Schmidt number of bipartite and multipartite states under local projections

  • Lin Chen
  • Yu Yang
  • Wai-Shing Tang


The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and local projections are fundamental operations in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive partial transpose entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states and explore its relation with the Schmidt number of bipartite reduced density operators.


Quantum states Local projections PPT entangled states Schmidt number Joint Schmidt number Reduced density operators 



Lin Chen was supported by the NSF of China (Grant No. 11501024), and the Fundamental Research Funds for the Central Universities (Grant Nos. 30426401, 30458601 and 29816133). Yu Yang acknowledged financial support from Department of Mathematics, National University of Singapore, for his Ph.D. study. Wai-Shing Tang was partially supported by Singapore Ministry of Education Academic Research Fund Tier 1 Grant (No. R-146-000-193-112). The authors would like to thank Farid Shahandeh for his comments.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.International Research Institute for Multidisciplinary ScienceBeihang UniversityBeijingChina
  3. 3.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

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