Unextendible Product Bases

✍ Scribed by N. Alon; L. Lovász

• Book ID: 102586054
• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 124 KB
• Volume: 95
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2000.3122

✦ Synopsis

Let C denote the complex field. A vector v in the tensor product } m i=1 C ki is called a pure product vector if it is a vector of the form v 1 v 2 } } } v m , with v i # C ki . A set F of pure product vectors is called an unextendible product basis if F consists of orthogonal nonzero vectors, and there is no nonzero pure product vector in } m i=1 C ki which is orthogonal to all members of F. The construction of such sets of small cardinality is motivated by a problem in quantum information theory.

Here it is shown that the minimum possible cardinality of such a set F is precisely 1+ m i=1 (k i &1) for every sequence of integers k 1 , k 2 , ..., k m 2 unless either (i) m=2 and 2 # [k 1 , k 2 ] or (ii) 1+ m i=1 (k i &1) is odd and at least one k i is even. In each of these two cases, the minimum cardinality of the corresponding F is strictly bigger than 1+ m i=1 (k i &1).