Quantum Information Theory with Gaussian Systems

1 Introduction
electromagnetic field ofbright2 laser beams. A field mode is described as a quantum
harmonic oscillator with field operators Q, P corresponding to the quadrature
components of the complex amplitude. Since Q and P have continuous eigenvalue
spectrum, the mode is a continuous-variable system, which cannot be represented
on a finite-dimensional Hilbert space. 3
Gaussian continuous-variable states are characterized by a Gaussian Wigner quasiprobability
function. They naturally arise as the ground and thermal states of
quadratic bosonic Hamiltonians, in particular for the standard harmonic oscillator,
H = 1
2 (Q2 + P 2 ).
(Throughout this thesis, we set = 1. Similarly, we do not distinguish different
modes by their frequency but always assume m, ω = 1. Units of physical quantities
are chosen accordingly.) Hence Gaussian states are relevant wherever quantum systems
are described by such Hamiltonians. Examples of Gaussian states in quantum
optics include coherent states (pure states with minimal uncertainty,displaced vacuum),
thermal states (coherent states with additional classical Gaussian noise) and
squeezed states (with reduced variance for Q or P). In particular, the output states
of lasers are approximated by coherent states.
Gaussian states are also mathematically appealing, because they can be described
by a finite number of parameters for each mode. The underlying phase space related
to the canonical commutation relation,
[Q, P] = i ,
provides a rich mathematical structure. This makes Gaussian states much easier
to handle than general continuous-variable states, which require tools for infinitedimensional
Hilbert spaces: Restricting questions to Gaussian states allows to investigate
problems which would otherwise be hardly tractable. Moreover, Gaussian
states are extremal among all states with the same first and second moments with
respect to certain functionals: It is a standard result of statistical mechanics that
Gaussian states maximize the von Neumann entropy S(ρ) = − tr[ρ log ρ] for fixed
energy. Only recently, Wolf et al. [3] have proved that a similar result holds for a
more general class of functionals, which comprises important examples from quantum
information theory (entanglement measures, key distillation rates, channel capacities).
One can thus assume an unknown quantum state to be Gaussian in order to
obtain reliable bounds on such quantities. For these reasons, Gaussian states are of
particular relevance for the study of continuous-variable systems.
While quantum information theory for finite-dimensional systems is quite far developed,
continuous-variable systems have not yet attracted equal attention. In this
2 This emphasizes the contrast to very weak laser pulses with approximately 0.1 photons per
pulse, which are used to emulate single-photon sources.
3 Consider e.g. position and momentum operators Q and P, which obey the canonical commutation
relation [Q, P] = i . If Q and P could be described by finite-dimensional matrices, the
trace of the commutator would vanish, tr[Q P −P Q] = tr[Q P]−tr[P Q] = tr[Q P]−tr[Q P] = 0.
This contradicts tr[ ] = dim H.
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