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arXiv:2308.16393 · Physical Review A

Parametrized multipartite entanglement measures

Hui Li, Ting Gao, Fengli Yan

Published Jan 16, 2024• 23 citations
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Abstract

We investigate parametrized multipartite entanglement measures from the perspective of $k$ nonseparability in this paper. We present two types of entanglement measures in $n$-partite systems, $q\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$ and $\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)$, which unambiguously detect all $k$-nonseparable states in arbitrary $n$-partite systems. Rigorous proofs show that the proposed $k$-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all $k$-separable states, and being strictly greater than zero for all $k$-nonseparable states. In particular, the $q\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}$ concurrence, renamed as $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}$ concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition $(k=2)$. The lower bounds of the two classes $k$-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between $q\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence $(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}$ concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the $q\text{\ensuremath{-}}\mathrm{GME}$ concurrence defined by us to other genuine entanglement measures.

DOI
10.1103/physreva.109.012213

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We investigate parametrized multipartite entanglement measures from the perspective of kkk nonseparability in this paper. We present two types of entanglement measures in nnn-partite systems, q\ensuremath-k\ensuremath-MEq\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}q\ensuremath-k\ensuremath-ME concurrence (q>1,2\ensuremath≤k\ensuremath≤n)(q>1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)(q>1,2\ensuremath≤k\ensuremath≤n) and \ensuremathα\ensuremath-k\ensuremath-ME\ensuremath{\alpha}\text{\ensuremath{-}}k\text{\ensuremath{-}}\mathrm{ME}\ensuremathα\ensuremath-k\ensuremath-ME concurrence (0\ensuremath≤\ensuremathα<1,2\ensuremath≤k\ensuremath≤n)(0\ensuremath{\le}\ensuremath{\alpha}<1,\phantom{\rule{0.28em}{0ex}}2\ensuremath{\le}k\ensuremath{\le}n)(0\ensuremath≤\ensuremathα<1,2\ensuremath≤k\ensuremath≤n), which unambiguously detect all kkk-nonseparable states in arbitrary nnn-partite systems. Rigorous proofs show that the proposed kkk-nonseparable measures satisfy all the requirements for being an entanglement measure including the entanglement monotone, strong monotone, convexity, vanishing on all kkk-separable states, and being strictly greater than zero for all kkk-nonseparable states. In particular, the q\ensuremath-2\ensuremath-MEq\text{\ensuremath{-}}\mathrm{2}\text{\ensuremath{-}}\mathrm{ME}q\ensuremath-2\ensuremath-ME concurrence and \ensuremathα\ensuremath-2\ensuremath-ME\ensuremath{\alpha}\text{\ensuremath{-}}2\text{\ensuremath{-}}\mathrm{ME}\ensuremathα\ensuremath-2\ensuremath-ME concurrence, renamed as q\ensuremath-GMEq\text{\ensuremath{-}}\mathrm{GME}q\ensuremath-GME concurrence and \ensuremathα\ensuremath-GME\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{GME}\ensuremathα\ensuremath-GME concurrence, respectively, are two kinds of genuine entanglement measures corresponding to the case where the systems are divided into bipartition (k=2)(k=2)(k=2). The lower bounds of the two classes kkk-nonseparable measures are obtained by employing the approach that takes into account the permutationally invariant part of a quantum state. Furthermore, the relations between q\ensuremath-n\ensuremath-MEq\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}q\ensuremath-n\ensuremath-ME concurrence (\ensuremathα\ensuremath-n\ensuremath-ME(\ensuremath{\alpha}\text{\ensuremath{-}}n\text{\ensuremath{-}}\mathrm{ME}(\ensuremathα\ensuremath-n\ensuremath-ME concurrence) and global negativity are established. In addition, we discuss the degree of separability and elaborate on an effective detection method with concrete examples. Moreover, we compare the q\ensuremath-GMEq\text{\ensuremath{-}}\mathrm{GME}q\ensuremath-GME concurrence defined by us to other genuine entanglement measures.
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.
Multipartite entanglement is the premier resource for quantum technologies. Yet, its exact quantification in the laboratory is notoriously challenging, typically requiring the full knowledge of high dimensional quantum states. Here, we construct observable bounds to multipartite entanglement for systems of arbitrary size, which are functions of the local and global state purities, and correlation functions. First, we derive experimentally accessible upper and lower limits to both the bipartite entanglement of formation and the squashed entanglement of bipartite systems, by leveraging cornerstone results of quantum information theory: the entropy strong subadditivity inequality and the Koashi-Winter monogamy relation. Then, we convert them into bounds to the entanglement up to degree k for arbitrary states, and to the genuine k-partite entanglement, by employing a recently proposed method. Finally, we analytically and numerically test these results, by bounding the multipartite entanglement of several relevant states and mixtures, including the important classes of GHZ, Dicke, W states, and random pure states.
Quantifying entanglement is vital to understand entanglement as a resource in quantum information processing, and many entanglement measures have been suggested for this purpose. When mathematically defining an entanglement measure, we should consider the distinguishability between entangled and separable states, the invariance under local transformation, the monotonicity under local operations and classical communication, and the convexity. These are reasonable requirements but may be insufficient, in particular when taking into account the usefulness of quantum states in multi-party quantum information processing. Therefore, if we want to investigate multipartite entanglement as a resource, then it can be necessary to consider the usefulness of quantum states in multi-party quantum information processing when we define a multipartite entanglement measure. In this paper, we define new multipartite entanglement measures for three-qubit systems based on the three-party teleportation capability, and show that these entanglement measures satisfy the requirements for being genuine multipartite entanglement measures. We also generalize our entanglement measures for N-qubit systems, where [Formula: see text], and discuss that these quantities may be good candidates to measure genuine multipartite entanglement.
Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, we give an approach of constituting genuine m -partite entanglement measures from any bipartite entanglement and any k -partite entanglement measure, 3 ⩽ k < m . In addition, as a complement to the three-qubit concurrence triangle proposed in ( Phys. Rev. Lett. 127 040403), we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated with tripartite and bipartite entanglement respectively. For multipartite system that contains more than four parties, there is no symmetric geometric structure as that of tri- and four-partite cases.
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.