We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons, and fermions. In each case properties of an entanglement measure are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entanglement measure reduces to the well-known many-particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a nontrivial lower bound of thus obtained generalized concurrence.
Via a multidimensional complementarity relation we derive an operational entanglement measure for any discrete quantum system, i.e., for any multidimensional and multipartite system. This measure admits a separation into different classes of entanglement obtained by using a flip operator
2,3,…,n times, defining a
m-flip concurrence. This operator sum has the practical feature to allow one to calculate for mixed states bounds on this
m-flip concurrence. Moreover, the information content of an
n-partite multidimensional system admits a simple and intuitive interpretation in terms of single particle obtainable information, entanglement, and information due to lack of classical knowledge of the quantum state under investigation. Explicitly, the three qubit system is analyzed and, e.g., the physical difference in entanglement of the W state, the Greenberger-Horne-Zeilinger (GHZ) state, or a biseparable state is revealed.
We investigate parametrized multipartite entanglement measures from the perspective of
k nonseparability in this paper. We present two types of entanglement measures in
n-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all
k-nonseparable states in arbitrary
n-partite systems. Rigorous proofs show that the proposed
k-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all
k-separable states, and being strictly greater than zero for all
k-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition
(k=2). The lower bounds of the two classes
k-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.
We present two sets of computable entanglement measures for multipartite systems where each subsystem can have different degrees of freedom (so-called qudits). One set, called ``separability'' measure, reveals which of the subsystems are separable or entangled. For that we have to extend the concept of
k separability for multipartite systems to a unambiguous separability concept which we call ${\ensuremath{\gamma}}_{k}$ separability. The second set of entanglement measures reveals the ``kind'' of entanglement, i.e., if it is bipartite, tripartite, \dots{},
n-partite entangled and is denoted as the ``physical'' measure. We show how lower bounds on both sets of measures can be obtained by the observation that any entropy may be rewritten via operational expressions known as
m concurrences. Moreover, for different classes of bipartite or multipartite qudit systems we compute the bounds explicitly and discover that they are often tight or equivalent to positive partial transposition.