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%I #19 Apr 19 2022 21:50:59
%S 2,4,4,8,6,7,8,9,10,12,16,6,8,9,10
%N Irregular triangle in which row n lists the possible sizes of n-qubit unextendible product bases.
%C 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.
%C 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).
%C The first entry in row n is A211390(n).
%C The last entry in row n is 2^n.
%C The next-to-last entry in row n is 2^n - 4 (for n >= 3).
%H R. Augusiak, T. Fritz, M. Kotowski, M. Kotowski, M. Pawlowski, M. Lewenstein, and A. Acín, <a href="http://arxiv.org/abs/1112.3238">Tight Bell inequalities with no quantum violation from qubit unextendible product bases</a>. arXiv:1112.3238 [quant-ph], 2011-2012.
%H R. Augusiak, T. Fritz, M. Kotowski, M. Kotowski, M. Pawlowski, M. Lewenstein, and A. Acín, <a href="http://dx.doi.org/10.1103/PhysRevA.85.042113">Tight Bell inequalities with no quantum violation from qubit unextendible product bases</a>. Phys. Rev. A, 85:042113, 2012.
%H L. Chen and D. Z. Djokovic, <a href="https://arxiv.org/abs/1709.01232">Nonexistence of n-qubit unextendible product bases of size 2^n - 5</a>, arXiv:1709.01232 [quant-ph], 2017.
%H D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, <a href="http://arxiv.org/abs/quant-ph/9908070">Unextendible product bases, uncompletable product bases and bound entanglement</a>, arXiv:quant-ph/9908070, 1999-2000.
%H D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal, <a href="http://dx.doi.org/10.1007/s00220-003-0877-6">Unextendible product bases, uncompletable product bases and bound entanglement</a>, Commun. Math. Phys., 238:379-410, 2003.
%H N. Johnston, <a href="http://www.njohnston.ca/2013/10/in-search-of-a-4-by-11-matrix/">In Search of a 4-by-11 Matrix</a>
%H N. Johnston, <a href="http://arxiv.org/abs/1401.7920">The Structure of Qubit Unextendible Product Bases</a>, arXiv:1401.7920 [quant-ph], 2014.
%e Triangle begins:
%e 2
%e 4
%e 4 8
%e 6 7 8 9 10 12 16
%e 6 8 9 10
%e 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.
%e 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.
%e 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:
%e 1 -1 2 -2
%e 1 -2 -1 2
%e 1 2 -2 -1
%Y Cf. A211390.
%K nonn,tabf,hard,more
%O 1,1
%A _Nathaniel Johnston_, Oct 03 2013
%E a(10)-a(15) from _Nathaniel Johnston_, Jan 30 2014
%E Updated comments and references by _Nathaniel Johnston_, Apr 19 2022
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