Czartowski Jakub, Goyeneche Dardo, Grassl Markus, Życzkowski Karol
Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland.
Depto de Física, Fac. de Cs. Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile.
Phys Rev Lett. 2020 Mar 6;124(9):090503. doi: 10.1103/PhysRevLett.124.090503.
Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled 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.
希尔伯特空间中的离散结构在寻找量子测量的最优方案中起着关键作用。我们解决了在四维空间中是否存在一组完整的五个等纠缠相互无偏基的问题，并给出了明确的解析构造。构成这种广义量子测量的这20个纯态的约化密度矩阵形成了一个正则十二面体，该十二面体内接于半径为(\sqrt{3/20})的球体，该球体位于半径为(1/2)的布洛赫球内部。这样的一组态形成了一个混合态2 - 设计——一组具有这样性质的量子态离散集，即密度矩阵的任何二次函数的平均值等于相对于平坦的希尔伯特 - 施密特测度在整个混合态集合上的积分。我们建立了混合态设计需要满足的充分必要条件，并给出了构造它们的通用方法。此外，还表明复合希尔伯特空间中投影设计的部分迹形成一个混合态设计，而投影设计元素的退相干产生经典概率单纯形中的一个设计。我们确定了一个特殊的两比特正交基，使得四个约化态均匀分布在布洛赫球内并形成一个混合态2 - 设计。
