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arXiv:1108.4001 · Physical Review A

Witnessing nonclassical multipartite states

A. Saguia, C. C. Rulli, Thiago Rodrigues de Oliveira, M. S. Sarandy

Published Oct 31, 2011• 27 citations
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Abstract

We investigate a witness for nonclassical multipartite states based on their disturbance under local measurements. The witness operator provides a sufficient condition for nonclassicality that coincides with a nonvanishing global quantum discord, but it does not demand an extremization procedure. Moreover, for the case of ${Z}_{2}$-symmetric systems, we rewrite the witness in terms of correlation functions so that classicality is found to necessarily require either vanishing magnetization in the invariant axis or isotropy of the two-point function in the transverse spin plane. We illustrate our results in quantum spin chains, where a characterization of factorized ground states (with spontaneously broken ${Z}_{2}$ symmetry) is achieved. As a by-product, the witness is also shown to indicate a second-order quantum phase transitions, which is illustrated for both the $XY$ and the Ashkin-Teller spin chains.

DOI
10.1103/physreva.84.042123

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We investigate a witness for nonclassical multipartite states based on their disturbance under local measurements. The witness operator provides a sufficient condition for nonclassicality that coincides with a nonvanishing global quantum discord, but it does not demand an extremization procedure. Moreover, for the case of Z2{Z}_{2}Z2​-symmetric systems, we rewrite the witness in terms of correlation functions so that classicality is found to necessarily require either vanishing magnetization in the invariant axis or isotropy of the two-point function in the transverse spin plane. We illustrate our results in quantum spin chains, where a characterization of factorized ground states (with spontaneously broken Z2{Z}_{2}Z2​ symmetry) is achieved. As a by-product, the witness is also shown to indicate a second-order quantum phase transitions, which is illustrated for both the XYXYXY and the Ashkin-Teller spin chains.
We propose a global measure for quantum correlations in multipartite systems, which is obtained by suitably recasting the quantum discord in terms of relative entropy and local von Neumann measurements. The measure is symmetric with respect to subsystem exchange and is shown to be nonnegative for an arbitrary state. As an illustration, we consider tripartite correlations in the Werner-GHZ (Greenberger-Horne-Zeilinger) state and multipartite correlations at quantum criticality. In particular, in contrast with the pairwise quantum discord, we show that the global quantum discord is able to characterize the infinite-order quantum phase transition in the Ashkin-Teller spin chain.
The ground state of a quantum spin chain is a natural playground for investigating correlations. Nevertheless, not all correlations are genuinely of quantum nature. Here we review the recent progress to quantify the "quantumness" of the correlations throughout the phase diagram of quantum spin systems. Focusing to one spatial dimension, we discuss the behavior of quantum discord (QD) close to quantum phase transitions (QPT). In contrast to the two-spin entanglement, pairwise discord is effectively long-ranged in critical regimes. Besides the features of QPT, QD is especially feasible to explore the factorization phenomenon, giving rise to nontrivial ground classical states in quantum systems. The effects of spontaneous symmetry breaking are also discussed as well as the identification of quantum critical points through correlation witnesses.
A bipartite quantum system in a mixed state can exhibit nonclassical correlations which can go beyond quantum entanglement. While quantum discord is the standard measure of quantifying such general quantum correlations, the nonclassicality can be determined by simpler means via the measurement of witness operators. We experimentally construct a positive map to witness nonclassicality of two qubits in an NMR system. The map can be decomposed in terms of measurable spin magnetizations so that a single run of an experiment on an ensemble of spins suffices to detect the nonclassicality in the state, if present. We let the state evolve in time and use the map to detect nonclassicality as a function of time. To evaluate the efficacy of the witness operator as a means to detect nonclassicality, we measure quantum discord by performing full quantum-state tomography at each time instant and obtain a fairly good match between the two methods.
Quantum correlations in composite and separable quantum systems are characterized by non-vanishing quantum discord. We demonstrate the necessary and sufficient conditions for existence of hermitian witness operators for quantum discord, bypassing the problem of non-convexity of the set of all classical states. We propose an example of a linear witness which can detect quantum correlations in large classes of separable bipartite systems. Decomposition of the witness in terms of a small number of locally measurable operators as shown here, paves the way for experimental detection of quantum correlations beyond entanglement.
We discuss the behavior of quantum and classical pairwise correlations in critical systems, with the quantumness of the correlations measured by the quantum discord. We analytically derive these correlations for general real density matrices displaying Z2{Z}_{2}Z2​ symmetry. As an illustration, we analyze both the XXZXXZXXZ and the transverse field Ising models. Finite size as well as infinite chains are investigated and the quantum criticality is discussed. Moreover, we identify the spin functions that govern the correlations. As a further example, we also consider correlations in the Hartree-Fock ground state of the Lipkin-Meshkov-Glick model. It is then shown that both classical correlation and quantum discord exhibit signatures of the quantum phase transitions.
We provide insight into the quantum correlations structure present in strongly correlated systems beyond the standard framework of bipartite entanglement. To this aim we first exploit rotationally invariant states as a test bed to detect genuine tripartite entanglement beyond the nearest neighbor in spin-1/2 models. Then we construct in a closed analytical form a family of entanglement witnesses which provides a sufficient condition to determine if a state of a many-body system formed by an arbitrary number of spin-1/2 particles possesses genuine tripartite entanglement, independently of the details of the model. We illustrate our method by analyzing in detail the anisotropic XXZXXZXXZ spin chain close to its phase transitions, where we demonstrate the presence of long-range multipartite entanglement near the critical point and the breaking of the symmetries associated with the quantum phase transition.