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.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.

arXiv:quant-ph/0502092Physical Review AMay 31, 200549 citations

Mutually Unbiased Bases

M. Kibler

Elsevier eBooksJan 1, 20170 citations

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

Gen Kimura, Hajime Tanaka, Masanao Ozawa

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.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.

arXiv:quant-ph/0604096Physical Review AMay 5, 200630 citations

On the mathematical foundations of mutually unbiased bases

Koen Thas

Quantum Information ProcessingDec 21, 20173 citations

Geometrically Underpinned Maximally Entangled States Bases

M. Revzen

Finite geometry is used to underpin finite, two d-dimensional particles Hilbert space, d=prime 6= 2. A central role is allotted to states with mutual unbiased bases (MUB) labeling. Dual affine plane geometry (DAPG) points underpin single particle, MUB labeled, product states. The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space. The relevance of mutually unbiased collective coordinates bases (MUCB) for dealing with maximally entangled states is discussed and shown to provide an economic alternative mode of study. These maximally entangled, geometrically reasoned states, provide the resource to a transparent solution to what may be termed tracking of the Mean King Problem (MKP): here Alice prepares a state measured by King along some orientation which Alice succeed in identifying with a subsequent measurement. Brief expositions of the topics considered: MUB, DAPG, MUCB and the MKP are included, rendering the paper self contained.Finite geometry is used to underpin finite, two d-dimensional particles Hilbert space, d=prime 6= 2. A central role is allotted to states with mutual unbiased bases (MUB) labeling. Dual affine plane geometry (DAPG) points underpin single particle, MUB labeled, product states. The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space. The relevance of mutually unbiased collective coordinates bases (MUCB) for dealing with maximally entangled states is discussed and shown to provide an economic alternative mode of study. These maximally entangled, geometrically reasoned states, provide the resource to a transparent solution to what may be termed tracking of the Mean King Problem (MKP): here Alice prepares a state measured by King along some orientation which Alice succeed in identifying with a subsequent measurement. Brief expositions of the topics considered: MUB, DAPG, MUCB and the MKP are included, rendering the paper self contained.

arXiv:1206.0356Jun 2, 20120 citations

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

d=10

dimensions even if the maximal number of MUB's exists in these dimensions.

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

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.