Quantum Information Theory with Gaussian Systems

2.1 Phase space
transform of χρ(ξ) with respect to the variable σ ·η is the Wigner function [9] Wρ(η)
of ρ,
Wρ(ξ) = (2π) −2f
dη e
Ξ
iξT ·σ·η
χρ(η). (2.14)
The Wigner function is related to expectation values of the parity operatorÈ[10].
For more than one mode,Èis a tensor product of single-mode parity operators,
which act on the respective field operators by inversion of sign. HenceÈRkÈ=−Rk.
Moreover, it is unitary and hermitian,È−1 =È∗ =È. With this,
Wρ(ξ) = π −f tr ρ WξÈW ∗ ξ . (2.15)
The description of a quantum state by the Wigner function as a quasi-probability
distribution on phase space is equivalent to the characteristic function. 5 However, we
mostly use the characteristic function χ(ξ) to describe states.
Similar to classical probability theory, the derivatives of the characteristic function
of a state yield the moments with respect to the field operators [6]. In particular, the
first and second moments are derived in terms of modified field operators R ′ = σ · R
as
1 ∂
χρ(ξ) = tr
i ∂ξk ξ=0 ρ R ′
k ,
− ∂2
χρ(ξ) =
∂ξk ∂ξl ξ=0 1
2 trρ {R ′ k, R ′
l}+ ,
where {R ′ k , R′ l }+ = R ′ kR′ l + R′ lR′ k denotes the anti-commutator of R′ k and R′ l . From
these moments we define the displacement vector d ′ by
and the covariance matrix γ ′ by
γ ′
k,l = tr ρ R ′ k − 〈R ′ k〉 , R ′ l − 〈R ′ l〉
d ′ k = trρ R ′
k
+
(2.16)
= tr ρ {R ′ k, R ′ ′
l}+ − 2〈R k〉〈R ′ l〉, (2.17)
where the prime indicates quantities with respect to the modified field operators.
Using the commutation relation (2.2), this is equivalent to
tr ρ R ′ k − 〈R ′ k〉 R ′ l − 〈R ′ l〉
= 1
2 γ′ k,l + i
2 σk,l . (2.18)
Note that necessarily γ + iσ ≥ 0: Consider the matrix
Ak,l = tr ρ Rk − 〈Rk〉 Rl − 〈Rl〉 = (γ + iσ)/2
5 Note that there exist other quasi-probability functions, namely the P- and the Q-function, which
give rise to other characteristic functions. These correspond to different orderings of the field
operators.
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