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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 (list; graph; refs; listen; history; text; internal format)
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

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 [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.

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, In Search of a 4-by-11 Matrix

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 or 27.

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: 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

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Last modified November 20 02:57 EST 2018. Contains 317371 sequences. (Running on oeis4.)