  Limits of nonlinear in quantum mechanics are studied. Impossibility of
physical implementation of the transformation $\varrho^{\otimes n} \to
\varrho^{n}$ in quantum mechanics is proved. For sake of further analysis the
simplest notion of structural completely positive approximation (SCPA) and
structural physical approximations (SPA) of unphysical map are introduced. Both
always exist for linear hermitian maps and can be optimised under natural
assumptions. However it is shown that some intuitively natural SPA of the
nonlinear operation $\varrho^{\otimes 2} \to \varrho^{2}$ that was already
proven to be unphysical is impossible. It is conjectured that there exist no
SPA of the operation $\varrho^{\otimes n} \to \varrho^{n}$ at all. It is
pointed out that, on the other hand, it is physically possible to measure the
trace of the second power of the state $Tr(\varrho^{2})$ if only two copies of
the system are available. This gives the interpretation of one of Tsallis
entropy as mean value of some ``multicopy'' observable. The (partial)
generalisation of this idea shows that each of higher order Tsallis entropies
can be measured with help of only two multicopy observables. Following this
observations the notion of multicopy entanglement witnesses is defined and
first example is provided. Finally, with help of multicopy observables simple
method of spectrum state estimation is pointed out and discussed.
