A Continuity Theorem for Stinespring's Dilation

We show a continuity theorem for Stinespring's dilation: two completely positive maps between arbitrary C*-algebras are close in cb-norm iff we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to quantum information theory.Comment: 18 pages, no figure

Informationsübertragung durch Quantenkanäle

This PhD thesis represents work done between Aug. 2003 and Dec. 2006 in Reinhard F. Werner's quantum information theory group at Technische Universität Braunschweig, and Artur Ekert's Centre for Quantum Computation at the University of Cambridge. Quantum information science combines ideas from physics, computer science and information theory to investigate how quintessentially quantum mechanical effects such as superposition and entanglement can be employed for the handling and transfer of information. My thesis falls into the field of abstract quantum information theory, which is concerned with the fundamental resources for quantum information processing and their interconversion and tradeoffs. Every such processing of quantum information can be represented as a quantum channel: a completely positive and trace-preserving map between observable algebras associated to physical systems. This work investigates both fundamental properties of quantum channels (mostly in Chs. 3 and 4) and their asymptotic capacities for classical as well as quantum information transfer (in Chs. 5 through 8).Diese Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) entstand zwischen August 2003 und Dezember 2006 in Prof. Reinhard F. Werners Arbeitsgruppe Quanteninformationstheorie an der Technischen Universität Braunschweig und Prof. Artur Ekerts Centre for Quantum Computation an der Universität Cambridge. Die Quanteninformationswissenschaft untersucht mit den Ideen und Methoden der Physik, der Informatik und der Informationstheorie, wie sich charakteristisch quantenphysikalische Effekte, beispielsweise Superposition und Verschränkung, zur Verarbeitung und Übertragung von Information nutzbar machen lassen. Die vorliegende Dissertation fällt in das Gebiet der abstrakten Quanteninformationstheorie, die die grundlegenden Ressourcen für die Verarbeitung von Quanteninformation sowie deren Wechselbeziehungen und Abhängigkeiten untersucht. Eine jede solche Verarbeitung von Quanteninformation läßt sich mathematisch beschreiben als sogenannter Quantenkanal, eine vollständig positive und spurerhaltende Abbildung zwischen den physikalischen Systemen zugeordneten Observablen-Algebren. In dieser Arbeit werden sowohl grundlegende Eigenschaften solcher Quantenkanäle (vor allem in den Kap. 3 und Kap. 4) als auch ihre asymptotischen Kapazitäten für die Übertragung von klassischer Information und Quanteninformation (in Kap. 5 bis 8) untersucht

Quantum Channels with Memory

We present a general model for quantum channels with memory, and show that it is sufficiently general to encompass all causal automata: any quantum process in which outputs up to some time t do not depend on inputs at times t' > t can be decomposed into a concatenated memory channel. We then examine and present different physical setups in which channels with memory may be operated for the transfer of (private) classical and quantum information. These include setups in which either the receiver or a malicious third party have control of the initializing memory. We introduce classical and quantum channel capacities for these settings, and give several examples to show that they may or may not coincide. Entropic upper bounds on the various channel capacities are given. For forgetful quantum channels, in which the effect of the initializing memory dies out as time increases, coding theorems are presented to show that these bounds may be saturated. Forgetful quantum channels are shown to be open and dense in the set of quantum memory channels.Comment: 21 pages with 5 EPS figures. V2: Presentation clarified, references adde

Complementarity of Private and Correctable Subsystems in Quantum Cryptography and Error Correction

We make an explicit connection between fundamental notions in quantum cryptography and quantum error correction. Error-correcting subsystems (and subspaces) for quantum channels are the key vehicles for contending with noise in physical implementations of quantum information-processing. Private subsystems (and subspaces) for quantum channels play a central role in cryptographic schemes such as quantum secret sharing and private quantum communication. We show that a subsystem is private for a channel precisely when it is correctable for a complementary channel. This result is shown to hold even for approximate notions of private and correctable defined in terms of the diamond norm for superoperators.Comment: 5 pages, 2 figures, preprint versio

Reexamination of Quantum Bit Commitment: the Possible and the Impossible

Bit commitment protocols whose security is based on the laws of quantum mechanics alone are generally held to be impossible. In this paper we give a strengthened and explicit proof of this result. We extend its scope to a much larger variety of protocols, which may have an arbitrary number of rounds, in which both classical and quantum information is exchanged, and which may include aborts and resets. Moreover, we do not consider the receiver to be bound to a fixed "honest" strategy, so that "anonymous state protocols", which were recently suggested as a possible way to beat the known no-go results are also covered. We show that any concealing protocol allows the sender to find a cheating strategy, which is universal in the sense that it works against any strategy of the receiver. Moreover, if the concealing property holds only approximately, the cheat goes undetected with a high probability, which we explicitly estimate. The proof uses an explicit formalization of general two party protocols, which is applicable to more general situations, and a new estimate about the continuity of the Stinespring dilation of a general quantum channel. The result also provides a natural characterization of protocols that fall outside the standard setting of unlimited available technology, and thus may allow secure bit commitment. We present a new such protocol whose security, perhaps surprisingly, relies on decoherence in the receiver's lab.Comment: v1: 26 pages, 4 eps figures. v2: 31 pages, 5 eps figures; replaced with published version; title changed to comply with puzzling Phys. Rev. regulations; impossibility proof extended to protocols with infinitely many rounds or a continuous communication tree; security proof of decoherence monster protocol expanded; presentation clarifie

Tema Con Variazioni: Quantum Channel Capacity

Channel capacity describes the size of the nearly ideal channels, which can be obtained from many uses of a given channel, using an optimal error correcting code. In this paper we collect and compare minor and major variations in the mathematically precise statements of this idea which have been put forward in the literature. We show that all the variations considered lead to equivalent capacity definitions. In particular, it makes no difference whether one requires mean or maximal errors to go to zero, and it makes no difference whether errors are required to vanish for any sequence of block sizes compatible with the rate, or only for one infinite sequence.Comment: 32 pages, uses iopart.cl