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arXiv:0911.5174

Unified (r,s)-relative entropy

Jiamei Wang, Junde Wu

Published Nov 26, 2009• 0 citations
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Abstract

In this paper, we introduce and study unified $(r,s)$-relative entropy and quantum unified $(r,s)$-relative entropy, in particular, our main results of quantum unified $(r,s)$-relative entropy are established on the separable complex Hilbert spaces. Moreover, the entanglement-measure of states due to the quantum unified $(r,s)$-relative entropy is considered, too. Our results improved a uncorrect statement on the monotone property of entanglement-measure function.

DOI
10.48550/arxiv.0911.5174

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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.
The unified entropy as a promotion of the von Neumann entropy exhibits distinct diversity which contains the Tsallis entropy, the Rényi entropy, the von Neumann entropy as special cases. The unified-(r,tr,tr,t) entropy entanglement with 0<r<10<r<10<r<1 and 0<t≤10< t\leq 10<t≤1 was shown to be an entanglement monotone in literature. In this paper, we explore unified-(q,sq,sq,s) entropy entanglement with q>1q>1q>1 and qs≥1qs\geq1qs≥1 and show that it is also an entanglement monotone and that both of them are monogamous. Going further, we present two kinds of global multipartite entanglement measures (GlMEMs) based on the unified entropy and each kind has two subclasses which are classified by the parameters (q,s)(q,s)(q,s) and (r,t)(r,t)(r,t). Consequently, from the view of the complete multipartite entanglement measure theory, we show that one of them is a complete multipartite entanglement monotone and is not only completely monogamous but also tightly completely monogamous, but the other three are even not complete. We also explore the genuine entanglement measures induced by the unified entropy and the relations with the bipartite entanglement and the global entanglement are discussed, respectively.
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