Quantum Information Theory with Gaussian Systems

Summary
mum is reached by a measurement and preparation of coherent states. The bound
on classical cloning is turned into a criterion for the successful transmission of a
coherent state by quantum teleportation.
4 Quantum Cellular Automata Quantum cellular automata (qcas) are a model
for universal quantum computation in translationally invariant lattice systems with
localized dynamics. They provide an alternative concept for experimental realization
of quantum computing as they do not require individual addressing of their
constituent systems but rather rely on global parameters for the dynamics. Quantum
cellular automata seem to be particularly fitted for implementation in optical
lattices as well as for the simulation of lattice systems from statistical mechanics.
For this purpose the qca should be able to reproduce the ground state of a different
dynamics, preferably by driving an initial state into a suitable stationary state in
the limit of large time.
This chapter investigates abstract Gaussian qcas with respect to irreversibility.
As a basis, it provides methods to deal with translationally invariant systems on
infinite lattices with localization conditions. A simple example of a reversible Gaussian
qca (a nonsqueezing dynamics with nearest-neighbor interaction on the infinite
linear chain of harmonic oscillators) proves that even reversible qcas show aspects of
irreversibility. In addition, we characterize the stationary states for this type of dynamics.
While reversible qcas exhibit properties which make their characterization
particularly convenient both for finite-dimensional and Gaussian continuous-variable
systems, the definition of irreversible qcas causes problems. Gaussian systems provide
a testbed to illuminate these difficulties. We present different concepts of localization
and their impact on the requirements in the definition of qcas.
5 Private Quantum Channels Besides the generation of classical keys for encryption,
quantum cryptography provides a scheme to encrypt quantum information by
a one-time pad with classical key. The elements of the key are in one-to-one correspondence
with the elements of a finite set of unitary encryption operations. A
sequence of input states is encrypted by applying the operations as determined by
the sequence of key elements. A receiver with the same key sequence can easily decipher
these states by applying the respective inverse unitary operations. However, to
an eavesdropper without knowledge about the key sequence, the output state of the
encryption looks like a random mixture of all encryption operations applied to the
input and weighted with the probability of the key elements. For a suitable set of
encryption operations, this output does not contain any information about the input
state. Hence any eavesdropping must remain unsuccessful and the encrypted state
can be safely sent over a public quantum channel. The encryption thus establishes
a private quantum channel for sender and receiver with the same key.
We construct a private quantum channel for the sequential encryption of coherent
states with a classical key, where the key elements have finite precision. This scheme
can be made arbitrarily secure, i.e. the trace norm distance of any two encrypted
states is bounded from above. The necessary precision of the key elements depends
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