The Significance of the C-Numerical Range and the Local C-Numerical Range in Quantum Control and Quantum Information

This paper shows how C-numerical-range related new strucures may arise from practical problems in quantum control—and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function | tr{C†UAU†}|. In quantum control of N qubits one may be interested (i) in having U ∈ SU(2 ) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e. to the N-fold tensor product SU(2)⊗ SU(2)⊗ · · · ⊗ SU(2). Interestingly, the latter then leads to a novel entity, the local C-numerical range Wloc(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying paper on Relative C-Numerical Ranges and Local C-Numerical Ranges for Application in Quantum Computing [1]. We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: they serve as powerful tools (1) for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn’s famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. We conclude by relating the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient flow algorithms.