The smooth entropy formalism for von Neumann algebras (Q2786618)

The goal of the paper is to generalize the smooth entropy formalism -- defined in [\textit{R. Renner}, Security of quantum key distribution. Zürich: ETH Zürich (Ph.D. Thesis) (2005), \url{arXiv:0512258}] for finite-dimensional systems -- to a more large field of quantum mechanics, like von Neumann algebras. The authors show that in this way some problems characteristic to infinite-dimensional systems (e.g., quantum fields of bosons and fermions) can be approached using a theoretical framework.NEWLINENEWLINEThe notions of min- and max- entropy (conditional and unconditional, smooth conditional, min- and max-relative entropy) are redefined and extended in Chapter 3 for the particular case of von Neumann algebras. Beside the new forms, some general properties are proved (the local operations cannot increase the knowledge about systems, smooth conditional min- and max- entropy are finite, relative entropy is monotone under quantum channels).NEWLINENEWLINEChapter 4 is dedicated to the study of some applications to quantum information theory for von Neumann algebras of the form \({\mathcal M}_{AB}={\mathcal B}(\mathbb{C}^n)\otimes {\mathcal M}(B)\) (that is the \(A\)-system is finite dimensional while the \(B\)-system is general). Different subsections prove short properties concerning: optimal fidelity (entanglement and decoupling) which generalize some other results from literature, the final result being an ordering of min-/max-entropies (Subsection B), conditional min- and max-entropy in quantum systems (Subsection C), entropic uncertainty relations with quantum side information (Subsection C), where is proved perhaps the most important result of the paper (Proposition 19) that gives a measurement of uncertainty in terms of the smooth conditional min- and max- entropy (results continued by authors in [the second author et al., J. Math. Phys. 55, No. 12, 122205, 30 p. (2014; Zbl 1309.81116)]), and Subsection D where a completely new direction towards information security (quantum key distribution, privacy, data compression) in infinite states quantum computing is open.