# MinUPBSize

MinUPBSize | |

Gives the minimum cardinality of an unextendible product basis in given dimensions | |

Other toolboxes required | none |
---|---|

Related functions | UPB |

Function category | Unextendible product bases |

` MinUPBSize` is a function that returns that minimum numbers of vectors required in an unextendible product basis (UPB), given the dimensions of the local Hilbert spaces. If the minimum size of a UPB in the given space is not known, and error will be produced.

## Syntax

`S = MinUPBSize(DIM)``S = MinUPBSize(DIM,VERBOSE)`

## Argument descriptions

`DIM`: A vector that specifies the local Hilbert space dimensions.`VERBOSE`(optional, default`1`): A flag (either 1 or 0) indicating that a reference to a journal article that provides a proof of the minimal size claimed by this script will or will not be displayed.

## Examples

The following code tells the user that the smallest UPB in \(\mathbb{C}^5 \otimes \mathbb{C}^6\) has \(10\) states, and points to the reference ^{[1]} for a proof of this fact:

```
>> MinUPBSize([5,6])
A proof of the minimal size in this case can be found in:
N. Alon and L. Lovasz. Unextendible product bases. J. Combinatorial Theory, Ser. A, 95:169-179, 2001.
ans =
10
```

The following code gives the same information, but without displaying the reference:

```
>> MinUPBSize([5,6],0)
ans =
10
```

The minimum size of UPBs is also known in many multipartite cases:

```
>> MinUPBSize([2,2,2,2,2,2,2,2])
A proof of the minimal size in this case can be found in:
N. Johnston. The minimum size of qubit unextendible product bases. In Proceedings of the 8th Conference on the
Theory of Quantum Computation, Communication and Cryptography (TQC 2013). E-print: arXiv:1302.1604 [quant-ph], 2013.
ans =
11
```

Unfortunately, there are also many cases where the minimum size of a UPB is not currently known. In these cases, and error is returned:

```
>> MinUPBSize([2,2,2,4])
??? Error using ==> MinUPBSize at 69
The minimum size of UPBs in the specified space is not currently known.
```

## Notes

The minimum size of UPBs is not known in full generality yet! If you know of another case that has been solved that isn't contained in this script, please e-mail me (my e-mail address is in the header of the source code below) the article reference.

## Source code

Click here to view this function's source code on github.

## References

- ↑ N. Alon and L. Lovasz. Unextendible product bases.
*J. Combinatorial Theory, Ser. A*, 95:169–179, 2001.