, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parameterization of the genuinely quantum solutions of

4 \times 4

SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8 and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet for this special dimension. In general, we proved that for any

N

it is possible to find an

N^2 \times N^2

SudoQ solution of cardinality

N^4

, which for a prime

N

is related to a set of

N

mutually unbiased bases of size

N^2

. Such a construction of

N^4

different vectors of size

N

yields a set of

N^3

orthogonal measurements.

DOI: 10.48550/arxiv.2106.02967

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N^2

that have cardinality greater than

N^2

, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parameterization of the genuinely quantum solutions of

4 \times 4

N

it is possible to find an

N^2 \times N^2

SudoQ solution of cardinality

N^4

, which for a prime

N

is related to a set of

N

mutually unbiased bases of size

N^2

. Such a construction of

N^4

different vectors of size

N

yields a set of

N^3

orthogonal measurements.

{N}^{2}

that have cardinality greater than

{N}^{2}

, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parametrization of the genuinely quantum solutions of a $4\ifmmode\times\else\texttimes\fi{}4$ SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8, and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet [I. Nechita and J. Pillet, Quantum Inf. Comput. 21, 781 (2021)] for this special dimension. In general, we proved that for any

N

it is possible to find an ${N}^{2}\ifmmode\times\else\texttimes\fi{}{N}^{2}$ SudoQ solution of cardinality

{N}^{4}

, which for a prime

N

is related to a set of

N

mutually unbiased bases of size

{N}^{2}

. Such a construction of

{N}^{4}

different vectors of size

N

yields a set of

{N}^{3}

orthogonal measurements.

We consider the notion of unitary transformations forming bases for subspaces of

M(d,\mathbb{C})

such that the square of Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case, weconstruct the maximal number of such bases for the 4 and 2 dimensional subspaces while proving the nonexistence of such a construction for the 3 dimensional case. Extending this to higher dimensions, we commit to such a construct for the case of qutrits and provide evidence for the existence of such unitaries for prime dimensional quantum systems. Focusing on the qubit case, we show that the average fidelity for estimating any of such a transformation is equal to the case for estimating a completely unknown unitary from

\text{SU}(2)

. This is then followed by a quick application for such unitaries in a quantum cryptographic setup.

We consider the notion of unitary transformations forming bases for subspaces of

M(d,\mathbb{C})

such that the square of the Hilbert-Schmidt inner product of matrices from the differing bases is a constant. Moving from the qubit case, we construct the maximal number of such bases for the four- and two-dimensional subspaces while proving the nonexistence of such a construction for the three-dimensional case. Extending this to higher dimensions, we commit to such a construct for the case of qutrits and provide evidence for the existence of such unitaries for prime dimensional quantum systems. Focusing on the qubit case, we show that the average fidelity for estimating any such transformation is equal to the case for estimating a completely unknown unitary from

\text{SU}(2)

. This is then followed by a quick application for such unitaries in a quantum cryptographic setup.