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arXiv:quant-ph/0602067 · Physical Review A

Entanglement in Gaussian matrix-product states

Gerardo Adesso, Marie Ericsson

Published Sep 15, 2006• 10 citations
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Abstract

Gaussian matrix-product states are obtained as the outputs of projection operations from an ancillary space of $M$ infinitely entangled bonds connecting neighboring sites, applied at each of $N$ 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.

DOI
10.1103/physreva.74.030305

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Gaussian matrix-product states are obtained as the outputs of projection operations from an ancillary space of MMM infinitely entangled bonds connecting neighboring sites, applied at each of NNN 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)(M=1)(M=1). Finally we discuss how these states can be experimentally engineered from NNN copies of a three-mode building block and NNN two-mode finitely squeezed states.
We study Gaussian valence bond states of continuous variable systems obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites applied at each of N sites of a harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2 M + 1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swapping from the chain of ancillary bonds, through the building blocks. We provide optical schemes to produce bisymmetric three-mode Gaussian building blocks (which correspond to a single bond, M = 1), and study the entanglement structure in the output Gaussian valence bond states. Finally, the usefulness of such states for quantum communication protocols with continuous variables, like telecloning and teleportation networks, is discussed.
Abstract. We study Gaussian valence bond states of continuous variable systems, obtained as the outputs of projection operations from an ancillary space of M infinitely entangled bonds connecting neighboring sites, applied at each of N sites of an harmonic chain. The entanglement distribution in Gaussian valence bond states can be controlled by varying the input amount of entanglement engineered in a (2M + 1)-mode Gaussian state known as the building block, which is isomorphic to the projector applied at a given site. We show how this mechanism can be interpreted in terms of multiple entanglement swapping from the chain of ancillary bonds, through the building blocks. We provide optical schemes to produce bisymmetric three-mode Gaussian building blocks (which correspond to a single bond, M = 1), and study the entanglement structure in the output Gaussian valence bond states. The usefulness of such states for quantum communication protocols with continuous variables, like telecloning and teleportation networks, is finally discussed. Optical implementation and entanglement distribution in Gaussian valence bond states 2 1.
We introduce Gaussian Matrix Product States (GMPS), a generalization of Matrix Product States (MPS) to lattices of harmonic oscillators. Our definition resembles the interpretation of MPS in terms of projected maximally entangled pairs, starting from which we derive several properties of GMPS, often in close analogy to the finite dimensional case: We show how to approximate arbitrary Gaussian states by MPS, we discuss how the entanglement in the bonds can be bounded, we demonstrate how the correlation functions can be computed from the GMPS representation, and that they decay exponentially in one dimension, and finally relate GMPS and ground states of local Hamiltonians.
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.
We investigate the correlation structure of pure N-mode Gaussian resources which can be experimentally generated by means of squeezers and beam splitters, whose entanglement properties are generic. We show that those states are specified (up to local unitaries) by N(N-1)/2 parameters, corresponding to the two-point correlations between any pair of modes. Our construction yields a practical scheme to engineer such generic-entangled N-mode pure Gaussian states by linear optics. We discuss our findings in the framework of Gaussian matrix product states of harmonic lattices, raising connections with entanglement frustration and the entropic area law.