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A229913
Irregular triangle in which row n lists the possible sizes of n-qubit unextendible product bases.
1
2, 4, 4, 8, 6, 7, 8, 9, 10, 12, 16, 6, 8, 9, 10
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 n-qubit 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 next-to-last entry in row n is 2^n - 4 (for n >= 3).
LINKS
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 [quant-ph], 2011-2012.
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.
L. Chen and D. Z. Djokovic, Nonexistence of n-qubit unextendible product bases of size 2^n - 5, arXiv:1709.01232 [quant-ph], 2017.
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:quant-ph/9908070, 1999-2000.
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.
N. Johnston, The Structure of Qubit Unextendible Product Bases, arXiv:1401.7920 [quant-ph], 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 12--26, 28, and 32, but it is unknown if it contains 11. Chen and Djokovic showed that it does not contain 27 (see links). More generally, they showed that the n-th row does not contain 2^n - 5.
The 3rd row of the triangle contains the value 4 because there is a 3-qubit 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: A338506 A353190 A005884 * A285326 A355467 A079890
KEYWORD
nonn,tabf,hard,more
AUTHOR
Nathaniel Johnston, Oct 03 2013
EXTENSIONS
a(10)-a(15) from Nathaniel Johnston, Jan 30 2014
Updated comments and references by Nathaniel Johnston, Apr 19 2022
STATUS
approved