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This article contains a survey of the geometric approach to quantum correlations. We focus mainly on the geometric measures of quantum correlations based on the Bures and quantum Hellinger distances., Comment: to be published as a chapter of the book "Lectures on general quantum correlations and their applications" edited by F. Fanchini, D. Soares-Pinto, and G. Adesso (Springer, 2017); 47 pages, 3 figures; second version: minor typos in the first version corrected

We analyse the nature of spontaneous symmetry breaking in complex quantum systems by investigating the long-standing conjecture that the maximally symmetry-breaking quantum ground states are the most classical ones corresponding to a globally ordered phase. We make this argument quantitatively precise by comparing different local and global indicators of classicality and quantumness, respectively in symmetry-breaking and symmetry-preserving quantum ground states. We first discuss how naively comparing local, pairwise entanglement and discord apparently leads to the opposite conclusion. Indeed, we show that in symmetry-preserving ground states the two-body entanglement captures only a modest portion of the total two-body quantum correlations, while, on the contrary, in maximally symmetry-breaking ground states it contributes the largest amount to the total two-body quantum correlations. We then put to test the conjecture by looking at the global, macroscopic correlation properties of quantum ground states. We prove that the ground states which realize the maximum breaking of the Hamiltonian symmetries, associated to a globally ordered phase, are the only ones that: I) are always locally convertible, i.e. can be obtained from all other ground states by only applying LOCC transformations (local operations and classical communication), while the reverse is never possible; II) minimize the monogamy inequality on the globally shared, macroscopic bipartite entanglement., Comment: 9 pagese, 6 figures. The present submission updates and supersedes our previous preprints arXiv:1408.1412 and arXiv:1412.1054

Local unitary operations allow for a unifying approach to the quantification of quantum correlations among the constituents of a bipartite quantum system. For pure states, the distance between a given state and its image under least-perturbing local unitary operations is a bona fide measure of quantum entanglement, the so-called entanglement of response, which can be extended to mixed states via the convex roof construction. On the other hand, when defined directly on mixed states perturbed by local unitary operations, such a distance turns out to be a bona fide measure of quantum correlations, the so-called discord of response. Exploiting this unified framework, we perform a detailed comparison between two-body entanglement and two-body quantum discord in infinite XY quantum spin chains both in symmetry-preserving and symmetry-breaking ground states as well as in thermal states at finite temperature. The results of the investigation show that in symmetry-preserving ground states the two-point quantum discord dominates over the two-point entanglement, while in symmetrybreaking ground states the two-point quantum discord is strongly suppressed and the two-point entanglement is essentially unchanged. In thermal states, for certain regimes of Hamiltonian parameters, we show that the pairwise quantum discord and the pairwise entanglement can increase with increasing thermal fluctuations., Comment: 10 pages, 12 figures. Submitted to Phys. Rev. A

The presence of quantum correlations in a quantum state is related to the state response to local unitary perturbations. Such response is quantified by the distance between the unperturbed and perturbed states, minimized with respect to suitably identified sets of local unitary operations. In order to be a bona fide measure of quantum correlations, the distance function must be chosen among those that are contractive under completely positive and trace preserving maps. The most relevant instances of such physically well behaved metrics include the trace, the Bures, and the Hellinger distance. To each of these metrics one can associate the corresponding discord of response, namely the trace, or Hellinger, or Bures minimum distance from the set of unitarily perturbed states. All these three discords of response satisfy the basic axioms for a proper measure of quantum correlations. In the present work we focus in particular on the Bures distance, which enjoys the unique property of being both Riemannian and contractive under completely positive and trace preserving maps, and admits important operational interpretations in terms of state distinguishability. We compute analytically the Bures discord of response for two-qubit states with maximally mixed marginals and we compare it with the corresponding Bures geometric discord, namely the geometric measure of quantum correlations defined as the Bures distance from the set of classically correlated quantum states. Finally, we investigate and identify the maximally quantum correlated two-qubit states according to the Bures discord of response. These states exhibit a remarkable nonlinear dependence on the global state purity., Comment: 10 pages, 2 figures. Improved and expanded version, to be published in J. Phys. A: Math. Gen

We show that only those composite quantum systems possessing nonvanishing quantum correlations have the property that any nontrivial local unitary evolution changes their global state. We derive the exact relation between the global state change induced by local unitary evolutions and the amount of quantum correlations. We prove that the minimal change coincides with the geometric measure of discord (defined via the Hilbert- Schmidt norm), thus providing the latter with an operational interpretation in terms of the capability of a local unitary dynamics to modify a global state. We establish that two-qubit Werner states are maximally quantum correlated, and are thus the ones that maximize this type of global quantum effect. Finally, we show that similar results hold when replacing the Hilbert-Schmidt norm with the trace norm., Comment: 5 pages, 1 figure. To appear in Physical Review A