Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces

doi:10.1007/s00220-017-3077-5

	Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces
	Li, Yinan 1; Qiao, Youming 1; Wang, Xin 1; Duan, Runyao 1,2
刊名	COMMUNICATIONS IN MATHEMATICAL PHYSICS
	2018-03-01
卷号	358 期号:2 页码:791-814
ISSN号	0010-3616
DOI	10.1007/s00220-017-3077-5
英文摘要	We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings, utilizing powerful results from the structure of matrix spaces. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation when multiple copies are provided. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate. Computing this rate turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under SLOCC, in the asymptotic setting. Applying the recent progress on the non-commutative rank problem, we can verify this condition in deterministic polynomial time.
资助项目	Australian Research Council[DP120103776] ; Australian Research Council[FT120100449] ; Australian Research Council[DE150100720]
WOS研究方向	Physics
语种	英语
出版者	SPRINGER
WOS记录号	WOS:000427467400009
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/29759]
专题	中国科学院数学与系统科学研究院
通讯作者	Li, Yinan
作者单位	1.Univ Technol Sydney, Fac Engn & Informat Technol, Ctr Quantum Software & Informat, Sydney, NSW 2007, Australia 2.Chinese Acad Sci, Acad Math & Syst Sci, UTS AMSS Joint Res Lab Quantum Computat & Quantum, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Li, Yinan,Qiao, Youming,Wang, Xin,et al. Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS,2018,358(2):791-814.
APA	Li, Yinan,Qiao, Youming,Wang, Xin,&Duan, Runyao.(2018).Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces.COMMUNICATIONS IN MATHEMATICAL PHYSICS,358(2),791-814.
MLA	Li, Yinan,et al."Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces".COMMUNICATIONS IN MATHEMATICAL PHYSICS 358.2(2018):791-814.