Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow us to recognize novel families of multiparticle bound entangled states.
Although numerous measures of entanglement have been proposed so far, the calculation of a given faithful entanglement measure is a hard work since it is always involved in some optimization process. It is, therefore, important to estimate the lower bound of a given entanglement measure for an arbitrary quantum state. This results in a subject of intensive mathematical research. In particular, along this line, the lower bounds of concurrence or other measures that are induced from concurrence have been explored a lot. Here, we investigate the lower bounds of two kinds of entanglement monotones, i.e.,
q-concurrence (
q>1) and
α-concurrence (
0<α<1), or termed the parameterized entanglement monotone together. We obtain, in the light of the informationally complete (
N,
M)-positive operator-valued measure [(
N,
M)-POVM], the lower bounds for the case of
21<α<1,
1<q<2 for two-qudit states, and the case of
2⩽q<3 for two-qubit states. We list several examples which show that the lower bounds based on (
N,
M)-POVM outperform that of GSIC-POVM and SIC-POVM, and all these measurement based bounds are better then the ones induced by positive partial transpose (PPT) and realignment criteria in literature. In addition, we obtain an analytical formula of the parameterized entanglement monotone with
21<α<1 and
1<q<2 for the isotropic state.
We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite mixed states. We show that this assisted entanglement, when measured in terms of the generalized concurrence (called the
G concurrence), is (tightly) bounded by an entanglement monotone, which we call the
G concurrence of assistance. The
G concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the geometric mean of the Schmidt numbers. For a large (nontrivial) class of $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional mixed states, we are able to generalize Wootters' formula for the concurrence into lower and upper bounds on the
G concurrence. Moreover, we have found an explicit formula for the
G concurrence of assistance that generalizes the expression for the concurrence of assistance for a large class of $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite pure states.
In this Brief Report, we prove that an analytically solvable two-qubit entanglement monotone introduced by Zhang et al. [Phys. Rev. A 76, 032306 (2007)] is no greater than the square of concurrence. This proof is based on the equivalence between Zhang et al.'s monotone and an observable entanglement measure for mixed states suggested by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)]. Our result may clarify the connections between several recently defined entanglement measures and help understand their possible applications in quantum-information theory.
Covariance matrices are a useful tool for investigating correlations and entanglement in quantum systems. They are widely used in continuous-variable systems, but recently also for finite-dimensional systems powerful entanglement criteria in terms of covariance matrices have been derived. We show how these results can be used for the quantification of entanglement in bipartite systems. To that aim we introduce an entanglement parameter that quantifies the violation of the covariance matrix criterion and can be used to give lower bounds on the concurrence. These lower bounds are easily computable and give entanglement estimates for many weakly entangled states.
We extend the definition of concurrence into a family of entanglement monotones, which we call concurrence monotones. We discuss their properties and advantages as computational manageable measures of entanglement, and show that for pure bipartite states all measures of entanglement can be written as functions of the concurrence monotones. We then show that the concurrence monotones provide bounds on quantum information tasks. As an example, we discuss their applications to remote entanglement distributions (RED) such as entanglement swapping and remote preparation of bipartite entangled states (RPBES). We prove a powerful theorem which states what kind of (possibly mixed) bipartite states or distributions of bipartite states cannot be remotely prepared. The theorem establishes an upper bound on the amount of
G-concurrence (one member in the concurrence family) that can be created between two single-qudit nodes of quantum networks by means of tripartite RED. For pure bipartite states the bound on the
G-concurrence can always be saturated by RPBES.
We derive a family of entanglement monotones, one member of which turns out to be the negativity. Two others are shown to be lower bounds on the I-concurrence, and on the I-tangle, respectively [P. Rungta and C. M. Caves, to appear in Phys. Rev. A]. We compare these bounds with the I-concurrence and I-tangle on the isotropic states, and on rank-two density operators resulting from a Tavis-Cummings interaction. Our results provide a global structure relating several different entanglement measures. Additionally, they possess analytic forms which are easily evaluated in the most general cases.