# Orthogonal Pairs and Mutually Unbiased Bases

- 41 Downloads
- 1 Citations

*The goal of our article is a study of related mathematical and physical objects: orthogonal pairs in* sl(*n*) *and mutually unbiased bases in* ℂ^{n}*. An orthogonal pair in a simple Lie algebra is a pair of Cartan subalgebras that are orthogonal with respect to the Killing form. The description of orthogonal pairs in a given Lie algebra is an important step in the classification of orthogonal decompositions, i.e., decompositions of the Lie algebra into a direct sum of Cartan subalgebras pairwise orthogonal with respect to the Killing form. One of the important notions of quantum mechanics, quantum information theory, and quantum teleportation is the notion of mutually unbiased bases in the Hilbert space* ℂ^{n}*. Two orthonormal bases* {*e*_{i}}_{i = 1}^{n}, {*f*_{j}}_{j = 1}^{n}*are mutually unbiased if and only if*\( {\left|\left\langle {e}_i\left|{f}_j\right.\right\rangle \right|}^2=\frac{1}{n} \)*for any i, j* = 1*,…, n. The notions of mutually unbiased bases in* ℂ^{n}*and orthogonal pairs in* sl(*n*) *are closely related. The problem of classification of orthogonal pairs in* sl(*n*) *and the closely related problem of classification of mutually unbiased bases in* ℂ^{n}*are still open even for the case n* = 6*. In this article, we give a sketch of our proof that there is a complex four-dimensional family of orthogonal pairs in* sl(6)*. This proof requires a lot of algebraic geometry and representation theory. Further, we give an application of the result on the algebraic geometric family to the study of mutually unbiased bases. We show the existence of a real four-dimensional family of mutually unbiased bases in* ℂ^{6}*, thus solving a long-standing problem. Bibliography:* 24 *titles.*

## Preview

Unable to display preview. Download preview PDF.

### References

- 1.A. Bondal and I. Zhdanovskiy, “Representation theory for system of projectors and discrete Laplace operator,” Preprint of IPMU, IPMU13-0001 (2013).Google Scholar
- 2.A. Bondal and I. Zhdanovskiy, “Simplectic geometry of unbiasedness and critical points of a potential,” arxiv:1507.00081.Google Scholar
- 3.A. Bondal and I. Zhdanovskiy, “Orthogonal pairs for Lie algebra
*sl*(6),” Preprint of IPMU, IPMU14-0296 (2014).Google Scholar - 4.P. O. Boykin, M. Sitharam, P. H. Tiep, and P. Wocjan, “Mutually unbiased bases and orthogonal decompositions of Lie algebras,”
*Quantum Inf. Comput.*,**7**, 371–382 (2007).MathSciNetMATHGoogle Scholar - 5.M. D. de Burgh, N. K. Landford, A. C. Doherty, and A. Gilchrist, “Choice of measurement sets in qubit tomography,”
*Phys. Rev. A*,**78**, 052122 (2008).CrossRefGoogle Scholar - 6.W. Crawley-Boevey, “Noncommutative deformations of Kleinian singularities,”
*Duke Math. J.*,**92**, No. 3, 605–635 (1998).MathSciNetCrossRefMATHGoogle Scholar - 7.W. Crawley-Boevey, “Geometry of the moment map for representations of quivers,”
*Compos. Math.*,**126**, 257–293 (2001).MathSciNetCrossRefMATHGoogle Scholar - 8.T. Durt, B.-G. Englert, I. Bengtsson, and K. Zyczkowski, “On mutually unbiased bases,”
*Int. J. Quantum Inform.*,**8**, 535–640 (2010).CrossRefMATHGoogle Scholar - 9.S. N. Filippov and V. I. Man’ko, “Mutually unbiased bases: tomography of spin states and the star-product scheme,”
*Phys. Scripta*,**2011**, T143.Google Scholar - 10.W. L. Gan and V. Ginzburg, “Deformed preprojective algebras and symplectic reflection for wreath products,”
*J. Algebra*,**283**, 350–363 (2005).MathSciNetCrossRefMATHGoogle Scholar - 11.U. Haagerup, “Orthogonal maximal abelian *-subalgebras of the
*n×n*matrices and cyclic*n*-roots,” in:*Operator Algebras and Quantum Field Theory*, International Press, Cambridge, Massachusetts (1996), pp. 296–322.Google Scholar - 12.C. D. Hacon, J. McKernan, and C. Xu, “On the birational automorphisms of varieties of general type,”
*Ann. Math.*,**177**, 1077–1111 (2013).MathSciNetCrossRefMATHGoogle Scholar - 13.D. N. Ivanov, “An analogue of Wagner’s theorem for orthogonal decompositions of the algebra of matrices
*M*_{n}(**C**),”*Uspekhi Mat. Nauk*,**49**, No. 1(225), 215–216 (1994).Google Scholar - 14.A. I. Kostrikin, T. A. Kostrikin, and V. A. Ufnarovskii, “Orthogonal decompositions of simple Lie algebras (type
*A*_{n}),”*Trudy Mat. Inst. Steklov.*,**158**, 105–120 (1981).MathSciNetMATHGoogle Scholar - 15.A. I. Kostrikin, I. A. Kostrikin, and V. A. Ufnarovskii, “On the uniqueness of orthogonal decompositions of Lie algebras of type
*A*_{n}and*C*_{n},”*Mat. Issled.*,**74**, 80–105 (1983).MathSciNetMATHGoogle Scholar - 16.A. I. Kostrikin and P. H. Tiep,
*Orthogonal Decompositions and Integral Lattices*, Walter de Gruyter (1994).Google Scholar - 17.G. Lima, L. Neves, R. Guzman, E. S. Gomez, W. A. T. Nogueira, A. Delgado, A. Vargas, and C. Saavedra, “Experimental quantum tomography of photonic qudits via mutually unbiased basis,”
*Optics Express*,**19**, No. 4, 3542–3552 (2011).Google Scholar - 18.M. Matolcsi and F. Szöllȍsi, “Towards a classification of 6
*×*6 complex Hadamard matrices,”*Open. Syst. Inf. Dyn.*,**15**, 93–108 (2008).MathSciNetCrossRefMATHGoogle Scholar - 19.K. Nomura, “Type II matrices of size five,”
*Graphs Combin.*,**15**, No. 1, 79–92 (1999).MathSciNetCrossRefMATHGoogle Scholar - 20.qig.itp.uni-hannover.de/qiproblems/Main Page.Google Scholar
- 21.M. B. Ruskai, “Some connections between frames, mutually unbiased bases, and POVM’s in quantum information theory,”
*Acta Appl. Math.*,**108**, No. 3, 709–719 (2009).MathSciNetCrossRefMATHGoogle Scholar - 22.F. Szöllȍsi, “Complex Hadamard matrices of order 6: a four-parameter family,”
*J. London Math. Soc.*,**85**, No. 3, 616–632 (2012).MathSciNetCrossRefMATHGoogle Scholar - 23.J. Thompson, “A conjugacy theorem for
*E*_{8},”*J. Algebra*,**38**, No. 2, 525–530 (1976).MathSciNetCrossRefMATHGoogle Scholar - 24.W. Tadej and K. Zyczkowski, “Defect of a unitary matrix,”
*Linear Algebra Appl.*,**429**, 447–481 (2008).MathSciNetCrossRefMATHGoogle Scholar