The entanglement between noncomplementary blocks of a many-body system, where a part of the system forms an ignored environment, is a largely untouched problem without analytic results. We rectify this gap by studying the logarithmic negativity between two macroscopic sets of spins in an arbitrary tripartition of a collection of mutually interacting spins described by the Lipkin-Meshkov-Glick Hamiltonian. This entanglement measure is found to be finite and universal at the critical point for any tripartition whereas it diverges for a bipartition. In this limiting case, we show that it behaves as the entanglement entropy, suggesting a deep relation between the scaling exponents of these two independently defined quantities which may be valid for other systems.
The entanglement can be localized between two noncomplementary parts of a many-body system by performing measurements on the rest of the system. This localized entanglement (LE) depends on the chosen basis set of measurement (BSM). We derive here a generic optimality condition for the LE, which, besides being helpful in studying tripartite systems in pure states, can also be of use in studying mixed states of general bipartite systems. We further discuss a canonical way of localizing entanglement, where the BSM is not chosen arbitrarily, but is fully determined by the properties of the system. The LE obtained in this way, we call the localized entanglement by canonical measurement (LECM), is not only operationally meaningful and easy to calculate in practice (without needing any demanding optimization procedure), it provides a nice way to define the entanglement length in many-body systems. For spin-1/2 systems, the LECM is shown to be optimal in some important cases. At the end, some numerical results are presented for
j1−j2 spin model to demonstrate how the LECM behaves.
We study the feasibility of extracting a pure entangled state of noncomplementary and potentially well-separated regions of a quantum many-body system. It is shown that this can indeed be accomplished in nonequilibrium scenarios as well as the ground state of the considered spin chain models when one locally measures observables such as the magnetization of separated blocks of spins. A general procedure which can search for the optimal way to extract a pure entangled state through local projections is presented. Our results indicate a connection of the projective extraction of entanglement to good quantum numbers of the underlying Hamiltonian.