PhD Defence Elias D. Theil

Title: Title: Unitary-equivariant and permutation-invariant quantum channels

Abstract: 

In this thesis we study quantum channels with unitary and permutation symmetries, their classification, implementation, and their performance for specific quantum information tasks.
We give a full classification of the set of unitary-equivariant and permutation-invariant quantum channels via their extremal points, together with efficient implementation algorithms for the examples of random permutations, quantum cloning and quantum purity amplification.
We also study the task of quantum purity amplification in detail, and we generalize the task to the amplification of an arbitrary eigenstate. We propose a quantum channel whose fidelity figure of merit has optimal asymptotics for both the all-site and one-site marginal outputs, and we provide non-asymptotic lower bounds that almost match the first-order asymptotics.
Finally, we embed the tasks of quantum cloning, quantum purity amplification and density matrix exponentiation into the larger framework of coherent inference. This framework describes quantum processes with quantum states as input and output, and we provide a sample-complexity separation between coherent and incoherent protocols for all three cases

Advisor:

Associate Professor Laura Mancinska, University of Copenhagen

Assessment committee:

Associate Professor Albert Werner (chair). University of Copenhagen

Professor Michael Walter, Ludwig Maximilian University of Munich, Germany

Professor Zoltán Zimborás University of Helsinki, Finland