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arXiv:quant-ph/0509015 · Journal of Optics B Quantum and Semiclassical Optics

Non-local properties of a symmetric two-qubit system

A. R. Usha Devi, M. S. Uma, R. Prabhu, Sudha

Published Nov 23, 2005• 12 citations
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Abstract

Non-local properties of symmetric two-qubit states are quantified in terms of a complete set of entanglement invariants. We prove that negative values of some of the invariants are signatures of quantum entanglement. This leads us to identify sufficient conditions for non-separability in terms of entanglement invariants. Non-local properties of two-qubit states extracted from (i) the Dicke state, (ii) a state generated by a one-axis twisting Hamiltonian, and (iii) a one-dimensional Ising chain with nearest neighbour interaction are analysed in terms of the invariants characterizing them.

DOI
10.1088/1464-4266/7/12/042

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Abstract For general symmetric multi-qubit systems, the behavior of one- and two-qubit entanglement for Dicke, spin coherent and parity-adapted (even and odd) spin coherent states is determined. These quantum correlations are quantified by linear and von Neumann entropies of the corresponding one- and two-qubit reduced density matrices of the multi-qubit system. These states play a fundamental role in the study of Hamiltonian systems written in terms of collective generators of the angular momentum algebra like, for example, the Lipkin–Meshkov–Glick (LMG) model. Here we shall use these entanglement measures as a signature to characterize the different quantum phases that appear in these models.
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We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite properties, such as positivity and positivity under partial transpose (PPT), can be simply characterized in terms of these vectors and their discrete Fourier transforms. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d ≤ 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from from the literature. Our results show that these states have a rich entanglement structure, even in the bosonic subspace.
We analyze entanglement and nonlocal properties of the convex set of symmetric NNN-qubit states which are diagonal in the Dicke basis. First, we demonstrate that within this set, semidefinite positivity of partial transposition (PPT) is necessary and sufficient for separability---which has also been reported recently by Yu [Phys. Rev. A 94, 060101(R) (2016)]. Furthermore, we show which states among the entangled diagonal symmetric are nonlocal under two-body Bell inequalities. The diagonal symmetric convex set contains a simple and extended family of states that violate the weak Peres conjecture, being PPT with respect to one partition but violating a Bell inequality in such partition. Our method opens directions to address entanglement and nonlocality on higher dimensional symmetric states, where presently very few results are available.