Quantum Information Theory with Gaussian Systems

Summary
This thesis applies ideas and concepts from quantum information theory to systems
of continuous-variables such as the quantum harmonic oscillator. In particular, it is
concerned with Gaussian states and Gaussian systems, which transform Gaussian
states into Gaussian states. While continuous-variable systems in general require an
infinite-dimensional Hilbert space, Gaussian states can be described by a finite set of
parameters. This reduces the complexity of many problems, which would otherwise
be hardly tractable. Moreover, Gaussian states and systems play an important role
in today’s experiments with continuous-variable systems, e.g. in quantum optics.
Examples of Gaussian states are coherent, thermal and squeezed states of a light field
mode. The methods utilized in this thesis are based on an abstract characterization
of Gaussian states, the results thus do not depend on the particular physical carriers
of information.
The focus of this thesis is on three topics: the cloning of coherent states, Gaussian
quantum cellular automata and Gaussian private channels. Correspondingly, the
main part of the thesis is divided into three chapters each of which presents the
results for one topic:
3 Cloning An unknown quantum state can in general not be duplicated perfectly.
This impossibility is a direct consequence of the linear structure of quantum mechan-
ics and enables quantum key distribution. The approximate copying orcloning
of quantum states is possible, though, and raises questions about optimal cloning.
Bounds on the fidelity of cloned states provide restrictions and benchmarks for other
tasks of quantum information: In quantum key distribution, bounds on cloning fidelities
allow to estimate the maximum information an eavesdropper can get from
intercepting quantum states in relation to noise detected by the receiver. Beyond
that, any communication task which aims at the complete transmission of quantum
states has to beat the respective cloning limits, because otherwise large amounts of
information either remain at the sender or are dissipated into the environment.
Cloning was investigated both for finite-dimensional and for continuous-variable
systems. However, results for the latter were restricted to covariant Gaussian operations.
This chapter presents a general optimization of cloning operations for coherent
input states with respect to fidelity. The optimal cloners are shown to be covariant
with respect to translations of the input states in phase space. In contrast to the
finite-dimensional case, optimization of the joint output state and of weighted combinations
of individual clones yields different cloners: while the former is Gaussian, the
latter is not. The optimal fidelities are calculated analytically for the joint case and
numerically for the individual judging of two clones. For classical cloning, the opti-
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