Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d+1 such bases are known for systems of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}.Mutually unbiased bases (MUBs) are a primitive used in quantum information processing to capture the principle of complementarity. While constructions of maximal sets of d+1 such bases are known for systems of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. In this paper we demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. We also give a new and simple proof that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}.

arXiv:quant-ph/0502031Feb 4, 20052 citations

Mutually unbiased bases, spherical designs, and frames

Andreas Klappenecker, Martin Roetteler

The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIEAug 18, 200524 citations

The algebraic structure of Mutually Unbiased Bases

Joanne L. Hall, Asha Rao

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in Copfdare known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that sets of complete MUBs only exist in Copfdif a projective plane of size d also exists. We investigate the structure of MUBs using two algebraic tools: relation algebras and group rings. We construct two relation algebras from MUBs and compare these to relation algebras constructed from projective planes. We show several examples of complete sets of MUBs in Copfd, that when considered as elements of a group ring form a commutative monoid. We conjecture that complete sets of MUBs will always form a monoid if the appropriate group ring is chosen.Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in Copfdare known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that sets of complete MUBs only exist in Copfdif a projective plane of size d also exists. We investigate the structure of MUBs using two algebraic tools: relation algebras and group rings. We construct two relation algebras from MUBs and compare these to relation algebras constructed from projective planes. We show several examples of complete sets of MUBs in Copfd, that when considered as elements of a group ring form a commutative monoid. We conjecture that complete sets of MUBs will always form a monoid if the appropriate group ring is chosen.

International Symposium on Information Theory and its ApplicationsDec 1, 20083 citations

Efficient 2-designs from bases exist

Gary McConnell, D. Gross

We show that in a complex $d$ -dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have many applications in quantum information theory (e.g.\ in state tomography, cloning, or cryptography) -- however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established, which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.We show that in a complex $d$-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have many applications in quantum information theory (e.g.\ in state tomography, cloning, or cryptography) -- however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established, which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.

arXiv:0710.1502Quantum Information and ComputationSep 1, 20089 citations

Mutually orthogonal Latin squares from the inner products of vectors in mutually unbiased bases

Joanne L. Hall, Asha Rao

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.

Journal of Physics A Mathematical and TheoreticalMar 10, 20109 citations

Mutually unbiased bases as minimal Clifford covariant 2-designs

Huangjun Zhu

Mutually unbiased bases (MUBs) are interesting for various reasons. The most attractive example of (a complete set of) MUBs is the one constructed by Ivanovi\ifmmode \acute{c}\else \'{c}\fi{} as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the orbit with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUBs provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.Mutually unbiased bases (MUBs) are interesting for various reasons. The most attractive example of (a complete set of) MUBs is the one constructed by Ivanovi\ifmmode \acute{c}\else \'{c}\fi{} as well as Wootters and Fields, which is referred to as the canonical MUB. Nevertheless, little is known about anything that is unique to this MUB. We show that the canonical MUB in any prime power dimension is uniquely determined by an extremal orbit of the (restricted) Clifford group except in dimension 3, in which case the orbit defines a special symmetric informationally complete measurement (SIC), known as the Hesse SIC. Here the extremal orbit is the orbit with the smallest number of pure states. Quite surprisingly, this characterization does not rely on any concept that is related to bases or unbiasedness. As a corollary, the canonical MUB is the unique minimal 2-design covariant with respect to the Clifford group except in dimension 3. In addition, these MUBs provide an infinite family of highly symmetric frames and positive-operator-valued measures (POVMs), which are of independent interest.

arXiv:1505.01123Physical Review AJun 30, 201524 citations

We show that in a complex

d

-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs). MUBs have many applications in quantum information theory (e.g.\ in state tomography, cloning, or cryptography) -- however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established, which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.