The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings (see Shimony [1] and Barnum and Linden [2]), is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary twoqubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of threequbit GHZ, W and inverted-W states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.

Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy.

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Spin-triplet superfluids can support exotic objects, such as half-quantum vortices characterized by the nontrivial winding of the spin structure. We present cantilever magnetometry measurements performed on mesoscopic samples of Sr(2)RuO(4), a spin-triplet superconductor. With micrometer-sized annular-shaped samples, we observed transitions between integer fluxoid states as well as a regime characterized by "half-integer transitions"--steps in the magnetization with half the height of the ones we observed between integer fluxoid states. These half-height steps are consistent with the existence of half-quantum vortices in superconducting Sr(2)RuO(4).

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Phase slips are topological fluctuation events that carry the superconducting order-parameter field between distinct current carrying states 1 . Owing to these phase slips low-dimensional superconductors acquire electrical resistance 2 . In quasi-one-dimensional nanowires it is well known that at higher temperatures phase slips occur via the process of thermal barrier-crossing by the orderparameter field. At low temperatures, the general expectation is that phase slips should proceed via quantum tunnelling events, which are known as quantum phase slips (QPS). However, resistive measurements have produced evidence both pro 3-6 and con [7][8][9] and hence the precise requirements for the observation of QPS are yet to be established firmly. Here we report strong evidence for individual quantum tunnelling events undergone by the superconducting order-parameter field in homogeneous nanowires. We accomplish this via measurements of the distribution of switching currents-the high-bias currents at which superconductivity gives way to resistive behaviour-whose width exhibits a rather counter-intuitive, monotonic increase with decreasing temperature. We outline a Quantum phenomena involving systems far larger than individual atoms are one of the most exciting fields of modern physics. Initiated by Leggett more than twentyfive years ago 14,15 , the field has seen widespread development, important realizations being furnished, e. g., by macroscopic quantum tunnelling (MQT) of the phase in Josephson junctions, and of the magnetization in magnetic nanoparticles [16][17][18][19] . More recently, the breakthrough recognition of the potential advantages of quantum-based computational methods has initiated the search for viable implementations of qubits 20 , several of which are rooted in MQT in superconducting systems. In particular, it has been recently proposed that superconducting nanowires (SCNWs) could provide a valuable setting for realizing qubits 12 . In this case, the essential behaviour needed of SCNWs that they undergo QPS, i.e., topological quantum fluctuations of the superconducting order-parameter field via which tunnelling occurs between currentcarrying states. It has also been proposed that QPS in nanowires could allow one to build a current standard, and thus could play a useful role in aspects of metrology 13 .Additionally, QPS are believed to provide the pivotal processes underpinning the 3 superconductor-insulator transition observed in nanowires 21-25, Observations of QPS have been reported previously on wires having high normal resistance (i.e., R N > R Q , where R Q = h/4e 2 ≈ 6,450 Ω) via low-bias resistance (R) vs. temperature (T) measurements 3,4 . Yet, low-bias measurements on short wires with normal resistance R N < R Q have been unable to reveal QPS 7,8 . Also, it has been suggested that some results ascribed to QPS could in fact have originated in inhomogeneity of the nanowires.Thus, no consensus exists about the conditions under which QPS occur, and qualitatively new evidence for QPS remains highl...

We show that the effective spin-spin interaction between three-level atoms confined in a multimode optical cavity is long-ranged and sign changing, like the RKKY interaction; therefore, ensembles of such atoms subject to frozen-in positional randomness can realize spin systems having disordered and frustrated interactions. We argue that, whenever the atoms couple to sufficiently many cavity modes, the cavity-mediated interactions give rise to a spin glass. In addition, we show that the quantum dynamics of cavity-confined spin systems is that of a Bose-Hubbard model with strongly disordered hopping but no on-site disorder; this model exhibits a random-singlet glass phase, absent in conventional optical-lattice realizations. We briefly discuss experimental signatures of the realizable phases.

As Charles Goodyear discovered in 1839, when he first vulcanised rubber, a macromolecular liquid is transformed into a solid when a sufficient density of permanent crosslinks is introduced at random. At this continuous equilibrium phase transition, the liquid state, in which all macromolecules are delocalised, is transformed into a solid state, in which a nonzero fraction of macromolecules have spontaneously become localised. This solid state is a most unusual one: localisation occurs about mean positions that are distributed homogeneously and randomly, and to an extent that varies randomly from monomer to monomer. Thus, the solid state emerging at the vulcanisation transition is an equilibrium amorphous solid state: it is properly viewed as a solid state that bears the same relationship to the liquid and crystalline states as the spin glass state of certain magnetic systems bears to the paramagnetic and ferromagnetic states, in the sense that, like the spin glass state, it is diagnosed by a subtle order parameter.In this article we give a detailed exposition of a theoretical approach to the physical properties of systems of randomly, permanently crosslinked macromolecules. Our primary focus is on the equilibrium properties of such systems, especially in the regime of Goodyear's vulcanisation transition. This approach rests firmly on techniques from the statistical mechanics of disordered systems pioneered by Edwards and co-workers in the context of macromolecular systems, and by Edwards and Anderson in the context of magnetic systems. We begin with a review of the semi-microscopic formulation of the statistical mechanics of randomly crosslinked macromolecular systems due to Edwards and co-workers, in particular discussing the role of crosslinks as quenched random variables. Then we turn to the issue of order parameters, and review a version capable, inter alia, of diagnosing the amorphous solid state. To develop some intuition, we examine the order parameter in an idealised situation, which subsequently turns out to be surprisingly relevant. Thus, we are motivated to hypothesise an explicit form for the order parameter in the amorphous solid state that is parametrised in terms of two physical quantities: the fraction of localised monomers, and the statistical distribution of localisation lengths of localised monomers. Next, we review the symmetry properties of the system itself, the liquid state and the amorphous solid state, and discuss connections with scattering experiments. Then, we review a representation of the statistical mechanics of randomly crosslinked macromolecular systems from which the quenched disorder has been eliminated via an application of the replica technique. We transform the statistical mechanics into a field-theoretic representation, which exhibits a close connection with the order parameter, and analyse this representation at the saddle-point level. This analysis reveals that sufficient crosslinking causes an instability of the liquid state, this state giving way to the amorphous solid s...