matrix-product states can possess unlimited, long-range entanglement even with minimum number of ancillary bonds

(M=1)

. Finally we discuss how these states can be experimentally engineered from

N

copies of a three-mode building block and

N

two-mode finitely squeezed states.

We begin by deriving bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated States (FCS). The bounds we derive become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our result permits a more efficient calculation, numerically and in some cases even analytically, of the entanglement in arbitrary finitely correlated quantum spin chains. We continue the study of entanglement in the setting of ground states of Hamiltonians with a spectral gap. In particular, for

V

a finite subset of

Z^d

, we let

H_V

denote a Hamiltonian on

V

with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap. For a density matrix

ρ_A

that describes the finite-volume restriction to a region

A

of the unique ground state, we provide a detailed version of a proof by M. Hastings, that the entropy of

ρ_A

is bounded by a uniform constant

H_V

. Moreover, we provide a detailed generalization of the 1-dimensional construction of Hastings' approximation to the ground state in dimensions 2 and higher. Finally, we turn our attention to the study of a conjecture central to Quantum Information Theory, the multiplicativity of the maximal output Schatten

p

-norm of quantum channels, for

p > 1

. In particular, we study the output 2-norm for a special class of quantum channels, the depolarized Werner-Holevo channels, and show that multiplicativity holds for a product of two identical channels in this class.

We consider entanglement properties of pure finitely correlated states (FCS). We derive bounds for the entanglement of a spin with an interval of spins in an arbitrary pure FCS. Finitely correlated states are also known as matrix product states or generalized valence-bond states. The bounds become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our bounds provide a proof of the recent conjecture by Benatti, Hiesmayr, and Narnhofer that their necessary condition for nonvanishing entanglement in terms of a single spin and the memory of the FCS is also sufficient. We also generalize the study of entanglement in the Affleck-Kennedy-Lieb-Tasaki model by Fan, Korepin, and Roychowdhury. Our result permits a more efficient calculation, numerically and in some cases analytically, of the entanglement of arbitrary finitely correlated quantum spin chains.