Mutually unbiased bases and Hadamard matrices of order six

被引:95
作者
Bengtsson, Ingemar [1 ]
Bruzda, Wojciech
Ericsson, Asa
Larsson, Jan-Ake
Tadej, Wojciech
Zyczkowski, Karol
机构
[1] Stockholm Univ, AlbaNova, S-10691 Stockholm, Sweden
[2] Jagiellonian Univ, Inst Fizyki Smoluchowskiego, PL-30059 Krakow, Poland
[3] Linkoping Univ, Inst Math, S-58183 Linkoping, Sweden
[4] Univ Kardynala Stefana, Szkola Nauk Scislych, Wydziat Matemat Przyrodinczy, PL-01815 Warsaw, Poland
[5] Polish Acad Sci, Ctr Fizyki Teoret, PL-02668 Warsaw, Poland
关键词
D O I
10.1063/1.2716990
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought. (c) 2007 American Institute of Physics.
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页数:21
相关论文
共 31 条
[1]   COMPLEX SEQUENCES WITH LOW PERIODIC CORRELATIONS [J].
ALLTOP, WO .
IEEE TRANSACTIONS ON INFORMATION THEORY, 1980, 26 (03) :350-354
[2]  
[Anonymous], 2004, PROC ERATO C QUANT I
[3]   There is no generalization of known formulas for mutually unbiased bases [J].
Archer, C .
JOURNAL OF MATHEMATICAL PHYSICS, 2005, 46 (02)
[4]   The limitations of nice mutually unbiased bases [J].
Aschbacher, Michael ;
Childs, Andrew M. ;
Wocjan, Pawel .
JOURNAL OF ALGEBRAIC COMBINATORICS, 2007, 25 (02) :111-123
[5]  
BEAUCHAMP K, COMMUNICATION
[6]  
BEAUCHAMP K, MATHOA0609076
[7]  
Bengtsson I., 2017, Geometry of quantum states: an introduction to quantum entanglement, V2nd, DOI DOI 10.1017/9781139207010
[8]   A FASTER WAY TO COUNT THE SOLUTIONS OF INHOMOGENEOUS SYSTEMS OF ALGEBRAIC EQUATIONS, WITH APPLICATIONS TO CYCLIC N-ROOTS [J].
BJORCK, G ;
FROBERG, R .
JOURNAL OF SYMBOLIC COMPUTATION, 1991, 12 (03) :329-336
[9]  
BJORCK G, 1995, CR ACAD SCI I-MATH, V320, P319
[10]   Z(4)-Kerdock codes, orthogonal spreads, and extremal euclidean line-sets [J].
Calderbank, AR ;
Cameron, PJ ;
Kantor, WM ;
Seidel, JJ .
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 1997, 75 :436-480