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Characterization of Combinatorially Independent Permutation Separability Criteria

Wocjan, Paweł and Horodecki, Michał (2005) Characterization of Combinatorially Independent Permutation Separability Criteria. Open Systems and Information Dynamics, 12 (4). pp. 331-345. ISSN 1230-1612. https://resolver.caltech.edu/CaltechAUTHORS:20190923-122512278

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Abstract

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to equivalent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorial independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combinatorially independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently a simple representative for every permutation. This method allows to check easily if two permutations are combinatorially equivalent or not.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/s11080-005-4483-2DOIArticle
https://arxiv.org/abs/quant-ph/0503129arXivDiscussion Paper
Additional Information:© 2005 World Scientific Publishing Co Pte Ltd. Received 14 June 2005. We would like to thank Asa Ericsson and Lieven Clarisse for useful comments. The paper has been written while M. H. was visiting the Institute for Quantum Information, California Institute of Technology. M. H. is supported by Polish Ministry of Scientific Research and Information Technology under the (solicited) grant no. PBZ-MIN-008/P03/2003 and by EC grants RESQ, contract no. IST-2001-37559 and QUPRODIS, contract no. IST-2001-38877. P.W. is supported by the National Science Foundation under the grant no. EIA 0086038.
Group:UNSPECIFIED, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
Ministry of Scientific Research and Information Technology (Poland)PBZ-MIN-008/P03/2003
European CommissionIST-2001-37559
European CommissionIST-2001-38877
NSFEIA-0086038
Issue or Number:4
Record Number:CaltechAUTHORS:20190923-122512278
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190923-122512278
Official Citation:Characterization of Combinatorially Independent Permutation Separability Criteria. Paweł Wocjan and Michał Horodecki. Open Systems & Information Dynamics 2005 12:04, 331-345. https://doi.org/10.1007/s11080-005-4483-2
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98800
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:23 Sep 2019 21:18
Last Modified:04 Jun 2020 10:14

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