Stochastic matrices generated by entangled states of qubit-qutrit systems

Abstract

Considering the spin-tomogram of entangled qubit-qutrit states, three sets of stochastic matrices are generated. These matrices realize representations of three semigroups, which are mutually related. The representations are characterized by different Bell numbers — one of them is the Cirelson bound \(2\sqrt 2 \), and the other one corresponds (up to accuracy 10−4) to \(\sqrt 5 \). The correlations corresponding to the latter number are stronger than the classical ones and weaker than purely quantum ones. The nonpositive Hermitian matrix obtained by the positive partial transpose density matrix of an entangled qubit-qutrit state is shown to create under local unitary transformations the semigroup of stochastic 4×4 matrices having the same bounds \(\sqrt 5 \) and \(2\sqrt 2 \), which are the characteristics of the semigroup obtained from the nonnegative density matrix of the entangled state.