# Summary

**Sum****mary**** of our Research Projects**

Necessary and sufficient conditions for Discord. We recently presented necessary and sufficient conditions for the existence of quantum discord, that also naturally lead to a new geometrical way of quantifying discord. Our results are general, holding for any number of subsystems of any dimensionality. This work also stimulated further experimental work on discord using optical implementations (both discrete and continuous variable). | |

Entropy and statistical mechanics. A thermodynamical interpretation of negative conditional entropies in quantum mechanics was presented. We are also modifying statistical mechanics to make it more suitable to nano- and quantum systems where fluctuations from the average are significant. | |

Geometric local invariants. We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or “gauge” invariant is associated with a distinct closed path (or plaquette) joining some or all of the qubits. In our approach the points of the lattice are qubits, the link transformations between the qubits are defined by the correlations between them, and the gauge invariant observable, the local invariants associated with a particular closed path, are also given by a Wilson looplike construction. The link transformations share many of the properties of parallel transporters, although they are not undone when one retraces one’s steps through the lattice. This feature is used to generate many of the invariants. We give an operational interpretation of these invariants in terms of observables. | |

Electron Transport in Complex I. We have been studying electron transport processes in the mitochondria both theoretically and experimentally. Electron transfer in biology is often mediated by molecules containing metallic redox centers, commonly known as metalloproteins. The cellular respiration of eukaryotes, for instance, involves electron transfer over several enzymes containing metallic cofactors. Respiratory complex I (RC-I) is one of four such membrane-bound enzymes involved in cellular respiration within mitochondria and respiring bacteria. Electron transport within RC-I is coupled to the partial generation of an electrochemical potential, which in turn is used in the synthesis of adenosine triphosphate (ATP), the molecule that provides energy for life. We have developed a new model to describe the electron transport as well as started to perform experiments to understand the vibrational modes better. | |

Quantum coherence in bird navigation. We showed that there is plenty of evidence that the quantum coherence time (and even entanglement) in the electron pair inside the Cryptochrome molecule in the retina of European Robins https://www.yourcanadianmeds.com/product/viagra/ pharmacy https://www.antibioticsonline.org/ medication tramadol online lasts for up to 100 micro-seconds. This result made an impact because it is difficult to keep an electron spin coherent for that long even artificially in highly insulated and controlled Carbon nanotubes. |