# UPB

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UPB | |

Generates an unextendible product basis | |

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` UPB` is a function that generates an unextendible product basis (UPB). The user may either request a specific UPB from the literature such as

`'Tiles'`or

`'Pyramid'`

^{[1]}, or they may request a UPB of specified dimensions.

## Syntax

`U = UPB(NAME)``U = UPB(NAME,OPT_PAR)``[U,V,W,...] = UPB(NAME,OPT_PAR)``U = UPB(DIM)``U = UPB(DIM,VERBOSE)``[U,V,W,...] = UPB(DIM,VERBOSE)`

## Argument descriptions

### Input arguments

**Important**: Do not specify both `NAME` and `DIM`: just one or the other!

`NAME`: The name of a UPB that is found in the literature. Accepted values are:`'Min4x4'`: A UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in^{[2]}.`'Pyramid'`: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in^{[1]}.`'QuadRes'`: A UPB in $\mathbb{C}^d \otimes \mathbb{C}^d$ (only valid when 2d-1 is prime and d is odd) constructed in^{[3]}. Note that you must set`OPT_PAR`equal to d (the local dimension) in this case.`'Tiles'`: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in^{[1]}.`'Shifts'`: A UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ constructed in^{[3]}(though a slightly different version appeared in^{[1]}).

`DIM`: A vector containing the local dimensions of the desired UPB. In all cases, the smallest known UPB of the desired dimensionality is returned. If no unextendible product basis is known for the specified dimensions, an error is produced.

## Examples

To be added.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}C.H. Bennett, D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal. Unextendible product bases and bound entanglement.*Phys. Rev. Lett.*82, 5385–5388, 1999. E-print: arXiv:quant-ph/9808030 - ↑ T.B. Pedersen.
*Characteristics of unextendible product bases*. Thesis, Aarhus Universitet, Datalogisk Institut, 2002. - ↑
^{3.0}^{3.1}D.P. DiVincenzo, T. Mor, P.W. Shor, J.A. Smolin, and B.M. Terhal. Unextendible product bases, uncompletable product bases and bound entanglement.*Commun. Math. Phys.*238, 379–410, 2003. E-print: arXiv:quant-ph/9908070