Tensoring Positive Maps on Operator Structures

Descrizione del progetto

Unire i problemi matematici della teoria dell’informazione quantistica sotto una teoria comune

Molti problemi importanti della teoria dell’informazione quantistica possono essere formulati in termini di comportamento delle mappe lineari tra le algebre di matrici sotto le forze tensoriali. Tra gli esempi figurano il problema della distillazione, l’annichilazione locale dell’entanglement e la congettura della trasposizione parziale positiva al quadrato. Finora mancava una teoria generale per risolverli. Il progetto TIPTOP, finanziato dal programma di azioni Marie Skłodowska-Curie, formulerà tali problemi in termini di sistemi di operatori astratti. L’obiettivo è quello di comprendere più a fondo come le proprietà delle strutture dei sistemi di operatori influenzano le proprietà delle mappe lineari sotto le forze tensoriali e trovare impostazioni in cui solo gli esempi triviali di mappe completamente positive e copositive rimangono positive sotto qualsiasi forza tensoriale. In definitiva, il progetto esplorerà le impostazioni in cui i problemi di tensorizzazione diventano più facili.

Obiettivo

Many important problems in quantum information theory can be formulated in terms of how linear maps between matrix algebras behave under tensor powers. Examples include the distillation problem (fundamental for quantum communication), the problem of local entanglement annihilation (important for entanglement distribution in quantum networks), and the PPT squared conjecture important for (quantum key repeaters). Despite their importance for quantum communication, these problems are wide open, and no general theory is known for solving them. I realized that these problems can be formulated in the framework of abstract operator systems. Here, they correspond to characterizing which linear maps stay positive under tensor powers with respect to different operator system structures over the matrix algebras at the input and output. Completely positive maps and completely copositive maps (compositions of completely positive maps with a transposition) are always trivial examples, corresponding to known examples in quantum information theory. The question is, whether other examples exist. So far this type of problem has only been studied (indirectly) in the few special cases of operator systems over the matrix algebras corresponding to the above problems. There is a much richer theory of abstract operator systems (even over the matrix algebras) and different tensor products to combine them. In my project, I want to study such tensorization problems for other operator system structures over the matrix algebras and beyond. I want to understand how properties of these structures affect properties of linear maps under tensor powers, and find settings where only the trivial examples of completely positive and completely copositive maps stay positive under any tensor power. Finally, I aim to identify settings where tensorization problems become easier, and where I can construct examples of positive maps with properties we are currently lacking in quantum information theory.