We establish relations between geometric quantum discord and entanglement quantifiers obtained by means of optimal witness operators. In particular, we prove a relation between negativity and geometric discord in the Hilbert-Schmidt norm, which is slightly different from a previous conjectured one [Phys. Rev. A 84, 052110 (2011)].We also show that, redefining the geometric discord with the trace norm, better bounds can be obtained. We illustrate our results numerically.Comment: 8 pages + 3 figures. Revised version with erratum for PRA 86, 024302 (2012). Simplified proof that discord is bounded by entanglement in any nor

We address the problem of characterizing the steerability of quantum states under restrictive measurement scenarios, i.e., the problem of determining whether a quantum state can demonstrate steering when subjected to N measurements of k outcomes. We consider the cases of either general positive operator-valued measures (POVMs) or specific kinds of measurements (e.g., projective or symmetric). We propose general methods to calculate lower and upper bounds for the white-noise robustness of a d-dimensional quantum state under different measurement scenarios that are also applicable to the study of the noise robustness of the incompatibility of sets of unknown qudit measurements. We show that some mutually unbiased bases, symmetric informationally complete measurements, and other symmetric choices of measurements are not optimal for steering isotropic states and provide candidates to the most incompatible sets of measurements in each case. Finally, we provide numerical evidence that nonprojective POVMs do not improve over projective ones for this task.

We investigate the quantum phase transitions of the extended Hubbard model at half-filling with periodic boundary conditions employing the entanglement of particles, as opposed to the more traditional entanglement of modes. Our results show that the entanglement has either discontinuities or local minima at the critical points. We associate the discontinuities to first order transitions, and the minima to second order ones. Thus we show that the entanglement of particles can be used to derive the phase diagram, except for the subtle transitions between the phases SDW-BOW, and the superconductor phases TS-SS.

Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with the missing information. This is the purpose of the Maximum Entropy estimation (MaxEnt) and the Variational Quantum Tomography (VQT). Here, we propose a variant of Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We prove their equivalence in case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.PACS number(s): 03.65.Wj, 03.67.-a *

We explore procedures to detect entanglement of unknown mixed states, which can be experimentally viable. The heart of the method is a hierarchy of simple feasibility problems, which provides sufficient conditions to entanglement. Our numerical investigations indicate that the entanglement is detected with a cost which is much lower than full state tomography. The procedure is applicable to both free and bound entanglement, and involves only single copy measurements.Comment: 8 pages, 9 figures, 4 table