Quantum Information Theory with Gaussian Systems

3 Optimal cloners for coherent states
This chapter is concerned with optimizing the deterministic cloning of coherent
states, i.e. the approximate duplication of such quantum states. A general feature
of quantum physics is the impossibility of perfect duplication of an unknown quantum
state. On the one hand, this is a direct consequence of the linear structure of
quantum mechanics [1,26,27]. On the other hand, it is also related to a whole set
of impossible tasks in quantum mechanics 1 [28]: Given two identical copies of the
same quantum state, one could in principle obtain perfect measurement results for
two noncommuting observables, which is impossible by virtue of a Heisenberg uncertainty
[29]. However, it is possible to turn an unknown input quantum state and a
fixed initial quantum state into two approximate duplicates of the input state. The
quality of these clones is inversely related to each other: the better one resembles
the input state, the worse does the other. This relation can be strictly quantified in
terms of bounds on the cloning quality.
The field of quantum information has turned the impossibility of perfect cloning
into a key feature of secure quantum communication, because it allows to detect
essentially any eavesdropping on a transmission line from the degradation of the
output. It is thus possible to give estimations of the security of the exchanged information,
which is an important element of quantum key distribution (see e.g.
[30,31,32,47] for qkd with coherent states). In addition, bounds on the cloning quality
provide criteria to determine the validity of other protocols, since they cannot
possibly imply a violation of these bounds. A positive example is given in Section 3.6,
where we argue that violation of the cloning bounds necessarily implies certain success
criteria for quantum teleportation.
A general cloning map, acloner, turns m identical copies, i.e. an m-fold tensor
product, of an input state into n > m output states orclones, which resemble the
input state. In contrast to the input state, the overall output state might contain
correlations between the clones. The quality of the output states is measured in
terms of a figure of merit, a functional which compares the output states to the
input state. Usually, this is the fidelity, i.e. the overlap between input and output
states. Depending on whether one considers individual clones or compares the joint
output of the cloner with an n-fold tensor product of perfect copies of the input
state, we call the respective figures of merit either single-copy or joint fidelity. In
case the quality of the output states is identical, the cloner is called symmetric. It
is universal if the quality of the clones does not depend on the input state.
The cloning of finite-dimensional pure states was investigated thoroughly, e.g.
in [33,34,35,36,37,38,39]. Optimal universal cloners exist [33,34,35], which replicate
1 The impossibility of these tasks is not limited to quantum mechanics, but prevails in any
nonsignaling theory with violation of Bell’s inequalities.
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