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A229913
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Irregular triangle in which row n lists the possible sizes of n-qubit unextendible product bases.
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1
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2, 4, 4, 8, 6, 7, 8, 9, 10, 12, 16, 6, 8, 9, 10
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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nonn,tabf,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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