### Abstract

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).

Original language | English (US) |
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Pages (from-to) | 169-179 |

Number of pages | 11 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 95 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2001 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory. Series A*,

*95*(1), 169-179. https://doi.org/10.1006/jcta.2000.3122