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A211390
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The minimum cardinality of an n-qubit unextendible product basis.
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1
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2, 4, 4, 6, 6, 8, 8, 11, 10, 12, 12, 16, 14, 16, 16, 20, 18, 20, 20, 24, 22, 24, 24, 28, 26, 28, 28, 32, 30, 32, 32, 36, 34, 36, 36, 40, 38, 40, 40, 44, 42, 44, 44, 48, 46, 48, 48, 52, 50, 52, 52, 56, 54, 56, 56, 60, 58, 60, 60, 64, 62, 64, 64, 68, 66, 68, 68
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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.
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LINKS
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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; Commun. Math. Phys., 238:379-410, 2003.
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FORMULA
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a(n) = n + 1 if n is odd.
a(n) = n + 2 if n = 2 (mod 4) or if n = 4.
a(n) = n + 3 if n = 8.
a(n) = n + 4 otherwise (i.e., if n >= 12 and n = 0 (mod 4)).
G.f.: -x*(x^12-x^11+x^8-x^7+2*x^4-2*x^3-2*x-2) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 16 2013
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EXAMPLE
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a(2) = 4 because there is no nontrivial UPB on two qubits -- any UPB spans the entire 2^2 = 4-dimensional space.
a(3) = 4 because there is a 4-state UPB in 3-qubit space. 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.
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MAPLE
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a := proc(n) if(n mod 2 = 1)then return n + 1; elif(n = 4 or n mod 4 = 2)then return n + 2; elif(n = 8)then return 11; else return n + 4; fi: end: seq(a(n), n=1..67);
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MATHEMATICA
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Join[{2, 4, 4, 6, 6, 8, 8, 11}, LinearRecurrence[{1, 0, 0, 1, -1}, {10, 12, 12, 16, 14}, 60]] (* Jean-François Alcover, Nov 29 2017 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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