The Significance of the CC-Numerical Range and the Local CC-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^\dagger UAU^\dagger\}|. In quantum control of n qubits one may be interested (i) in having U\in SU(2^n) for the entire dynamics, or (ii) in restricting the dynamics to {\em local} operations on each qubit, i.e. to the n-fold tensor product SU(2)\otimes SU(2)\otimes >...\otimes SU(2). Interestingly, the latter then leads to a novel entity, the {\em local} C-numerical range W_{\rm loc}(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 (math-ph/0702005). We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools 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.Comment: update relating to math-ph/070200

Plant species with extremely small populations (PSESP) in China: A seed and spore biology perspective

AbstractApproximately one fifth of the world's plants are at risk of extinction. Of these, a significant number exist as populations of few individuals, with limited distribution ranges and under enormous pressure due to habitat destruction. In China, these most-at-risk species are described as ‘plant species with extremely small populations’ (PSESP). Implementing conservation action for such listed species is urgent. Storing seeds is one of the main means of ex situ conservation for flowering plants. Spore storage could provide a simple and economical method for fern ex situ conservation. Seed and spore germination in nature is a critical step in species regeneration and thus in situ conservation. But what is known about the seed and spore biology (storage and germination) of at-risk species? We have used China's PSESP (the first group listing) as a case study to understand the gaps in knowledge on propagule biology of threatened plant species. We found that whilst germination information is available for 28 species (23% of PSESP), storage characteristics are only known for 8% of PSESP (10 species). Moreover, we estimate that 60% of the listed species may require cryopreservation for long-term storage. We conclude that comparative biology studies are urgently needed on the world's most threatened taxa so that conservation action can progress beyond species listing