# Iso-entangled mutually unbiased bases, symmetric quantum measurements and mixed-state designs

## TL;DR

This paper constructs and analyzes special sets of quantum states called iso-entangled mutually unbiased bases in dimension four, demonstrating their properties as mixed-state 2-designs and exploring their applications in quantum measurements.

## Contribution

It provides an explicit construction of a complete set of five iso-entangled mutually unbiased bases in dimension four and establishes their role as mixed-state 2-designs.

## Key findings

- Constructed a complete set of five iso-entangled MUBs in dimension four.
- Showed that the reduced states form a mixed-state 2-design.
- Linked projective designs to mixed-state designs through partial traces and decoherence.

## Abstract

Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem whether a complete set of five iso-entangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these $20$ pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius $\sqrt{3/20}$ located inside the Bloch ball of radius $1/2$. Such a set forms a mixed-state $2$-design --- a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state $2$-design.