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We analyse orthogonal bases in a composite $N\times N$ Hilbert spacedescribing a bipartite quantum system and look for a basis with optimalsingle-sided mutual state distinguishability. This condition implies that ineach subsystem the $N^2$ reduced states form a regular simplex of a maximaledge length, defined with respect to the trace distance. In the case $N=2$ of atwo-qubit system our solution coincides with the elegant joint measurementintroduced by Gisin. We derive explicit expressions of an analogousconstellation for $N=3$ and provide a general construction of $N^2$ statesforming such an optimal basis in ${\cal H}_N \otimes {\cal H}_N$. Ourconstruction is valid for all dimensions for which a symmetric informationallycomplete (SIC) generalized measurement is known. Furthermore, we show that theone-party measurement that distinguishes the states of an optimal basis of thecomposite system leads to a local quantum state tomography with a linearreconstruction formula. Finally, we test the introduced tomographical scheme ona complete set of three mutually unbiased bases for a single qubit using twodifferent IBM machines.

Bipartite quantum measurements with optimal single-sided distinguishability