UPB - Revision history

Nathaniel: Fixed scaling so that bound entangled state has trace 1

2024-12-09T13:51:46Z

Fixed scaling so that bound entangled state has trace 1

← Older revision		Revision as of 13:51, 9 December 2024
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	rho = rho - v(:,j)*v(:,j)';		rho = rho - v(:,j)*v(:,j)';
	end		end
	rho = rho/7; % we are now done constructing the bound entangled state		rho = rho/8; % we are now done constructing the bound entangled state

	>> IsSeparable(rho,[3,5]) % show that the state is indeed entangled		>> IsSeparable(rho,[3,5]) % show that the state is indeed entangled

Nathaniel: Markup fix

2022-05-09T17:30:28Z

Markup fix

← Older revision		Revision as of 17:30, 9 May 2022
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	** <tt>'QuadRes'</tt>: A minimal UPB in <math>\mathbb{C}^d \otimes \mathbb{C}^d</math> (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 minimal UPB in <math>\mathbb{C}^d \otimes \mathbb{C}^d</math> (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>'Shifts'</tt>: A minimal UPB in <math>\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2</math> introduced in <ref name="BDM99"></ref>.		** <tt>'Shifts'</tt>: A minimal UPB in <math>\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2</math> introduced in <ref name="BDM99"></ref>.
	** <tt>'SixParam'</tt>: The six-parameter UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3</math> 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 <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> 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>'Tiles'</tt>: A minimal UPB in <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> constructed in <ref name="BDM99"></ref>.		** <tt>'Tiles'</tt>: A minimal UPB in <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> constructed in <ref name="BDM99"></ref>.
	** <tt>'GenTiles1'</tt>: A non-minimal UPB in <math>\mathbb{C}^n \otimes \mathbb{C}^n</math> when n is even from <ref name="BDM99"></ref>. Note that <tt>OPT_PAR</tt> must be n in this case.		** <tt>'GenTiles1'</tt>: A non-minimal UPB in <math>\mathbb{C}^n \otimes \mathbb{C}^n</math> when n is even from <ref name="BDM99"></ref>. Note that <tt>OPT_PAR</tt> must be n in this case.

Nathaniel: Added GenTiles1 and GenTiles2

2022-05-09T17:29:57Z

Added GenTiles1 and GenTiles2

Nathaniel at 18:10, 12 November 2014

2014-11-12T18:10:44Z

← Older revision		Revision as of 18:10, 12 November 2014
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	{{Function		{{Function
	\|name=UPB		\|name=UPB
	\|desc=Generates an [[unextendible product basis]]		\|desc=Generates an unextendible product basis
	\|rel=[[IsUPB]]<br />[[MinUPBSize]]<br />[[UPBSepDistinguishable]]		\|rel=[[IsUPB]]<br />[[MinUPBSize]]<br />[[UPBSepDistinguishable]]
	\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]		\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]
	\|upd=November 6, 2014		\|upd=November 12, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''UPB'''</tt> is a [[List of functions\|function]] that generates an [[unextendible product basis]] (UPB). The user may either request a specific UPB from the literature such as <tt>'Tiles'</tt> or <tt>'Pyramid'</tt><ref name="BDM99">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: [http://arxiv.org/abs/quant-ph/9808030 arXiv:quant-ph/9808030]</ref>, or they may request a minimal UPB of specified dimensions.		<tt>'''UPB'''</tt> is a [[List of functions\|function]] that generates an unextendible product basis (UPB). The user may either request a specific UPB from the literature such as <tt>'Tiles'</tt> or <tt>'Pyramid'</tt><ref name="BDM99">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: [http://arxiv.org/abs/quant-ph/9808030 arXiv:quant-ph/9808030]</ref>, or they may request a minimal UPB of specified dimensions.

	==Syntax==		==Syntax==

Nathaniel at 14:40, 7 November 2014

2014-11-07T14:40:39Z

← Older revision		Revision as of 14:40, 7 November 2014
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	\|rel=[[IsUPB]]<br />[[MinUPBSize]]<br />[[UPBSepDistinguishable]]		\|rel=[[IsUPB]]<br />[[MinUPBSize]]<br />[[UPBSepDistinguishable]]
	\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]		\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]
	\|upd=~~June 26~~, ~~2013~~		\|upd=November 6, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''UPB'''</tt> is a [[List of functions\|function]] that generates an [[unextendible product basis]] (UPB). The user may either request a specific UPB from the literature such as <tt>'Tiles'</tt> or <tt>'Pyramid'</tt><ref name="BDM99">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: [http://arxiv.org/abs/quant-ph/9808030 arXiv:quant-ph/9808030]</ref>, or they may request a minimal UPB of specified dimensions.		<tt>'''UPB'''</tt> is a [[List of functions\|function]] that generates an [[unextendible product basis]] (UPB). The user may either request a specific UPB from the literature such as <tt>'Tiles'</tt> or <tt>'Pyramid'</tt><ref name="BDM99">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: [http://arxiv.org/abs/quant-ph/9808030 arXiv:quant-ph/9808030]</ref>, or they may request a minimal UPB of specified dimensions.

Nathaniel at 23:57, 18 October 2014

2014-10-18T23:57:39Z

← Older revision		Revision as of 23:57, 18 October 2014
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	\|name=UPB		\|name=UPB
	\|desc=Generates an [[unextendible product basis]]		\|desc=Generates an [[unextendible product basis]]
	\|rel=[[IsUPB]]<br />[[MinUPBSize]]		\|rel=[[IsUPB]]<br />[[MinUPBSize]]<br />[[UPBSepDistinguishable]]
	\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]		\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]
	\|upd=June 26, 2013		\|upd=June 26, 2013

Nathaniel at 16:34, 29 September 2014

2014-09-29T16:34:48Z

← Older revision		Revision as of 16:34, 29 September 2014
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	\|desc=Generates an [[unextendible product basis]]		\|desc=Generates an [[unextendible product basis]]
	\|rel=[[IsUPB]]<br />[[MinUPBSize]]		\|rel=[[IsUPB]]<br />[[MinUPBSize]]
	\|~~req~~=[[~~FourierMatrix]]<br />[[IsTotallyNonsingular]]<br />[[MinUPBSize]]<br />[[normalize_cols]]<br />[[opt_args]]<br />[[opt_disp]]<br />[[perm_inv]]<br />[[PermuteSystems]]<br />[[sporth]]<br />[[Swap~~]]		\|cat=[[List of functions#Unextendible_product_bases\|Unextendible product bases]]
	\|upd=June 26, 2013		\|upd=June 26, 2013
	\|v=1.00}}		\|v=0.50}}
	<tt>'''UPB'''</tt> is a [[List of functions\|function]] that generates an [[unextendible product basis]] (UPB). The user may either request a specific UPB from the literature such as <tt>'Tiles'</tt> or <tt>'Pyramid'</tt><ref name="BDM99">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: [http://arxiv.org/abs/quant-ph/9808030 arXiv:quant-ph/9808030]</ref>, or they may request a minimal UPB of specified dimensions.		<tt>'''UPB'''</tt> is a [[List of functions\|function]] that generates an [[unextendible product basis]] (UPB). The user may either request a specific UPB from the literature such as <tt>'Tiles'</tt> or <tt>'Pyramid'</tt><ref name="BDM99">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: [http://arxiv.org/abs/quant-ph/9808030 arXiv:quant-ph/9808030]</ref>, or they may request a minimal UPB of specified dimensions.

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	===Generating the "Shifts" UPB===		===Generating the "Shifts" UPB===
	The following code returns the "Shifts" UPB <ref name="BDM99"></ref>, which is a UPB of 4 states on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$:		The following code returns the "Shifts" UPB <ref name="BDM99"></ref>, which is a UPB of 4 states on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$:
	<~~pre~~>		<syntaxhighlight>
	>> v = UPB('Shifts')		>> v = UPB('Shifts')

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	0 0.5000 -0.5000 0.0000		0 0.5000 -0.5000 0.0000
	0 0.5000 0.5000 0.5000		0 0.5000 0.5000 0.5000
	</~~pre~~>		</syntaxhighlight>

	Alternatively, we can request that the local vectors on each copy of $\mathbb{C}^2$ are returned, rather than the total product vectors on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$:		Alternatively, we can request that the local vectors on each copy of $\mathbb{C}^2$ are returned, rather than the total product vectors on $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$:
	<~~pre~~>		<syntaxhighlight>
	>> [u,v,w] = UPB('Shifts')		>> [u,v,w] = UPB('Shifts')

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	1.0000 0.7071 -0.7071 0.0000		1.0000 0.7071 -0.7071 0.0000
	0 0.7071 0.7071 1.0000		0 0.7071 0.7071 1.0000
	</~~pre~~>		</syntaxhighlight>

	===Generating [[bound entangled]] states===		===Generating [[bound entangled]] states===
	As noted in <ref name="BDM99"></ref>, if <math>\big\{\|v_i\rangle\big\}</math> is an unextendible product basis, then $I - \sum_i \|v_i\rangle\langle v_i\|$ is (up to scaling) a [[bound entangled]] state. The following code illustrates this fact in $\mathbb{C}^3 \otimes \mathbb{C}^5$ by first constructing a UPB in this space, then constructing the corresponding state, and then verifying that this state is bound entangled.		As noted in <ref name="BDM99"></ref>, if <math>\big\{\|v_i\rangle\big\}</math> is an unextendible product basis, then $I - \sum_i \|v_i\rangle\langle v_i\|$ is (up to scaling) a [[bound entangled]] state. The following code illustrates this fact in $\mathbb{C}^3 \otimes \mathbb{C}^5$ by first constructing a UPB in this space, then constructing the corresponding state, and then verifying that this state is bound entangled.
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> v = UPB([3,5]);		>> v = UPB([3,5]);
	Generated a minimal 7-state UPB from:		Generated a minimal 7-state UPB from:
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	rho = rho/7; % we are now done constructing the bound entangled state		rho = rho/7; % we are now done constructing the bound entangled state

	>> ~~[[IsSeparable\|~~IsSeparable(rho,[3,5])]] % show that the state is indeed entangled		>> IsSeparable(rho,[3,5]) % show that the state is indeed entangled
	Determined to be entangled by not having a 2-copy symmetric extension. Reference:		Determined to be entangled by not having a 2-copy symmetric extension. Reference:
	A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. A complete family of separability criteria.		A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. A complete family of separability criteria.
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	0		0
	>> ~~[[IsPPT\|~~IsPPT(rho,2,[3,5])]] % verify that this state has positive partial transpose and is thus bound entangled		>> IsPPT(rho,2,[3,5]) % verify that this state has positive partial transpose and is thus bound entangled

	ans =		ans =

	1		1
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	===Generating multipartite UPBs===		===Generating multipartite UPBs===
	Many multipartite minimal UPBs can be constructed via this script. For example, the following code generates a minimal UPB (of 6 states) in <math>\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3</math>:		Many multipartite minimal UPBs can be constructed via this script. For example, the following code generates a minimal UPB (of 6 states) in <math>\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3</math>:
	<~~pre~~>		<syntaxhighlight>
	>> [u,v,w] = UPB([2,2,3])		>> [u,v,w] = UPB([2,2,3])
	Generated a minimal 6-state UPB from:		Generated a minimal 6-state UPB from:
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	0 0 -0.8018 0.5774 0.3162 1.0000		0 0 -0.8018 0.5774 0.3162 1.0000
	0 0.7071 -0.2673 -0.5774 -0.9487 0		0 0.7071 -0.2673 -0.5774 -0.9487 0
	</~~pre~~>		</syntaxhighlight>

	However, the minimum size of UPBs is still unknown in many multipartite cases – an error is returned in these cases:		However, the minimum size of UPBs is still unknown in many multipartite cases – an error is returned in these cases:
	<~~pre~~>		<syntaxhighlight>
	>> [u,v,w,x] = UPB([2,2,3,7])		>> [u,v,w,x] = UPB([2,2,3,7])
	??? Error using ==> UPB at 132		??? Error using ==> UPB at 132
	No minimal UPB is currently known in the specified dimensions.		No minimal UPB is currently known in the specified dimensions.
	</~~pre~~>		</syntaxhighlight>

			{{SourceCode\|name=UPB}}

	==References==		==References==
	<references />		<references />

Nathaniel at 22:12, 26 June 2013

2013-06-26T22:12:43Z

Show changes

Nathaniel: updated ref

2012-12-20T20:10:23Z

updated ref

← Older revision		Revision as of 20:10, 20 December 2012
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	** <tt>'Feng4x4'</tt>: A minimal UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref name="Fen06"></ref>.		** <tt>'Feng4x4'</tt>: A minimal UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref name="Fen06"></ref>.
	** <tt>'GenShifts'</tt>: A minimal 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>'GenShifts'</tt>: A minimal 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 minimal 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 minimal UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref>T.B. Pedersen. ''Characteristics of unextendible product bases''. Master's Thesis, Aarhus Universitet, Datalogisk Institut, 2002.</ref>.
	** <tt>'Pyramid'</tt>: A minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>.		** <tt>'Pyramid'</tt>: A minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>.
	** <tt>'QuadRes'</tt>: A minimal 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 minimal 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.

Nathaniel at 17:08, 7 December 2012

2012-12-07T17:08:54Z

← Older revision		Revision as of 17:08, 7 December 2012
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	* <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>'Feng2x2x2x2'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ constructed in <ref name="Fen06">K. Feng. Unextendible product bases and 1-factorization of complete graphs. ''Discrete Applied Mathematics'', 154:942–949, 2006.</ref>.		** <tt>'Feng2x2x2x2'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ constructed in <ref name="Fen06">K. Feng. Unextendible product bases and 1-factorization of complete graphs. ''Discrete Applied Mathematics'', 154:942–949, 2006.</ref>.
			** <tt>'Feng2x2x2x4'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^4$ constructed in <ref name="Fen06"></ref>.
			** <tt>'Feng2x2x2x2x5'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^5$ constructed in <ref name="Fen06"></ref>.
	** <tt>'Feng2x2x3'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3$ constructed in <ref name="Fen06"></ref>.		** <tt>'Feng2x2x3'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^3$ constructed in <ref name="Fen06"></ref>.
	** <tt>'Feng2x2x5'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^5$ constructed in <ref name="Fen06"></ref>.		** <tt>'Feng2x2x5'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^5$ constructed in <ref name="Fen06"></ref>.
	** <tt>'Feng4m2'</tt>: A minimal UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p = 2 (mod 4)) constructed in <ref name="Fen06"></ref>. Note that <tt>OPT_PAR</tt> must be the number of parties (i.e., the integer p) in this case.		** <tt>'Feng4m2'</tt>: A minimal UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p = 2 (mod 4)) constructed in <ref name="Fen06"></ref>. Note that <tt>OPT_PAR</tt> must be the number of parties (i.e., the integer p) in this case.
			** <tt>'Feng4x4'</tt>: A minimal UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref name="Fen06"></ref>.
	** <tt>'GenShifts'</tt>: A minimal 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>'GenShifts'</tt>: A minimal 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 minimal 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 minimal 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>.

← Older revision		Revision as of 17:29, 9 May 2022
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	===Input arguments===		===Input arguments===
	'''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 include:
	** <tt>'GenShifts'</tt>: A minimal 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>'GenShifts'</tt>: A minimal UPB in <math>(\mathbb{C}^2)^{\otimes p}</math> (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 minimal UPB in $\mathbb{C}^4 \otimes \mathbb{C}^4$ constructed in <ref>T.B. Pedersen. ''Characteristics of unextendible product bases''. Master's Thesis, Aarhus Universitet, Datalogisk Institut, 2002.</ref>.		** <tt>'Min4x4'</tt>: A minimal UPB in <math>\mathbb{C}^4 \otimes \mathbb{C}^4</math> constructed in <ref>T.B. Pedersen. ''Characteristics of unextendible product bases''. Master's Thesis, Aarhus Universitet, Datalogisk Institut, 2002.</ref>.
	** <tt>'Pyramid'</tt>: A minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>.		** <tt>'Pyramid'</tt>: A minimal UPB in <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> constructed in <ref name="BDM99"></ref>.
	** <tt>'QuadRes'</tt>: A minimal 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 minimal UPB in <math>\mathbb{C}^d \otimes \mathbb{C}^d</math> (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>'Shifts'</tt>: A minimal UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ introduced in <ref name="BDM99"></ref>.		** <tt>'Shifts'</tt>: A minimal UPB in <math>\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2</math> 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</math> 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>'Tiles'</tt>: A minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>.		** <tt>'Tiles'</tt>: A minimal UPB in <math>\mathbb{C}^3 \otimes \mathbb{C}^3</math> constructed in <ref name="BDM99"></ref>.
			** <tt>'GenTiles1'</tt>: A non-minimal UPB in <math>\mathbb{C}^n \otimes \mathbb{C}^n</math> when n is even from <ref name="BDM99"></ref>. Note that <tt>OPT_PAR</tt> must be n in this case.
			** <tt>'GenTiles2'</tt>: A non-minimal UPB in <math>\mathbb{C}^m \otimes \mathbb{C}^n</math> from <ref name="BDM99"></ref>. Note that <tt>OPT_PAR</tt> must be a 2-entry vector containing m and n 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.
	* <tt>VERBOSE</tt> (optional, default <tt>1</tt>): A flag (either 1 or 0) indicating whether or not a reference to a journal article that contains the UPB (or a description of how to construct the UPB) returned by this script will be displayed.		* <tt>VERBOSE</tt> (optional, default <tt>1</tt>): A flag (either 1 or 0) indicating whether or not a reference to a journal article that contains the UPB (or a description of how to construct the UPB) returned by this script will be displayed.