Quantum Information Theory with Gaussian Systems

4.1 Quantum cellular automata
ued squeezing (see below) and symplectic rotations. An experimental realization of
a Gaussian qca might use the vibrational degrees of freedom of atoms in an optical
lattice.
Our principle aim in this chapter is to discover and access irreversibility in qcas for
the case of Gaussian systems. We prove that conceptually simple reversible Gaussian
qcas exhibit signs of irreversibility. Moreover, we examine the conceptual problems
in the definition of irreversible qcas, which become especially clear in the Gaussian
case. In the long run, such qcas could be employed to simulate ground states of other
systems; by tuning global parameters of their dynamics, they could robustly drive a
range of initial states into a limit state corresponding to a different Hamiltonian. We
set out with a brief discussion of the definition and properties of a quantum analog
of (deterministic) cas along the lines of [70], including the problem of quantizing
them in the first place. The remaining part of this chapter is devoted to Gaussian
quantum cellular automata and the special instance of a one-dimensional chain of
harmonic oscillators complete with Gaussian dynamics and Gaussian initial states.
We present methods to deal with an infinite number of modes and investigate this
system by decomposition into plane-wave modes. As a result, we show that the
system exhibits properties typically related to irreversibility: Although the system
evolves from a pure, uncorrelated state under a reversible dynamics, the correlation
function describing the state converges. Moreover, this implies convergence in trace
norm of the density operators describing the state for finite regions of the lattice.
The reflection symmetric limit states are thermal equilibrium states determined by
the correlation function of a pure state and a modewise temperature parameter. The
last section examines the conceptual problems in the definition of irreversible qcas,
even in the Gaussian case. In particular, we present different concepts of localization
and their impact on the definition of qcas.
The contents of this chapter have in part been published in [b].
4.1 Quantum cellular automata
This section introduces the concept of qcas formally and briefly presents some
general results. In both we closely follow Schumacher and Werner [70].
Repeating the above characterization, a cellular automaton (ca) is a discrete, regular
system with uniform dynamics arising from a local interaction. Abstractly, it is
realized as an infinite lattice of identical, finite systems, where each cell is coupled to
the sites in its neighborhood by a uniform dynamics called local transition rule. The
neighboring cells are determined from a uniform, finite neighborhood scheme relative
to any cell. While this scheme can be arbitrarily complex, it is mostly defined in the
usual sense as the nearest or next-nearest neighbors of a cell. The time evolution
of the whole system, the global rule, is discrete and synchronous. These properties
imply a finite propagation speed. While a qca is essentially a ca where the cells are
(identical) quantum systems, there are some points to clarify.
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