On asymptotically optimal hypotheses testing in quantum statistics

The mathematical formulation of a hypothesis testing problem in quantum statistics under consideration reduces to the following. Let $\{\psi_j\}$ be a given basis in a $d$-dimensional unitary space. Find an orthonormal basis $\{e_j\}$ which approximates the basis $\{\psi_j\}$ in the sense that the value of (1) is minimal. An asymptotic solution to this problem is given for «almost orthogonal» vectors $\psi_j$. An asymptotically optimal basis is $\widehat\psi_j=\Gamma^{-1/2}\psi_j$, where $\Gamma$ is the Gram operator of the system $\{\psi_j\}$.