TY - BOOK
T1 - An Information-Theoretic Framework for Quantum Repeaters
AU - Ferrara, Roberto
PY - 2018
Y1 - 2018
N2 - Pure maximally entangled states are the most powerful resource provided by quantum mechanics. Entanglement distillation is the process of producing these states with highfidelity between two distant parties, starting from a source of noisy entanglement. The rate at which this can be done is called distillable entanglement. The maximally entangled states can be used for teleportation of quantum information, Bell inequality violation, and, most importantly from the point of view of this thesis, generation of perfect key. Maximally entangled states are pure and in product with the environment, and thus they guarantee that a simple local measurement done by the parties will produce perfect key, namely perfectly equal and perfectly random strings shared between the two distant parties, with the absolute guarantee that they will be secret to anybody else. The process of using quantum states to share perfect key between distant parties is called quantum key distribution, and the rate at which it can be done is called distillable key. It turns out that there exist noisy entangled states, the private states, that also lead to perfect key just by measurement, and that, most surprisingly the distillable key equals the distillation rate of such states. Proving this equivalence allowed the authors to show that distillable entanglement and distillable key can be very different. There even exists a low-dimensional experimental realization. However these are very peculiar correlations that have been shown only for parties that interact together directly on the noisy entanglement. In the future of quantum processing, and in the not so distant future of quantum key distribution, parties will be distributed in a network, where the entanglement will have to be generated by parties that are directly connected, and then relayed to arbitrary nodes using quantum teleportation. The intermediate parties are then known as quantum repeater stations, or simply quantum repeaters. In light of this, it is natural to ask how much the separation between distillable key and distillable entanglement extends to general network scenarios, and in particular whether it persists if we insert a repeater station between the two parties. In this thesis, we provide a new perspective on key distillation, and thus quantum key distribution, by relating private states to quantum data hiding, the phenomenon of having perfect classical correlations that are not accessible by separated parties. This provides a tool for the study of quantum key distribution involving intermediate repeater stations, where for the first time we are able to show a close connection with entanglement distillation. We show the first bounds on the distillable key in quantum repeaters in terms of the distillable entanglement of the nodes, holding in particular for private states, and thus for states that do indeed provide perfect key between the nodes. To develop the tools, we expand the understanding of private states. For them we provide a simplification, that allows us to connect the distillable entanglement and the repeater distillable key to the recoverability of classical information by local parties, when these private states are used as encoders. We, also show that in general, most private states will have low recoverability of this classical information, which is the intuition behind private states with low distillable entanglement, and show that, under mild assumptions, this implies a low key in some realistic repeater. Our results add toward the intuition that the distillable entanglement is the only relevant resource that survives the relay of quantum information.
AB - Pure maximally entangled states are the most powerful resource provided by quantum mechanics. Entanglement distillation is the process of producing these states with highfidelity between two distant parties, starting from a source of noisy entanglement. The rate at which this can be done is called distillable entanglement. The maximally entangled states can be used for teleportation of quantum information, Bell inequality violation, and, most importantly from the point of view of this thesis, generation of perfect key. Maximally entangled states are pure and in product with the environment, and thus they guarantee that a simple local measurement done by the parties will produce perfect key, namely perfectly equal and perfectly random strings shared between the two distant parties, with the absolute guarantee that they will be secret to anybody else. The process of using quantum states to share perfect key between distant parties is called quantum key distribution, and the rate at which it can be done is called distillable key. It turns out that there exist noisy entangled states, the private states, that also lead to perfect key just by measurement, and that, most surprisingly the distillable key equals the distillation rate of such states. Proving this equivalence allowed the authors to show that distillable entanglement and distillable key can be very different. There even exists a low-dimensional experimental realization. However these are very peculiar correlations that have been shown only for parties that interact together directly on the noisy entanglement. In the future of quantum processing, and in the not so distant future of quantum key distribution, parties will be distributed in a network, where the entanglement will have to be generated by parties that are directly connected, and then relayed to arbitrary nodes using quantum teleportation. The intermediate parties are then known as quantum repeater stations, or simply quantum repeaters. In light of this, it is natural to ask how much the separation between distillable key and distillable entanglement extends to general network scenarios, and in particular whether it persists if we insert a repeater station between the two parties. In this thesis, we provide a new perspective on key distillation, and thus quantum key distribution, by relating private states to quantum data hiding, the phenomenon of having perfect classical correlations that are not accessible by separated parties. This provides a tool for the study of quantum key distribution involving intermediate repeater stations, where for the first time we are able to show a close connection with entanglement distillation. We show the first bounds on the distillable key in quantum repeaters in terms of the distillable entanglement of the nodes, holding in particular for private states, and thus for states that do indeed provide perfect key between the nodes. To develop the tools, we expand the understanding of private states. For them we provide a simplification, that allows us to connect the distillable entanglement and the repeater distillable key to the recoverability of classical information by local parties, when these private states are used as encoders. We, also show that in general, most private states will have low recoverability of this classical information, which is the intuition behind private states with low distillable entanglement, and show that, under mild assumptions, this implies a low key in some realistic repeater. Our results add toward the intuition that the distillable entanglement is the only relevant resource that survives the relay of quantum information.
UR - https://rex.kb.dk/primo-explore/fulldisplay?docid=KGL01011939904&context=L&vid=NUI&search_scope=KGL&tab=default_tab&lang=da_DK
M3 - Ph.D. thesis
BT - An Information-Theoretic Framework for Quantum Repeaters
PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
ER -