From limits of quantum operations to multicopy entanglement witnesses and state-spectrum estimation

Abstract

Limits of state transformations in quantum mechanics are studied. Impossibility of physical implementation of the transformation ${\ensuremath{\varrho}}^{\ensuremath{\bigotimes}n}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{n}$ in quantum mechanics is proved. The most natural notions of structural completely positive approximation and structural physical approximations of nonphysical map are introduced. It is shown that these always exist for linear Hermitian maps and can be optimized under natural conditions. It is pointed out that it is physically possible to measure in a simple way the traces of $n\mathrm{th}$ power of quantum state $\mathrm{Tr}({\ensuremath{\varrho}}^{n})$ if only joint measurements on n copies of the system are allowed. This gives the interpretation of Tsallis entropies as values of ``multicopy'' observables. Following this observations, the notion of multicopy entanglement witnesses is defined and examples are provided. Finally, using notion of multicopy observable, a simple method of spectrum state estimation is discussed.