2015/12/10 by Guillaume Aubrun, Aubrun, Guillaume, Stanisław J. Szarek +1
Computer Science · Physics and Astronomy · #15B48 #46N50 #47L07 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.1512.03293
openalex publication_date 2015/12/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The structure of the set of positivity-preserving maps between matrix algebras is notoriously difficult to describe. The notable exceptions are the results by Størmer and Woronowicz from 1960s and 1970s settling the low dimensional cases. By duality, these results are equivalent to the Peres-Horodecki positive partial transpose criterion being able to unambiguously establish whether a state in a 2 x 2 or 2 x 3 quantum system is entangled or separable. However, even in these low dimensional cases, the existing arguments (known to the authors) were based on long and seemingly ad hoc computations. We present a simple proof, based on Brouwer's fixed point theorem, for the 2 x 2 case (Størmer's theorem). For completeness, we also include another argument (following the classical outline, but highly streamlined) based on a characterization of extreme self-maps of the Lorentz cone and on a link - noticed by R. Hildebrand - to the S-lemma, a well-known fact from control theory and quadratic/semi-definite programming.