Bipartite quantum measurements with optimal single-sided
  distinguishability
We analyse orthogonal bases in a composite $N\times N$ Hilbert space
describing a bipartite quantum system and look for a basis with optimal
single-sided mutual state distinguishability. This condition implies that in
each subsystem the $N^2$ reduced states form a regular simplex of a maximal
edge length, defined with respect to the trace distance. In the case $N=2$ of a
two-qubit system our solution coincides with the elegant joint measurement
introduced by Gisin. We derive explicit expressions of an analogous
constellation for $N=3$ and provide a general construction of $N^2$ states
forming such an optimal basis in ${\cal H}_N \otimes {\cal H}_N$. Our
construction is valid for all dimensions for which a symmetric informationally
complete (SIC) generalized measurement is known. Furthermore, we show that the
one-party measurement that distinguishes the states of an optimal basis of the
composite system leads to a local quantum state tomography with a linear
reconstruction formula. Finally, we test the introduced tomographical scheme on
a complete set of three mutually unbiased bases for a single qubit using two
different IBM machines.
Jakub Czartowski Karol Życzkowski