Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
This chapter provides basic tools and notions for the handling of continuous-variable
systems in general and for Gaussian systems in particular. It does not strive to extensively
introduce this field but rather tries to provide common prerequisites for the
rest of this thesis in a concise way. For a more thorough treatment of the matter see
the forthcoming review [5] on quantum information with Gaussian systems and the
book by Holevo [6] for topics regarding phase space and Gaussian states. Fundamental
aspects from functional analysis are covered in [7]. For various other topics the
reader is referred to the references mentioned below. The following sections deal, in
this order, with the general concepts of phase space for continuous-variable quantum
systems, Gaussian states and Gaussian quantum channels.
Throughout this chapter, we implicitly refer to a preview version of [5]; a supplementary
source was [d].
Remark on notation: We denote the adjoint of an operator A with respect to a
scalar product by a star, i.e. as A ∗ . Complex conjugation of scalars or matrices is
indicated by a bar, e.g. as α or A. For simplicity, we generally set = 1; units
of physical quantities are understood to be chosen accordingly. The identity operator
and the identity matrix are denoted by the symbol . In some instances, the
dimension of matrices is specified by a single index, e.g. f.
2.1 Phase space
As in classical mechanics (cf. e.g. [4]), a system of f degrees of freedom (or modes)
can be described in a phase space (Ξ, σ), which consists of a real vector space Ξ of
dimension 2f equipped with a symplectic form σ: Ξ × Ξ →Ê. This antisymmetric
bilinear form gives rise to a symplectic scalar product σ(ξ, η) = 2f
k=1 ξT k · σk,l · ηl implemented by the symplectic matrix σk,l = σ(ek, el), where {ek} is an orthonormal
basis in Ξ. We will only deal with cases where σ is nondegenerate, i.e. if σ(ξ, η) = 0
for all ξ ∈ Ξ, then η = 0. To keep notation simple, we will not distinguish between
bilinear forms and their implementing matrices in a particular basis. For translationally
invariant systems, we will also identify any matrix γ of entries γx,y with the
function γ(x − y) = γx,y yielding these entries. Similarly, we will refer to the linear
space Ξ alone as the phase space if the symplectic form is of secondary importance
in a particular context.
We introduce the symplectic adjoint A +
of a matrix A with respect to the symplectic
scalar product by
σ(Aξ, η) = σ(ξ, A +
η). (2.1)
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