# Difference between revisions of "UPB"

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'''Important''': Do not specify both <tt>NAME</tt> and <tt>DIM</tt>: just one or the other! | '''Important''': Do not specify both <tt>NAME</tt> and <tt>DIM</tt>: just one or the other! | ||

* <tt>NAME</tt>: The name of a UPB that is found in the literature. Accepted values are: | * <tt>NAME</tt>: The name of a UPB that is found in the literature. Accepted values are: | ||

+ | ** <tt>'GenShifts'</tt>: A UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p ≥ 3 is odd) constructed in <ref name="DMS03"></ref>. Note that <tt>OPT_PAR</tt> must be the number of parties (i.e., the integer p) in this case. | ||

** <tt>'Min4x4'</tt>: A UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref>T.B. Pedersen. ''Characteristics of unextendible product bases''. Thesis, Aarhus Universitet, Datalogisk Institut, 2002.</ref>. | ** <tt>'Min4x4'</tt>: A UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref>T.B. Pedersen. ''Characteristics of unextendible product bases''. Thesis, Aarhus Universitet, Datalogisk Institut, 2002.</ref>. | ||

** <tt>'Pyramid'</tt>: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>. | ** <tt>'Pyramid'</tt>: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>. | ||

** <tt>'QuadRes'</tt>: A UPB in $\mathbb{C}^d \otimes \mathbb{C}^d$ (only valid when 2d-1 is prime and d is odd) constructed in <ref name="DMS03">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: [http://arxiv.org/abs/quant-ph/9908070 arXiv:quant-ph/9908070]</ref>. Note that you must set <tt>OPT_PAR</tt> equal to d (the local dimension) in this case. | ** <tt>'QuadRes'</tt>: A UPB in $\mathbb{C}^d \otimes \mathbb{C}^d$ (only valid when 2d-1 is prime and d is odd) constructed in <ref name="DMS03">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: [http://arxiv.org/abs/quant-ph/9908070 arXiv:quant-ph/9908070]</ref>. Note that you must set <tt>OPT_PAR</tt> equal to d (the local dimension) in this case. | ||

** <tt>'Tiles'</tt>: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>. | ** <tt>'Tiles'</tt>: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>. | ||

− | ** <tt>'Shifts'</tt>: A UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ | + | ** <tt>'Shifts'</tt>: A UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ introduced in <ref name="BDM99"></ref>. |

** <tt>'SixParam'</tt>: The six-parameter UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ introduced in Section IV.A of <ref name="DMS03"></ref>. Note that <tt>OPT_PAR</tt> must be a vector containing the six parameters in this case. | ** <tt>'SixParam'</tt>: The six-parameter UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ introduced in Section IV.A of <ref name="DMS03"></ref>. Note that <tt>OPT_PAR</tt> must be a vector containing the six parameters in this case. | ||

* <tt>DIM</tt>: 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. | * <tt>DIM</tt>: 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. |

## Revision as of 22:05, 30 November 2012

UPB | |

Generates an unextendible product basis | |

Other toolboxes required | none |
---|

` 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:`'GenShifts'`: A UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p ≥ 3 is odd) constructed in^{[2]}. Note that`OPT_PAR`must be the number of parties (i.e., the integer p) in this case.`'Min4x4'`: A UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in^{[3]}.`'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^{[2]}. 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 $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ introduced in^{[1]}.`'SixParam'`: The six-parameter UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ introduced in Section IV.A of^{[2]}. Note that`OPT_PAR`must be a vector containing the six parameters in this case.

`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 - ↑
^{2.0}^{2.1}^{2.2}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 - ↑ T.B. Pedersen.
*Characteristics of unextendible product bases*. Thesis, Aarhus Universitet, Datalogisk Institut, 2002.