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.
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.
The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one cannot expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher-order structure characterizing that is needed. In this paper we make some steps in this direction. First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. K\"ok\'enyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It also turns out that, physically, this structure is related to the local operations and classical communication convertibility: If a state from a class can be mapped into another one, then that class can be found higher in the hierarchy. Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some ``unified'' notion of entanglement.
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.
The main concern of this paper is how to define proper measures of multipartite entanglement for mixed quantum states. Since the structure of partial separability and multipartite entanglement is getting complicated if the number of subsystems exceeds two, one can not expect the existence of an ultimate scalar entanglement measure, which grasps even a small part of the rich hierarchical structure of multipartite entanglement, and some higher order structure characterizing that is needed. In this paper we make some steps towards this direction.
First, we reveal the lattice-theoretic structure of the partial separability classification, introduced earlier [Sz. Szalay and Z. Kokenyesi, Phys. Rev. A 86, 032341 (2012)]. It turns out that, mathematically, the structure of the entanglement classes is the up-set lattice of the structure of the different kinds of partial separability, which is the down-set lattice of the lattice of the partitions of the subsystems. It turns also out that, physically, this structure is related to the LOCC convertibility: if a class can completely be mapped into another one by the use of LOCC, then it can be found higher in the hierarchy.
Second, we introduce the notion of multipartite monotonicity, expressing that a given set of entanglement monotones, while measuring the different kinds of entanglement, shows also the same hierarchical structure as the entanglement classes. Then we construct such hierarchies of entanglement measures, and propose a physically well-motivated one, being the direct multipartite generalization of the entanglement of formation based on the entanglement entropy, motivated by the notion of statistical distinguishability. The multipartite monotonicity shown by this set of measures motivates us to consider the measures to be the different manifestations of some unified notion of entanglement.
Entanglement concurrence is an important bipartite entanglement measure that has found wide applications in quantum technologies. In this work, inspired by unified entropy, we introduce a two-parameter family of entanglement measures, referred to as the unified (q,s)-concurrence. Both the standard entanglement concurrence and the recently proposed q-concurrence emerge as special cases within this family. By combining the positive partial transposition and realignment criteria, we derive an analytical lower bound for this measure for arbitrary bipartite mixed states, revealing a connection to strong separability criteria. Explicit expressions are obtained for the unified (q,s)-concurrence in the cases of isotropic and Werner states under the constraint q>1 and qs≥1. Furthermore, we explore the monogamy properties of the unified (q,s)-concurrence for q≥2, 0≤s≤1 and 1≤qs≤3, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified (q,s)-concurrence with q≥1 and qs≥1, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
In this paper, we present a measure of multipartite entanglement (k-nonseparable), k-ME concurrence ${C}_{k\ensuremath{-}\mathrm{ME}}(\ensuremath{\rho})$, that unambiguously detects all k-nonseparable states in arbitrary dimensions, where the special case 2-ME concurrence ${C}_{2\ensuremath{-}\mathrm{ME}}(\ensuremath{\rho})$ is a measure of genuine multipartite entanglement. The measure k-ME concurrence satisfies important characteristics of an entanglement measure, including the entanglement monotone, vanishing on k-separable states, convexity, subadditivity, and being strictly greater than zero for all k-nonseparable states. Two powerful lower bounds on this measure are given. These lower bounds are experimentally implementable without quantum state tomography and are easily computable as no optimization or eigenvalue evaluation is needed. We illustrate detailed examples in which the given bounds perform better than other known detection criteria.
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.