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arXiv:quant-ph/0604096 · Physical Review A

Solution to the mean king’s problem with mutually unbiased bases for arbitrary levels

Gen Kimura, Hajime Tanaka, Masanao Ozawa

Published May 5, 2006• 30 citations
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Abstract

The mean king's problem with mutually unbiased bases is reconsidered for arbitrary $d$-level systems. Hayashi et al. [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of $d\ensuremath{-}1$ mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when, e.g., $d=6$ or $d=10$. In contrast to their result, we show that the king's problem always has a solution for arbitrary levels if we also allow positive operator-valued measures. In constructing the solution, we use orthogonal arrays in combinatorial design theory.

DOI
10.1103/physreva.73.050301

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The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension $d$ is investigated. It is shown that a solution of the problem exists if and only if the maximal number $(d+1)$ of orthogonal Latin squares exists. This implies that there is no solution in $d=6$ or $d=10$ dimensions even if the maximal number of MUB's exists in these dimensions.

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Comment on “Mutually unbiased bases, orthogonal Latin squares, and hidden-variable models”

Joanne L. Hall, Asha Rao

In a recent article Paterek, Daki\ifmmode \acute{c}\else \'{c}\fi{}, and Brukner [Phys. Rev. A 79, 012109 (2009)] show an algorithm for generating mutually unbiased bases from sets of orthogonal Latin squares. They claim that this algorithm works for every set of orthogonal Latin squares. We show that the algorithm only works for particular sets of orthogonal Latin squares. Furthermore, the algorithm is a more readable version of work previously published [Phys. Rev. A 70, 062101 (2004)].

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On the Maximal Number of Real Mutually Unbiased Bases

Paweł Wocjan

In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.

•Feb 3, 2005•1 citation

New construction of mutually unbiased bases in square dimensions

WocjanPaweł, BethThomas

We show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines th...

Quantum Information and Computation•Mar 1, 2005•1 citation

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.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 18, 2005•24 citations
The mean king's problem with mutually unbiased bases is reconsidered for arbitrary ddd-level systems. Hayashi et al. [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of d\ensuremath−1d\ensuremath{-}1d\ensuremath−1 mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when, e.g., d=6d=6d=6 or d=10d=10d=10. In contrast to their result, we show that the king's problem always has a solution for arbitrary levels if we also allow positive operator-valued measures. In constructing the solution, we use orthogonal arrays in combinatorial design theory.
The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension ddd is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1)(d+1)(d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6d=6d=6 or d=10d=10d=10 dimensions even if the maximal number of MUB's exists in these dimensions.
In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.
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.