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arXiv:1005.4448 · Physical Review A

Application of entanglement conditions to spin systems

Hongjun Zheng, Ho Trung Dung, Mark S. Hillery

Published Jun 11, 2010• 7 citations
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Abstract

There have been numerous studies of entanglement in spin systems. These have usually focused on examining the entanglement between individual spins or determining whether the state of the system is completely separable. Here, we present conditions that allow us to determine whether blocks of spins are entangled. We show that sometimes these conditions can detect entanglement better than conditions involving individual spins. We apply these conditions to study entanglement in spin-wave states, both when there are only a few magnons present and also at finite temperature.

DOI
10.1103/physreva.81.062311

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There have been numerous studies of entanglement in spin systems. These have usually focused on examining the entanglement between individual spins or determining whether the state of the system is completely separable. Here, we present conditions that allow us to determine whether blocks of spins are entangled. We show that sometimes these conditions can detect entanglement better than conditions involving individual spins. We apply these conditions to study entanglement in spin-wave states, both when there are only a few magnons present and also at finite temperature.
There is no single test that will determine whether a general quantum state is separable or entangled. There are, however, a number of sufficient conditions, including the partial-transpose condition and entanglement witnesses. Here we present a class of sufficient conditions based on expectation values of non-hermitian operators. Initially we examine bipartite systems. The application of these conditions to multimode states of the electromagnetic field and to spin systems is discussed. We show that our original conditions can be strengthened if we make them invariant under certain classes of local unitary transformations. We then go on to study multipartite systems and present conditions that can be used to show that a state is not totally separable. Some simple applications of these conditions are also presented.
Entanglement between individual spins can be detected by using thermodynamics quantities as entanglement witnesses. This applies to collective spins also, provided that their internal degrees of freedom are frozen, as in the limit of weakly coupled nanomagnets. Here, we extend such approach to the detection of entanglement between subsystems of a spin cluster, beyond such weak-coupling limit. The resulting inequalities are violated in spin clusters with different geometries, thus allowing the detection of zero- and finite-temperature entanglement. Under relevant and experimentally verifiable conditions, all the required expectation values can be traced back to correlation functions of individual spins, which are now made selectively available by four-dimensional inelastic neutron scattering.
What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized spin squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular momentum components. They can be used for the experimental detection of entanglement in a system of spin‐12 particles in which the spins cannot be individually addressed. We also show how to obtain the complete set of entanglement conditions if the mean spin and the spin correlation matrix are known. Finally, we apply our criteria for entanglement detection in spin models, showing that they can be used to detect bound entanglement in the thermal state of these systems.
We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.
Magnons can serve as a bridge between spin, phonon, and photon systems, which renders them suitable for constructing hybrid systems. An important application of such hybrid systems is generating entanglement between different platforms. As magnons can support a broad variety of states, e.g., Fock states, squeezed states, or coherent states, and hybrid states can be produced with cavities or spins, there are many different kinds of entangled states in magnon systems. In this paper, we consider the entanglement of cat-state-like throughout states, which can be generated in magnon systems with parametric pumps beyond the parametric stable intensities. However, estimating the entanglement in such states is challenging due to their multiphoton and non-Gaussian properties. Here, we apply a modular variable-based projection, which maps the catlike states to spin states, preserving the encoded information. After the projection, Bell's inequality is employed to detect the entanglement in the effective spin states. Our numerical analysis provides the conditions for generating catlike entanglement in magnon systems and can be conveniently extended to other entangled states that may be formed by magnon and spin systems.