Citation

N. Alon et L. Lovasz, Unextendible product bases, J COMB TH A, 95(1), 2001, pp. 169-179

Citations number

11

Language

INGLESE

art.tipo

Article

Categorie Soggetti

Mathematics

Journal title

JOURNAL OF COMBINATORIAL THEORY SERIES A

ISSN journal

0097-3165
→ ACNP

Volume

95

Issue

1

Year of publication

2001

Pages

169 - 179

Database

ISI

SICI code

0097-3165(200107)95:1<169:UPB>2.0.ZU;2-Y

Abstract

Let C denote the complex field. A vector upsilon in the tensor product circ
le times (m)(i=1) C-k,C- is called a pure produc vector if it is a vector o
f the form nu (1) circle times upsilon (2) ... circle times nu (m), with nu
is an element of C-k,C-. A set F of pure product vectors is called an unex
tendible product basis if F consists of orthogonal nonzero vectors. and the
re is no nonzero pure product vector in circle times (m)(i=1) C-k,C- which
is orthogonal to all members of F. The construction of such sets of small c
ardinality is motivated by a problem in quantum information theory. Here it
is shown that the minimum possible cardinality of such a set F is precisel
y 1 + Sigma (m)(i=1)(k(i-1)) for every sequence of integers k(1), k(2)....
k(m) greater than or equal to 2 unless either (i) m = 2 and 2 is an element
of {k(1), k(2)} or (ii) 1 +Sigma (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 c
orresponding F is strictly bigger than 1 + Sigma (m)(i=1)(k(i) - 1). (C) 20
01 Academic Press.