

A229913


Irregular triangle in which row n lists the possible sizes of nqubit unextendible product bases.


1



2, 4, 4, 8, 6, 7, 8, 9, 10, 12, 16, 6, 8, 9, 10
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OFFSET

1,1


COMMENTS

An unextendible product basis (UPB) is a set of mutually orthogonal product states such that there is no product state orthogonal to every member of the set. An nqubit UPB is a UPB on the space C^2 tensored with itself n times, where C is the field of complex numbers.
Row n also gives the values of m such that there exists an n X m matrix M with the following three properties: (1) every entry of M is a nonzero integer; (2) the sum of any two columns of M contains a 0 entry; and (3) there is no way to append an (m+1)st column to M so that M still has property (2).
The first entry in row n is A211390(n).
The last entry in row n is 2^n.
The nexttolast entry in row n is 2^n  4 (for n >= 3).


LINKS

Table of n, a(n) for n=1..15.
R. Augusiak, T. Fritz, M. Kotowski, M. Kotowski, M. Pawlowski, M. Lewenstein, and A. Acín, Tight Bell inequalities with no quantum violation from qubit unextendible product bases. arXiv:1112.3238 [quantph], 20112012.
R. Augusiak, T. Fritz, M. Kotowski, M. Kotowski, M. Pawlowski, M. Lewenstein, and A. Acín, Tight Bell inequalities with no quantum violation from qubit unextendible product bases. Phys. Rev. A, 85:042113, 2012.
D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, Unextendible product bases, uncompletable product bases and bound entanglement, arXiv:quantph/9908070, 19992000.
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., 238:379410, 2003.
N. Johnston, In Search of a 4by11 Matrix
N. Johnston, The Structure of Qubit Unextendible Product Bases, arXiv:1401.7920 [quantph], 2014.


EXAMPLE

Triangle begins:
2
4
4 8
6 7 8 9 10 12 16
6 8 9 10
The 5th row of the triangle also contains 1226, 28, and 32, but it is unknown if it contains 11 or 27.
The 3rd row of the triangle contains the value 4 because there is a 3qubit unextendible product basis consisting of 4 states. If we use "ket" notation from quantum mechanics, then one such UPB is: 0>0>0>, +>1>>, 1>>+>, >+>1>. This is the "shifts" UPB from the DiVincenzo et al. paper.
Equivalently, the 3rd row of the triangle contains the value 4 because there is a 3 X 4 matrix M with the three properties given in the Comments section:
1 1 2 2
1 2 1 2
1 2 2 1


CROSSREFS

Cf. A211390.
Sequence in context: A140434 A107748 A005884 * A285326 A079890 A223592
Adjacent sequences: A229910 A229911 A229912 * A229914 A229915 A229916


KEYWORD

nonn,tabf,hard,more


AUTHOR

Nathaniel Johnston, Oct 03 2013


EXTENSIONS

a(10)a(15) from Nathaniel Johnston, Jan 30 2014


STATUS

approved



