2000/08/06 by Jean-Luc Brylinski, Brylinski, Jean-Luc
Computer Science · Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #Tensor decomposition and applications #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0008031
10 pages, Latex
arxiv created 2000/08/06 · openalex publication_date 2000/08/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the rank of a general tensor u in a tensor product H1\ot...\ot Hk. The rank of u is the minimal number p of pure states v1,...,vp such that u is a linear combination of the vj's. This rank is an algebraic measure of the degree of entanglement of u. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in (\C2)\ot 3 and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for (\C2)\ot 4; in particular, we show that the maximal rank of a tensor in (\C2)\ot 4 is equal to 4.