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>'Feng2x2x2x2'</tt>: A 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&ndash;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&ndash;949, 2006.</ref>.
** <tt>'Feng2x2x3'</tt>: A 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 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>'GenShifts'</tt>: A UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p &ge; 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>'Feng4m2'</tt>: A minimal UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p = 2 (mod 4)) constructed in <ref name="Fen06"></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>'GenShifts'</tt>: A minimal UPB in $(\mathbb{C}^2)^{\otimes p}$ (only valid when p &ge; 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>'Pyramid'</tt>: A UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></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>.
** <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&ndash;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>'Pyramid'</tt>: A minimal 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>'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&ndash;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 UPB in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ introduced in <ref name="BDM99"></ref>.
+
** <tt>'Tiles'</tt>: A minimal UPB in $\mathbb{C}^3 \otimes \mathbb{C}^3$ constructed in <ref name="BDM99"></ref>.
 +
** <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>'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 07:57, 7 December 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:
    • '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.

Examples

To be added.

References

  1. 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. 2.0 2.1 2.2 2.3 K. Feng. Unextendible product bases and 1-factorization of complete graphs. Discrete Applied Mathematics, 154:942–949, 2006.
  3. 3.0 3.1 3.2 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
  4. T.B. Pedersen. Characteristics of unextendible product bases. Thesis, Aarhus Universitet, Datalogisk Institut, 2002.