Hermitian Schmidt decomposition and twin observables of bipartite mixed states

The study of mixed-state quantum correlations in terms of opposite-subsystem observables - the measurement of one of which amounts to the same as that of the other - of so-called twins is continued. Twin events that imply biorthogonal mixing of states, called 'strong twin events', are studied. It is shown that for each mixed state there exists a Schmidt (super-state-vector) decomposition in terms of Hermitian operators, and that it can be the continuation of the above-mentioned biorthogonal mixing due to strong twins. The case of weak twins and non-Hermitian Schmidt decomposition is also investigated. For separable states a necessary and sufficient condition for the existence of nontrivial twins is derived. Utilization of the Hermitian Schmidt decomposition for finding all twins is illustrated in full detail for the case of the two spin-half-particle states with maximally disordered subsystems (mixtures of Bell states). It is shown that only rank-two mixtures have nontrivial twins. (author)