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.

Dense coding is arguably the protocol that launched the field of quantum communication 1 . Today, however, more than a decade after its initial experimental realization 2 , the channel capacity remains fundamentally limited as conceived for photons using linear elements. Bob can only send to Alice three of four potential messages owing to the impossibility of carrying out the deterministic discrimination of all four Bell states with linear optics 3,4 , reducing the attainable channel capacity from 2 to log 2 3 ≈ 1.585 bits. However, entanglement in an extra degree of freedom enables the complete and deterministic discrimination of all Bell states 5-7 . Using pairs of photons simultaneously entangled in spin and orbital angular momentum 8,9 , we demonstrate the quantum advantage of the ancillary entanglement. In particular, we describe a dense-coding experiment with the largest reported channel capacity and, to our knowledge, the first to break the conventional linear-optics threshold. Our encoding is suited for quantum communication without alignment 10 and satellite communication. The first realization of quantum dense coding was optical, using pairs of photons entangled in polarization 2 . Dense coding has since been realized in various physical systems and broadened theoretically to include high-dimension quantum states with multiparties 11 , and even coding of quantum states 12 . The protocol extension to continuous variables 13,14 has also been experimentally explored optically, using superimposed squeezed beams 15 . Other physical approaches include a simulation in nuclear magnetic resonance with temporal averaging 16 , and an implementation with atomic qubits on demand without postselection 17 . However, photons remain the optimal carriers of information given their resilience to decoherence and ease of creation and transportation.Quantum dense coding was conceived 1 such that Bob could communicate 2 bits of classical information to Alice with the transmission of a single qubit, as follows. Initially, each party holds one spin-1/2 particle of a maximally entangled pair, such as one of the four Bell states. Bob then encodes his 2-bit message by applying one of four unitary operations on his particle, which he then transmits to Alice. Finally, Alice decodes the 2-bit message by discriminating the Bell state of the pair.Alice's decoding step, deterministically resolving the four Bell states, is known as Bell-state analysis (BSA). Although in principle attainable with nonlinear interactions, such BSA with photons is very difficult to achieve with present technology, yielding extremely low efficiencies and low discrimination fidelities 18 . Therefore, current fundamental studies and technological developments demand the use of linear optics. However, for quantum communication, standard BSA with linear optics is fundamentally impossible 3,4 . At best, only two Bell states can be discriminated; for quantum communication, the other two are considered together for a three-message encoding. Consequently, the maximu...

Complete and precise characterization of a quantum dynamical process can be achieved via the method of quantum process tomography. Using a source of correlated photons, we have implemented several methods, each investigating a wide range of processes, e.g., unitary, decohering, and polarizing. One of these methods, ancilla-assisted process tomography (AAPT), makes use of an additional "ancilla system," and we have theoretically determined the conditions when AAPT is possible. Surprisingly, entanglement is not required. We present data obtained using both separable and entangled input states. The use of entanglement yields superior results, however.

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.

Universal quantum computation can be achieved by simply performing single-qubit measurements on a highly entangled resource state, such as cluster states. The family of Affleck-Kennedy-LiebTasaki (AKLT) states has recently been intensively explored and shown to provide restricted computation. Here, we show that the two-dimensional AKLT state on a honeycomb lattice is a universal resource for measurement-based quantum computation.

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...

Entanglement in quantum XY spin chains of arbitrary length is investigated via a recentlydeveloped global measure suitable for generic quantum many-body systems. The entanglement surface is determined over the phase diagram, and found to exhibit structure richer than expected. Near the critical line, the entanglement is peaked (albeit analytically), consistent with the notion that entanglement-the non-factorization of wave functions-reflects quantum correlations. Singularity does, however, accompany the critical line, as revealed by the divergence of the field-derivative of the entanglement along the line. The form of this singularity is dictated by the universality class controlling the quantum phase transition. [5,6,7,8], where it can play the role of a diagnostic of quantum correlations. Quantum phase transitions [9] are transitions between qualitatively distinct phases of quantum many-body systems, driven by quantum fluctuations. In view of the connection between entanglement and quantum correlations, one anticipates that entanglement will furnish a dramatic signature of the quantum critical point. From the viewpoint of quantum information, the more entangled a state, the more resources it is likely to possess. It is thus desirable to study and quantify the degree of entanglement near quantum phase transitions. By employing entanglement to diagnose many-body quantum states one may obtain fresh insight into the quantum many-body problem.To date, progress in quantifying entanglement has taken place primarily in the domain of bipartite systems [10]. Much of the previous work on entanglement in quantum phase transitions has been based on bipartite measures, i.e., focus has been on entanglement either between pairs of parties [5,6] or between a part and the remainder of a system [7]. For multipartite systems, however, the complete characterization of entanglement requires the consideration of multipartite entanglement, for which a consensus measure has not yet emerged.Singular and scaling behavior of entanglement near quantum critical points was discovered in important work by Osterloh and co-workers [6], who invoked Wootters' bipartite concurrence [11] as a measure of entanglement. In the present letter, we apply a recently-developed global measure that provides a holistic, rather than bipartite, characterization of the entanglement of quantum manybody systems. Our focus is on one-dimensional spin systems, specifically ones that are exactly solvable and

Is the closest product state to a symmetric entangled multiparticle state also symmetric? This question has appeared in the recent literature concerning the geometric measure of entanglement. First, we show that a positive answer can be derived from results concerning symmetric multilinear forms and homogeneous polynomials, implying that the closest product state can be chosen to be symmetric. We then prove the stronger result that the closest product state to any symmetric multiparticle quantum state is necessarily symmetric. Moreover, we discuss generalizations of our result and the case of translationally invariant states, which can occur in spin models.