Gemma de las Cuevas: Quantum theory: From the whole to the parts, magic squares and shadows of infinity

Quantum Information and Quantum Computing Seminars CTP PAS

The foundations of quantum theory rest on non-commutative spaces, where composition is given by the tensor product, positive elements play a distinguished role, and so do complex numbers. I will present our recent investigations on the interplay of these characters from three perspectives. First, I will present a framework to go from the whole to the parts, that is, to decompose elements of tensor product spaces while transferring invariance and positivity to the parts, including approximations. Second, I will talk about quantum magic squares and the fact that they cannot be “purified”, that is, dilated to quantum permutation matrices. Finally, I will talk about shadows of infinity: one is the shadow of the hyperreal numbers to the reals, where I will show that the existence of bound entangled states with a negative partial transpose can be solved in the hyperreals, and the other is the undecidability of a related problem, which can be seen as an infinite set not admitting a finite description.