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arXiv:2206.07913 · Journal of Physics A Mathematical and Theoretical

Parameterized bipartite entanglement measure

Zhi-Wei Wei, Shao-Ming Fei

Published Jun 15, 2022• 14 citations
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Abstract

Abstract We propose a novel parameterized entanglement measure α -concurrence for bipartite systems. By employing positive partial transposition and realignment criteria, we derive analytical lower bounds for the α -concurrence. Moreover, we calculate explicitly the analytic expressions of the α -concurrence for isotropic states and Werner states.

DOI
10.1088/1751-8121/ac7592

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Abstract We propose a novel parameterized entanglement measure α -concurrence for bipartite systems. By employing positive partial transposition and realignment criteria, we derive analytical lower bounds for the α -concurrence. Moreover, we calculate explicitly the analytic expressions of the α -concurrence for isotropic states and Werner states.
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)(q,s)(q,s)-concurrence. Both the standard entanglement concurrence and the recently proposed qqq-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)(q,s)(q,s)-concurrence in the cases of isotropic and Werner states under the constraint q>1q>1q>1 and qs≥1qs\geq 1qs≥1. Furthermore, we explore the monogamy properties of the unified (q,s)(q,s)(q,s)-concurrence for q≥2q\geq 2q≥2, 0≤s≤10\leq s\leq 10≤s≤1 and 1≤qs≤31\leq qs\leq 31≤qs≤3, in qubit systems. In addition, we derive an entanglement polygon inequality for the unified (q,s)(q,s)(q,s)-concurrence with q≥1q\geq 1q≥1 and qs≥1qs\geq 1qs≥1, which manifests the relationship among all the marginal entanglements in any multipartite qudit system.
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., qqq-concurrence (q>1q>1q>1) and ααα-concurrence (0<α<10<α<10<α<1), or termed the parameterized entanglement monotone together. We obtain, in the light of the informationally complete (NNN, MMM)-positive operator-valued measure [(NNN, MMM)-POVM], the lower bounds for the case of 12<α<1\frac12<α<121​<α<1, 1<q<21<q<21<q<2 for two-qudit states, and the case of 2⩽q<32\leqslant q<32⩽q<3 for two-qubit states. We list several examples which show that the lower bounds based on (NNN, MMM)-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 12<α<1\frac12<α<121​<α<1 and 1<q<21<q<21<q<2 for the isotropic state.
Quantum entanglement is a crucial resource in quantum information processing, advancing quantum technologies. The greater the uncertainty in subsystems' pure states, the stronger the quantum entanglement between them. From the dual form of qqq-concurrence (q≥2q\geq 2q≥2) we introduce the total concurrence. A bona fide measure of quantum entanglement is introduced, the Cqt\mathcal{C}^{t}_qCqt​-concurrence (q≥2q \geq 2q≥2), which is based on the total concurrence. Analytical lower bounds for the Cqt\mathcal{C}^{t}_qCqt​-concurrence are derived. In addition, an analytical expression is derived for the Cqt\mathcal{C}^{t}_qCqt​-concurrence in the cases of isotropic and Werner states. Furthermore, the monogamy relations that the Cqt\mathcal{C}^{t}_qCqt​-concurrence satisfies for qubit systems are examined. Additionally, based on the parameterized ααα-concurrence and its complementary dual, the Cαt\mathcal{C}^{t}_αCαt​-concurrence (0≤α≤12)(0\leqα\leq\frac{1}{2})(0≤α≤21​) is also proposed.
Based on the realignment moments of density matrix, we study parameterized entanglement criteria for bipartite and multipartite states. By adjusting the different parameter values, our criterion can detect not only bound entangled states, but also non-positive partial transpose entangled states for bipartite quantum systems. Moreover, we propose the definition of multipartite realignment moments and generalize the result of bipartite systems to obtain a sufficient criterion to detect entanglement for multipartite quantum states in arbitrary dimensions. And we further improve the conclusion to obtain another new entanglement criterion. The new method can detect more entangled states than previous methods as backed by detailed examples.
Employing a recently proposed separability criterion we develop analytical lower bounds for the concurrence and for the entanglement of formation of bipartite quantum systems. The separability criterion is based on a nondecomposable positive map which operates on state spaces with even dimension N >= 4, and leads to a class of nondecomposable optimal entanglement witnesses. It is shown that the bounds derived here complement and improve the existing bounds obtained from the criterion of positive partial transposition and from the realignment criterion.