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Register a new userOn the geometric measure of entanglement for pure states
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The geometric measure of entanglement is the distance or angle between an entangled target state and the nearest unentangled state. Often one considers the geometric measure of entanglement for highly symmetric entangled states because it simplifies the calculations and allows for analytic solutions. Although some symmetry is required in order to deal with large numbers of qubits, we are able to loosen significantly the restrictions on the highly symmetric states considered previously, and consider several generalizations of the coefficients of both target and unentangled states. This allows us to compute the geometric entanglement measure for larger and more relevant classes of states.
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As two of the most important entanglement measures--the entanglement of formation and the entanglement of distillation--have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures--the relative entropy of entanglement and the geometric measure of entanglement--are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
We define a negative entanglement measure for separable states which shows that how much entanglement one should compensate the unentangled state at least for changing it into an entangled state. For two-qubit systems and some special classes of states in higher-dimensional systems, the explicit formula and the lower bounds for the negative entanglement measure have been presented, and it always vanishes for bipartite separable pure states. The negative entanglement measure can be used as a useful quantity to describe the entanglement dynamics and the quantum phase transition. In the transverse Ising model, the first derivatives of negative entanglement measure diverge on approaching the critical value of the quantum phase transition, although these two-site reduced density matrices have no entanglement at all. In the 1D Bose-Hubbard model, the NEM as a function of $t/U$ changes from zero to negative on approaching the critical point of quantum phase transition.
Entanglement swapping has played an important role in quantum information processing, and become one of the necessary core technologies in the future quantum network. In this paper, we study entanglement swapping for multi-particle pure states and maximally entangled states in qudit systems. We generalize the entanglement swapping of two pure states from the case where each quantum system contains two particles to the case of containing any number of particles, and consider the entanglement swapping between any number of systems. We also generalize the entanglement swapping chain of bipartite pure states to the one of multi-particle pure states. In addition, we consider the entanglement swapping chains for maximally entangled states.
We show that not all 4-party pure states are GHZ reducible (i.e., can be generated reversibly from a combination of 2-, 3- and 4-party maximally entangled states by local quantum operations and classical communication asymptotically) through an example, we also present some properties of the relative entropy of entanglement for those 3-party pure states that are GHZ reducible, and then we relate these properties to the additivity of the relative entropy of entanglement.
We show that genuine multipartite entanglement of all multipartite pure states in arbitrary finite dimension can be detected in a device-independent way by employing bipartite Bell inequalities on states that are deterministically generated from the initial state via local operations. This leads to an efficient scheme for large classes of multipartite states that are relevant in quantum computation or condensed-matter physics, including cluster states and the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model. For cluster states the detection of genuine multipartite entanglement involves only measurements on a constant number of systems with an overhead that scales linear with the system size, while for the AKLT model the overhead is polynomial. In all cases our approach shows robustness against experimental imperfections.
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