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.
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.
Despite the great importance of quantum entanglement, a computationally practical, physically motivated, and experimentally implementable way to estimate the amount of entanglement associated with a general (mixed) state of a bipartite quantum system is still lacking. An entanglement lower bound recently advanced by Mintert and Buchleitner [Phys. Rev. Lett. 98, 140505 (2007)] was proposed as a possible tool for the aforementioned task. Mintert and Buchleitner presented evidence indicating that, due to its tightness, in some important cases the alluded bound can be used as an entanglement measure. From the physical point of view, what makes this bound particularly appealing as an entanglement measure is that it is based on an observable quantity. For this last reason, and also because of its other attractive feature, mathematical simplicity, this promising bound and its possible use as an entanglement estimator certainly deserve to be the focus of careful attention. The usefulness of the bound as an entanglement measure, however, depends on its degree of tightness. Here we perform a systematic survey of the behavior of the alluded bound in state space, in order to determine its typical properties (in particular its tightness) and thus assess its value as an entanglement estimator.
A so-called quasientanglement measure was obtained [J. Zhang et al., Phys. Rev. A 73, 022319 (2006)] by extending Jaeger's Minkowskian norm entanglement measure to general multipartite mixed states. The obtained quasientanglement measure has good properties, but it is still left open whether it is nonincreasing under general local operations and classical communications (LOCC). We show that the obtained quasientanglement measure is indeed nonincreasing on average under LOCC at least for two-qubit states. Thus, it is a good candidate for the two-qubit entanglement monotone. Further, it is shown that this entanglement measure is less than the concurrence for a large class of mixed 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 derive measurable lower bounds on concurrence of arbitrary mixed states, for both bipartite and multipartite cases. First, we construct measurable lower bonds on the purely algebraic bounds of concurrence [F. Mintert et al. (2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a lower bound of the squared concurrence, we sum over our measurable bounds to achieve a measurable lower bound on concurrence. With two typical examples, we show that our method can detect more entangled states and also can give sharper lower bonds than the similar ones.
We derive measurable lower bounds on concurrence of arbitrary mixed states, for both bipartite and multipartite cases. First, we construct measurable lower bonds on the \textit{purely algebraic} bounds of concurrence [F. Mintert \textit{et al.} (2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a lower bound of the squared concurrence, we sum over our measurable bounds to achieve a measurable lower bound on concurrence. With two typical examples, we show that our method can detect more entangled states and also can give sharper lower bonds than the similar ones.
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.