# UPB

 Other toolboxes required UPB Generates an unextendible product basis 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:
• 'Feng2x2x2x2': A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ constructed in [2].
• 'Feng2x2x3': A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3$ constructed in [2].
• 'Feng2x2x5': A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^5$ constructed in [2].
• 'Feng4m2': A minimal UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p = 2 (mod 4)) constructed in [2].
• 'GenShifts': A minimal UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p ≥ 3 is odd) constructed in [3]. Note that OPT_PAR must be the number of parties (i.e., the integer p) in this case.
• 'Min4x4': A minimal UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in [4].
• 'Pyramid': A minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in [1].
• 'QuadRes': A minimal 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 minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in [1].
• 'Shifts': A minimal 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 [3]. 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.