We define a new measure of quantum correlations in bipartite quantum systems given by the Bures distance of the system state to the set of classical states with respect to one subsystem, that is, to the states with zero quantum discord. Our measure is a geometrical version of the quantum discord. As the latter it quantifies the degree of non-classicality in the system. For pure states it is identical to the geometric measure of entanglement. We show that for mixed states it coincides with the optimal success probability of an ambiguous quantum state discrimination task. Moreover, the closest zero-discord states to a state ρ are obtained in terms of the corresponding optimal measurements.

A survey of various concepts in quantum information is given, with a main emphasis on the distinguishability of quantum states and quantum correlations. Covered topics include generalized and least square measurements, state discrimination, quantum relative entropies, the Bures distance on the set of quantum states, the quantum Fisher information, the quantum Chernoff bound, bipartite entanglement, the quantum discord, and geometrical measures of quantum correlations. The article is intended both for physicists interested not only by collections of results but also by the mathematical methods justifying them, and for mathematicians looking for an up-to-date introductory course on these subjects, which are mainly developed in the physics literature.Comment: Review article, 103 pages, to appear in J. Math. Phys. 55 (special issue: non-equilibrium statistical mechanics, 2014

We consider a random walk on the support of an ergodic stationary simple point process on R d , d ≥ 2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.

Let us note that Axioms (iii) and (iv) imply that, when n A ≤ n B , D is maximum on maximally entangled pure states, i.e., if ̺ is a maximally entangled pure state then D(̺) = D max [16]. This follows from the facts that a function D on S AB satisfying (iii) is maximal on pure states if n A ≤ n B [33] and that any pure state can be obtained from a maximally entangled pure state via a LOCC [15]. Thus, if Axioms (i-iv) are satisfied, the additional requirement in Axiom (v) is essentially that D(̺) = D max holds only for the maximally entangled states ̺.It has been shown in previous works [11,15] that the geometric discord D G Bu and discord of response D R Bu satisfy Axioms (i)-(iv) for the Bures distance, and hence are bona fide measures of quantum correlations. In this paper, we will prove that this is also the case for the three measures D G He , D M He , and D R He based on the Hellinger distance, as well as for the Bures measurement-induced geometric discord D M Bu and trace discord of response D R Tr . In contrast, it is known that D G HS = D M HS and D R HS do not fulfill Axiom (iii) because of the lack of monotonicity of the Hilbert-Schmidt distance under CPTP maps (an explicit counter-example is given in Ref. [34] for D G HS and applies to D R HS as well, see below). Therefore, the use of the Hilbert-Schmidt distance in the definitions of Eqs. (5)-(7) can and does lead to unphysical predictions. Considering the distances d p associated to the p-norms X p ≡ (Tr |X| p ) 1/p , one has that for p > 1, d p is not contractive under CPTP maps [35] (see also Ref.[36] for a counter-example for p = 2, which also holds for any p > 1). This is why the distances d p cannot be used to define measures of quantumness apart from the case p = 1, corresponding to the contractive trace distance, while for p = 2 the non-contractive Hilbert-Schmidt distance is well tractable and thus used to establish bounds on the bona fide geometric measures.Regarding our last Axiom (v), the only result established so far in the literature concerns the Bures geometric discord [9,15]. We will demonstrate below that all the other measures based on the trace, Bures, and Hellinger distances also satisfy this axiom. Our proofs are valid for arbitrary (finite) space dimensions n A and n B of subsystems A and B, excepted for D G He , for which they are restricted to the special cases n A = 2, 3, and for D M He , D G Tr , and D M Tr , for which they are restricted to n A = 2.The paper is organized as follows. Given its length and the wealth of mathematical relations and bounds that we have determined, we begin by summarizing the main results in Section II. We first give general expressions of the geometric measures for the Bures and Hellinger distances, which are convenient starting points to compare them (see Sec. II A). We then report in some synoptic Tables the various relations and bounds satisfied by D G , D M , and D R for the trace, Hilbert-Schmidt, Bures, and Hellinger distances (see Sec. II B). Closed expressions for the Hellinger geometric ...

The minimal Bures distance of a quantum state of a bipartite system AB to the set of classical states for subsystem A defines a geometric measure of quantum discord. When A is a qubit, we show that this geometric quantum discord is given in terms of the eigenvalues of a (2nB ) × (2nB ) hermitian matrix, nB being the Hilbert space dimension of the other subsystem B. As a first application, we calculate the geometric discord for the output state of the DQC1 algorithm. We find that it takes its highest value when the unitary matrix from which the algorithm computes the trace has its eigenvalues uniformly distributed on the unit circle modulo a symmetry with respect to the origin. As a second application, we derive an explicit formula for the geometric discord of a two-qubit state ρ with maximally mixed marginals and compare it with other measures of quantum correlations. We also determine the closest classical states to ρ.

The spectral fluctuations of a quantum Hamiltonian system with time-reversal symmetry are studied in the semiclassical limit by using periodic-orbit theory. It is found that, if long periodic orbits are hyperbolic and uniformly distributed in phase space, the spectral form factor K(τ ) agrees with the GOE prediction of random-matrix theory up to second order included in the time τ measured in units of the Heisenberg time (leading off-diagonal approximation). Our approach is based on the mechanism of periodic-orbit correlations discovered recently by Sieber and Richter [1]. By reformulating the theory of these authors in phase space, their result on the free motion on a Riemann surface with constant negative curvature is extended to general Hamiltonian hyperbolic systems with two degrees of freedom.PACS numbers: 05.45.Mt, 03.65.Sq Hyperbolic Hamiltonian systemsWe consider a particle moving in a Euclidean plane (f = two degrees of freedom), with Hamiltonian H(q, p) = H(q, −p) invariant under time-reversal symmetry. We assume the existence of a compact two-dimensional Poincaré surface of section Σ in the (four-dimensional) phase space Γ, contained in an energy shell H(q, p) = E and invariant under time reversal (TR) [4,12]. Every classical orbit of energy E intersects Σ transversally. The classical dynamics can then be described by an area-preserving map φ on Σ, together with a first-return time map x ∈ Σ → t x ∈ [0, ∞] (see [12]). In what follows, letters in normal and bold fonts are assigned

In a two-mode Bose Josephson junction the dynamics induced by a sudden quench of the tunnel amplitude leads to the periodic formation of entangled states. For instance, squeezed states are formed at short times and macroscopic superpositions of phase states at later times. The two modes of the junction can be viewed as the two arms of an interferometer; use of entangled states allows to perform atom interferometry beyond the classical limit. Decoherence due to the presence of noise degrades the quantum correlations between the atoms, thus reducing phase sensitivity of the interferometer. We consider the noise induced by stochastic fluctuations of the energies of the two modes of the junction. We analyze its effect on squeezed states and macroscopic superpositions and study quantitatively the amount of quantum correlations which can be used to enhance the phase sensitivity with respect to the classical limit. To this aim we compute the squeezing parameter and the quantum Fisher information during the quenched dynamics. For moderate noise intensities we show that these useful quantum correlations increase on time scales beyond the squeezing regime. This suggests multicomponent superpositions as interesting candidates for high-precision atom interferometry.

Squeezed states and macroscopic superpositions of coherent states have been predicted to be generated dynamically in Bose Josephson junctions. We solve exactly the quantum dynamics of such a junction in the presence of a classical noise coupled to the population-imbalance number operator (phase noise), accounting for e.g. the experimentally relevant fluctuations of the magnetic field. We calculate the correction to the decay of the visibility induced by the noise in the non-markovian regime. Furthermore, we predict that such a noise induces an anomalous rate of decoherence among the components of the macroscopic superpositions, which is independent on the total number of atoms, leading to potential interferometric applications.