Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
For a proof of this theorem, see e.g. [6]. An extended discussion of Fourier transforms
between operators and functions can be found in [8]. A useful application of the
Parseval relation (2.11) is the computation of the overlap |〈ψ|φ〉| 2 between pure
states |ψ〉 and |φ〉:
〈ψ|φ〉 2 −f
= tr |ψ〉〈ψ| |φ〉〈φ| = (2π) d 2f ξ χψ(ξ) χφ(ξ),
where χψ and χφ denote the characteristic functions of the two states.
The relations (2.10) connect properties of the density operator ρ with those of the
characteristic function χρ:
⊲ Boundedness: χρ(ξ) ≤ ρ1.
⊲ Normalization: tr[ρ] = tr[ρ W0] = 1 ⇐⇒ χρ(0) = 1.
⊲ Purity: χρ(ξ) corresponds to a pure state4 if and only if ρ2 = ρ or tr[ρ2 ] = 1
and hence if
d
Ξ
2f ξ χρ(ξ) 2 = (2π) f . (2.12)
⊲ Symmetry: Since ρ is hermitian, χρ(ξ) = χρ(−ξ).
⊲ Continuity: χ(ξ) is continuous if and only if it corresponds to a normal state,
i.e. to a state which can be described by a density matrix.
A given function χ(ξ) is the characteristic function of a quantum state if and
only if it obeys a quantum version of the Bochner-Khinchin criterion [6]: χ(ξ) has
to be normalized to χ(0) = 1, continuous at ξ = 0 and σ-positive definite, i.e.
for any number n ∈Æof phase space vectors ξ1, ξ2, . . .,ξn ∈ Ξ and coefficients
c1, c2, . . .,cn ∈it has to fulfill
Ξ
n
ck cl χ(ξk − ξl) exp iσ(ξk, ξl)/2 ≥ 0 . (2.13)
k,l=1
The characteristic function in (2.10b) can be taken as the classical Fourier transform
of a function. With this interpretation, the result of a classical inverse Fourier
4 A density matrix ρ corresponds to a pure state if and only if ρ2 = ρ, i.e. if ρ is a projector; due
to the normalization tr[ρ] = 1, this projector is of rank one, ρ = |ψ〉〈ψ|. If the state is not pure,
it is mixed and can be written as a convex combination of pure states |ψi〉〈ψi|:
ρ = P
iλi |ψi〉〈ψi|, where λi ≥ 0 and P
iλi = 1 .
14
For continuous-variable states, this convex combination might be continuous, i.e. an integral
over a classical probability density λ(z):
Z
Z
ρ = dz λ(z) |ψz〉〈ψz|, where λ(z) ≥ 0 and dz λ(z) = 1 .