Entropic constraints in quantum resource theories of magic and asymmetry

This thesis describes, quantifies and applies quantum information-processing resources that go beyond observables and generalised measurements (POVMs), and must instead be defined as monotones on the pre-order established within a resource theory. The research is divided into two parts, which examine monotones expressed in terms of generalised quantum entropies for the resource theories of magic and asymmetry respectively. These resources display counterintuitive behaviour signalling the breakdown of classical descriptions for quantum computing (in terms of statistical mechanics) and symmetry-constrained open system dynamics (in terms of Noether's Theorem).

Part I of the thesis concerns magic distillation. Previous work has cast universal fault-tolerant quantum computing by magic state injection for odd-prime-dimensional systems within a phase space setting wherein distillable magic states acquire negative alpha-Renyi entropies. We extend this statistical mechanics framework to the technologically important qubit case, from which we derive fundamental trade-off relations on parameters governing the performance of an elementary family of protocols that project onto CSS codes. These trade-off relations are tuned to physics specific to code projection protocols and can outperform previous monotone bounds on protocol parameters in regimes of practical interest.

Part II of this thesis concerns symmetry-constrained open system dynamics. Recently, a complete but infinite set of entropic monotones was found for the resource theory of asymmetry, given in terms of correlations with every state on a spontaneously emergent quantum reference system. We show that one can restrict to reference frames forming any surface enclosing the maximally mixed state, which implied that the possibility of state transition under symmetry-constrained general quantum channels can be determined by a single entropic minimality condition at the maximally mixed state. Building on this analysis, we provide simple, closed conditions on the minimal depolarization needed to make a quantum state accessible under channels covariant with symmetry groups whose multidimensional representations are multiplicity-free.