MinUPBSize - Revision history

Nathaniel at 16:36, 29 September 2014

2014-09-29T16:36:43Z

Nathaniel at 19:29, 27 November 2013

2013-11-27T19:29:06Z

Nathaniel: updated reference

2013-11-27T17:32:45Z

updated reference

Nathaniel at 22:16, 26 June 2013

2013-06-26T22:16:13Z

Nathaniel: Created page with "{{Function |name=MinUPBSize |desc=Gives the minimum cardinality of an unextendible product basis in given dimensions |req=opt_args, opt_disp |rel=UPB |upd=June..."

2013-06-22T19:01:34Z

Created page with "{{Function |name=MinUPBSize |desc=Gives the minimum cardinality of an unextendible product basis in given dimensions |req=opt_args, opt_disp |rel=UPB |upd=June..."

New page

{{Function
|name=MinUPBSize
|desc=Gives the minimum cardinality of an [[unextendible product basis]] in given dimensions
|req=[[opt_args]], [[opt_disp]]
|rel=[[UPB]]
|upd=June 22, 2013
|v=1.02}}
<tt>'''MinUPBSize'''</tt> is a [[List of functions|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==
* <tt>S = MinUPBSize(DIM)</tt>
* <tt>S = MinUPBSize(DIM,VERBOSE)</tt>

==Argument descriptions==
* <tt>DIM</tt>: A vector that specifies the local Hilbert space dimensions.
* <tt>VERBOSE</tt> (optional, default <tt>1</tt>): 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 <math>\mathbb{C}^5 \otimes \mathbb{C}^6</math> has <math>10</math> states, and points to the reference <ref>N. Alon and L. Lovasz. Unextendible product bases. ''J. Combinatorial Theory, Ser. A'', 95:169–179, 2001.</ref> for a proof of this fact:
<pre>
>> 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
</pre>

The following code gives the same information, but without displaying the reference:
<pre>
>> MinUPBSize([5,6],0)

ans =

10
</pre>

The minimum size of UPBs is also known in many multipartite cases:
<pre>
>> 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. E-print: arXiv:1302.1604 [quant-ph], 2013.

ans =

11
</pre>

Unfortunately, there are also many cases where the minimum size of a UPB is not currently known. In these cases, and error is returned:
<pre>
>> MinUPBSize([2,2,2,4])
??? Error using ==> MinUPBSize at 69
The minimum size of UPBs in the specified space is not currently known.
</pre>

==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 [http://www.qetlab.com/source/MinUPBSize.m the script file]) the article reference.

==References==
<references />

@@ Line 41: / Line 41: @@
 >> 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.
+N. Johnston. The minimum size of qubit unextendible product bases. In Proceedings of the 8th Conference on the
 ans =

@@ Line 2: / Line 2: @@
 |name=MinUPBSize
 |desc=Gives the minimum cardinality of an [[unextendible product basis]] in given dimensions
 |rel=[[UPB]]
 |upd=June 22, 2013
-|v=1.02}}
+|v=0.50}}
 <tt>'''MinUPBSize'''</tt> is a [[List of functions|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.
@@ Line 18: / Line 18: @@
 ==Examples==
 The following code tells the user that the smallest UPB in <math>\mathbb{C}^5 \otimes \mathbb{C}^6</math> has <math>10</math> states, and points to the reference <ref>N. Alon and L. Lovasz. Unextendible product bases. ''J. Combinatorial Theory, Ser. A'', 95:169&ndash;179, 2001.</ref> for a proof of this fact:
-<pre>
+<syntaxhighlight>
 >> MinUPBSize([5,6])
 A proof of the minimal size in this case can be found in:
@@ Line 26: / Line 26: @@
-</pre>
+</syntaxhighlight>
 The following code gives the same information, but without displaying the reference:
-<pre>
+<syntaxhighlight>
 >> MinUPBSize([5,6],0)
@@ Line 35: / Line 35: @@
-</pre>
+</syntaxhighlight>
 The minimum size of UPBs is also known in many multipartite cases:
-<pre>
+<syntaxhighlight>
 >> MinUPBSize([2,2,2,2,2,2,2,2])
 A proof of the minimal size in this case can be found in:
@@ Line 47: / Line 47: @@
-</pre>
+</syntaxhighlight>
 Unfortunately, there are also many cases where the minimum size of a UPB is not currently known. In these cases, and error is returned:
-<pre>
+<syntaxhighlight>
 >> MinUPBSize([2,2,2,4])
 ??? Error using ==> MinUPBSize at 69
 The minimum size of UPBs in the specified space is not currently known.
-</pre>
+</syntaxhighlight>
 ==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 [http://www.qetlab.com/source/MinUPBSize.m the script file]) the article reference.
+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.
 ==References==
 <references />