Isaac Newton Institute for Mathematical Sciences

Mixed state decompositions: Tensor Networks and beyond

2013-09-02

Presenter: Gemma De las Cuevas (Max Planck Institute for Quantum Optics, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany)

Co-authors: Norbert Schuch (Institut fuer Quanteninformation, RWTH Aachen, Aachen, Germany), David Pérez-García (Departamento de Análisis Matemático, Facultad de CC Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain ), J. Ignacio Cirac (Max Planck Institute for Quantum Optics, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany)

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.