Talk 3.    Iso-entangled Mutually Unbiased Bases and Mixed-state t-designs

A Euclidean design is a finite set of points in Rn with certain approximation properties. This concept can be generalized to give the notion of pure-state designs: vectors selected in such a way that the average values of the first t moments averaged over the design coincide with the moment  obtained by integrating over the set of all states. We introduce a related notion of a mixed-state t-design, which consist of a finite number of density matrices of order N with analogous properties. Mixed-state designs can be obtained by partial trace of pure states of size N2 forming a pure state design. We construct a set of five iso-entangled Mutually Unbiased Bases (MUBs) in H4, consisting of 20 pure states with the same degree of entanglement. Performing a partial trace on any of both subsystems leads to 20 single-qubit mixed states which form a regular dodecahedron inside the Bloch ball and a mixed 3-design [3].

[3] S. Denisov, T. Laptyeva, W. Tarnowski, D. Chruściński and  K. Życzkowski, Universal spectra of random Lindblad operators, Phys. Rev. Lett. 123, 140403 (2019).