Quantum Information Theory with Gaussian Systems

1 Introduction
As quantum systems can behave radically different from classical systems, the concept
of information based on quantum mechanics opens up new possibilities for
the manipulation, storage and transmission of information. Quantum information
theory [1] explores these possibilities and transforms them into applications such as
quantum computation and quantum cryptography. For suitable problems, these concepts
can perform better than their classical counterparts. A prominent example is
the Shor algorithm [2], which factorizes integers efficiently on a quantum computer;
it is thus exponentially faster than the known classical algorithms.
Quantum information is encoded in the state of a quantum system. To obtain
results which are independent of a physical realization, quantum information theory
usually refers to the physical carriers of information only by an abstract description
based on quantum mechanics. The basic unit of quantum information is the qubit,
which in analogy to a classical bit is modeled as a generic two-level quantum system.
Fundamental features of quantum mechanics are linearity and the tensor product
structure of the Hilbert space formalism, which allow for coherent superpositions of
quantum states and entanglement, i.e. correlations which are stronger than classically
1 possible. Hence in contrast to a classical bit, a qubit can take on not only
logical values0and1, corresponding to the ground state and excited state, but
also any coherent superposition. While such effects enable an exponential speedup
in quantum computation, some tasks pose difficulties. In particular, it is impossible
to perfectly duplicate a quantum state. However, an approximate copying or
cloningcan be achieved, where the quality of the clones is strictly limited. This
implies that quantum information cannot be completely transformed into classical
information, because otherwise the classical information could be used to generate
multiple copies of the respective quantum state. However, quantum teleportation can
transmit quantum information by sending only classical information if in addition
sender and receiver share entangled states, which are used to restore the quantum
states from the classical data.
For the processing of quantum information, finite-dimensional systems, i.e. qubits
and generalizations to d-level systems, are perfectly suited. Moreover, they can be
implemented in a large variety of physical systems, without a leading contender
so far. The transmission of quantum information over large macroscopic distances,
however, is usually implemented by means of an optical scheme. In principle, single
photons can be used to carry qubits in their polarization degree of freedom. Unfortunately,
single photons are fragile objects which have to be treated with care and tend
to get lost. As an alternative, the information can be encoded into a mode of the
1 Read: in a local realistic model.
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