PIRSA:13020149 Scientific Series Quantum Information

DOI10.48660/13020149

Series: Perimeter Institute Quantum Discussions

De les Coves, G. (2013). Bounding the purification rank of mixed states. Perimeter Institute. https://pirsa.org/13020149

De les Coves, Gemma. Bounding the purification rank of mixed states. Perimeter Institute, Feb. 25, 2013, https://pirsa.org/13020149 

@misc{ pirsa_PIRSA:13020149,
  doi = {10.48660/13020149},
  url = {https://pirsa.org/13020149},
  author = {De les Coves, Gemma},
  keywords = {Quantum Information},
  language = {en},
  title = {Bounding the purification rank of mixed states},
  publisher = {Perimeter Institute},
  year = {2013},
  month = {feb},
  note = {PIRSA:13020149 see, \url{https://pirsa.org}}
}

Abstract

A mixed state can be expressed as a sum of D tensor product matrices, where D is its operator Schmidt rank, or as the result of a purification with a purifying state of Schmidt rank D', where D' is its purification rank. The question whether D' can be upper bounded by D is important theoretically (to establish a description of mixed states with tensor networks), as well as numerically (as the first decomposition is more efficient, but the second one guarantees positive-semidefiniteness after truncation). Here we show that no upper bounds of the purification rank that depend only on operator Schmidt rank exist, but provide upper bounds that also depend on the number of eigenvalues. In addition, we formulate the approximation problem as a Semidefinite Program.
Joint work with N. Schuch, D. Perez-Garcia, and J. I. Cirac.

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