Structures in Quantum Systems

The localizable entanglement (LE) is the maximal amount of entanglement that can be concentrated, on average, onto part of a multipartite system via local projective measurements on its complement. LE has found many applications in the study of quantum phase transitions and entanglement in spin systems. Traditionally, the LE has been defined with respect to bipartite entanglement measures, such as the entanglement entropy or the two-qubit concurrence. More recent work has analyzed specific quantum states using the LE defined by multipartite entanglement measures. Motivated by a GHZ state extraction problem, we study the LE given by multipartite entanglement measures known as the n-tangle, the GME-concurrence, and the concentratable entanglement (CE). For generic states, we derive a set of easily computable and experimentally accessible bounds on the LE given by these measures in terms of either spin correlation functions or functions of marginal states. We also derive a set of uniform continuity bounds on the LE. Using these results, we study concentration phenomena of the LE in high-dimensional Hilbert spaces. To illustrate the utility of our bounds, we use them to analyze allowable transformations on graph states. We also supplement our analytical results with numerics for random states to investigate the tightness of our bounds. Finally, we numerically study how the LE reveals critical behavior in spin-half systems by computing the LE and its bounds for ground states of the transverse-field Ising model.