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arXiv:2101.04256 · Physical Review A

Parametrized entanglement monotone

Xue Yang, Ming-Xing Luo, Yan-Han Yang, Shao-Ming Fei

Published May 20, 2021• 35 citations
arXivHTMLPDF

Abstract

Entanglement concurrence has been widely used for featuring entanglement in quantum experiments. As an entanglement monotone it is related to specific quantum Tsallis entropy. Our goal in this paper is to propose a parametrized bipartite entanglement monotone which is named as $q$-concurrence inspired by general Tsallis entropy. We derive an analytical lower bound for the $q$-concurrence of any bipartite quantum entanglement state by employing positive partial transposition criterion and realignment criterion, which shows an interesting relationship to the strong separability criteria. The parametrized entanglement monotone is used to characterize bipartite isotropic states. Finally, we provide a computational method to estimate the $q$-concurrence for any entanglement by superposing two bipartite pure states. It shows that the superposition operations can at most increase one ebit for the $q$-concurrence in the case that the two states being superposed are biorthogonal or one-sided orthogonal. These results reveal a series of phenomena about the entanglement, which may be interesting in quantum communication and quantum information processing.

DOI
10.1103/physreva.103.052423

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Entanglement concurrence has been widely used for featuring entanglement in quantum experiments. As an entanglement monotone it is related to specific quantum Tsallis entropy. Our goal in this paper is to propose a parametrized bipartite entanglement monotone which is named as qqq-concurrence inspired by general Tsallis entropy. We derive an analytical lower bound for the qqq-concurrence of any bipartite quantum entanglement state by employing positive partial transposition criterion and realignment criterion, which shows an interesting relationship to the strong separability criteria. The parametrized entanglement monotone is used to characterize bipartite isotropic states. Finally, we provide a computational method to estimate the qqq-concurrence for any entanglement by superposing two bipartite pure states. It shows that the superposition operations can at most increase one ebit for the qqq-concurrence in the case that the two states being superposed are biorthogonal or one-sided orthogonal. These results reveal a series of phenomena about the entanglement, which may be interesting in quantum communication and quantum information processing.
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.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)(q,s)(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ)E_{q,s}^{A|B}(ρ)Eq,sA∣B​(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) for quantifying kkk-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters qqq and sss are in certain ranges, Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) with respect to the parameters qqq and sss for fixed quantum states. In addition, for fixed qqq and sss, we also give their order relations for any two pure states.
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.
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.