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arXiv:quant-ph/0412099 · Physical Review A

Entanglement observables and witnesses for interacting quantum spin systems

Lian-Ao Wu, Somshubhro Bandyopadhyay, M. S. Sarandy, Daniel A. Lidar

Published Sep 7, 2005• 78 citations
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Abstract

We discuss the detection of entanglement in interacting quantum spin systems. First, thermodynamic Hamiltonian-based witnesses are computed for a general class of one-dimensional spin-$1∕2$ models. Second, we introduce optimal bipartite entanglement observables. We show that a bipartite entanglement measure can generally be associated with a set of independent two-body spin observables whose expectation values can be used to witness entanglement. The number of necessary observables is ruled by the symmetries of the model. Illustrative examples are presented.

DOI
10.1103/physreva.72.032309

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We discuss the detection of entanglement in interacting quantum spin systems. First, thermodynamic Hamiltonian-based witnesses are computed for a general class of one-dimensional spin-1∕21∕21∕2 models. Second, we introduce optimal bipartite entanglement observables. We show that a bipartite entanglement measure can generally be associated with a set of independent two-body spin observables whose expectation values can be used to witness entanglement. The number of necessary observables is ruled by the symmetries of the model. Illustrative examples are presented.
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Quantum science and engineering increasingly target the exploitation and control of large-scale dynamical systems. As system size grows, novel phenomena and quantum advantages emerge, but the resulting many-body dynamics become correspondingly harder to monitor and characterize. In particular, the underlying Hamiltonian often lacks symmetries, and the induced many-body dynamics quickly become hard to track. Focusing on one of the most important quantum features---entanglement---we demonstrate that, for a broad class of common Hamiltonian models, one can design effective witnesses that detect both the presence and structure of entanglement over relatively long evolution times. Inspired by stabilizer-based simulation, we track the evolution of generalized stabilizers of the underlying state and use them to construct linear entanglement witnesses. Through numerical experiments on Heisenberg-type Hamiltonians with up to twelve qubits, we show that the required number of measurement settings can be kept at a modest scale as the system size increases, and entanglement detection remains effective over evolution times that significantly exceed those accessible to naive approaches. The corresponding calculations can be carried out largely symbolically. Furthermore, unlike accurate state simulation, the stabilizer-based design of entanglement witnesses can be performed in a much more coarse-grained, digitized manner. We hope these results will stimulate further exploration of the quantum control of large systems.
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