Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
Theorem 2.1 (Stone, von Neumann):
Two families W (1) and W (2) of unitary operators satisfying the Weyl relations
(2.7) over a finite-dimensional phase space which are
(i) strongly continuous, i.e. ∀ψ ∈ H: limψ→0
(ii) irreducible, i.e.
∀ξ ∈ Ξ: W (i)
ξ , A = 0
ψ − W (i)
ξ ψ = 0, and
=⇒ A ∝ ,
are unitarily equivalent, i.e. there exists a unitary operator U mapping one system
to the other by W (1)
ξ = U ∗ W (2)
ξ U.
Note that the statement of this theorem is definitely not true for an infinite-dimensional
phase space. We will only consider Weyl systems which are strongly continuous
and irreducible. For finitely many degrees of freedom, these systems are thus equivalent
to the Schrödinger representation, where the Weyl operator of a single mode
acts on the Hilbert space H = L 2 (Ê, dx) of square-integrable functions by
Wq,p ψ(x) = e i(qP −pQ) ψ(x) = e −iqp/2−ipx ψ(x + q).
Note that by this definition the Weyl operators act on the field operators as a shift
by −ξ, i.e.
W ∗ ξ R k W ξ = R k − ξ k . (2.8)
Since by (2.7b) and (2.3) Weyl operators of different modes commute, they can be
decomposed into a tensor product of Weyl operators on single modes
Wξ1,ξ2,...,ξf =
f
j=1
Wξj , (2.9)
where the Weyl operators on different phase spaces are distinguished only by the
dimension of their argument. The unitary equivalence to the Schrödinger representation
can thus be established for each mode separately. Note that by this decomposition
Weyl operators act on each mode locally.
As the Weyl relations (2.7) give rise to the ccr (2.2), the family of Weyl operators
is a sufficient basis to describe a continuous-variable system for a given phase space
(Ξ, σ). In order to gain more structure, the Weyl operators are used to constitute
an algebra whose norm closure is the ccr algebra ccr(Ξ, σ) of the phase space. This
provides powerful algebraic tools for the description of continuous-variable quantum
systems. Since by (2.9) the Weyl operators can be decomposed into tensor factors
representing single modes, the ccr algebra can be represented by bounded operators
on a tensor product of representation Hilbert spaces for single modes, i.e. by B(H ⊗f )
for systems with f degrees of freedom.
Irreducible representations of the Weyl operators allow for a convenient result,
namely that operators which commute with all Weyl operators are multiples of the
identity:
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