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arXiv:0901.0882 · Mathematica Slovaca

A glance at singlet states and four-partite correlations

Maria Schimpf, Karl Svozil

Published Oct 1, 2010• 5 citations
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Abstract

Abstract Group theoretic methods to construct all N-particle singlet states by iterative recursion are presented. These techniques are applied to the quantum correlations of four spin one-half particles in their singlet states. Multipartite quantized systems can be partitioned, and their observables grouped and redefined into condensed correlations.

DOI
10.2478/s12175-010-0041-7

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Abstract Group theoretic methods to construct all N-particle singlet states by iterative recursion are presented. These techniques are applied to the quantum correlations of four spin one-half particles in their singlet states. Multipartite quantized systems can be partitioned, and their observables grouped and redefined into condensed correlations.
Simulation of two spin-sss singlet correlations for all sss involving spin measurements
In a recent paper [A. Ahanj et al., quant-ph/0603053], we gave a classical protocol to simulate quantum correlations corresponding to the spin sss singlet state for the infinite sequence of spins satisfying 2s+1=2n2s+1 = 2^{n}2s+1=2n. In the present paper, we have generalized this result by giving a classical protocol to exactly simulate quantum correlations implied by the spin-sss singlet state corresponding to all integer as well as half-integer spin values sss. The class of measurements we consider here are only those corresponding to spin observables, as has been done in the above-mentioned paper. The required amount of communication is found to be ⌈log2(s+1)⌉\lceil {\rm log}_{2} (s + 1) \rceil⌈log2​(s+1)⌉ in the worst case scenario, where ⌈x⌉\lceil x \rceil⌈x⌉ is the least integer greater than or equal to xxx.
We introduce an analytical iterative method, the density matrix recursion method, to generate arbitrary reduced density matrices of superpositions of short-range dimer coverings on periodic or non-periodic quantum spin-1/2 ladder lattices, with an arbitrary number of legs. The method can be used to calculate bipartite as well as multipartite physical properties, including bipartite and multi-partite entanglement. We apply this technique to distinguish between even- and odd-legged ladders. Specifically, we show that while genuine multi-partite entanglement decreases with increasing system size for the even-legged ladder states, it does the opposite for odd-legged ones.
Bell inequality is a vital tool to detect the nonlocal correlations, but the construction of it for multipartite systems is still a complicated problem. In this work, inspired via a decomposition of (n+1)(n+1)(n+1)-partite Bell inequalities into nnn-partite ones, we present a generalized iterative formula to construct nontrivial (n+1)(n+1)(n+1)-partite ones from the nnn-partite ones. Our iterative formulas recover the well-known Mermin-Ardehali-Belinsk\ifmmode \breve{\imath}\else \u{\i}\fi{}-Klyshko (MABK) and other families in the literature as special cases. Moreover, a family of ``dual-use'' Bell inequalities is proposed, in the sense that for the generalized Greenberger-Horne-Zeilinger states these inequalities lead to the same quantum violation as the MABK family and, at the same time, the inequalities are able to detect the nonlocality in the entire entangled region. Furthermore, we present a generalization of the I3322 inequality to any nnn-partite case which is still tight, and of the 46 \ifmmode \acute{S}\else \'{S}\fi{}liwa's inequalities to the four-partite tight ones, by applying our iteration method to each inequality and its equivalence class.