Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
for complex vectors ξ ∈2f together with the operator L = 2f
k=1 ξk Rk. Then A is
positive-semidefinite due to 〈ξ|A|ξ〉 = tr[ρ L ∗ L] ≥ 0. Since γ is real, this is equivalent
to γ − iσ ≥ 0. Moreover, as σ is antisymmetric, the inequality implies γ ≥ 0.
Due to the symplectic scalar product ξ T · σ · R in the definition (2.5) of the Weyl
operators, these relations are written in terms of modified field operators R ′ = σ · R.
In the standard basis of (2.3), this transformation is local to each mode,
′ Q j = σ0 ·
P ′ j
Qj
Since we are usually not concerned with specific physical realizations but rather with
qualitative results for all continuous-variable systems, we mostly drop the distinction
between R ′ k and Rk. Note, however, the effect of displacing a state ρ with Weyl
operators, ρ ′ = W η ρ W ∗ η, on the characteristic function:
χ ′ ρ(ξ) = tr[W η ρ W ∗ η Wξ] = tr[ρ W ∗ η W η Wξ] = χρ(ξ)e −iξT ·σ·η . (2.19)
In field operators R ′ k , the state is displaced by the vector −σ · η, which corresponds
to a translation by −η in Rk; cf. also Eq. (2.8).
2.1.2 Symplectic transformations
While an orthogonal transformation leaves the scalar product over a (real) vector
space unchanged, a real symplectic or canonical transformation S preserves the symplectic
scalar product of a phase space,
Pj
.
σ(S ξ, S η) = σ(ξ, η) for all ξ, η ∈ Ξ.
By this definition, a symplectic transformation for f degrees of freedom is a real
2f × 2f matrix such that S T · σ · S = σ. The group of these transformations is
the real symplectic group, denoted as Sp(2f,Ê). Moreover, with S ∈ Sp(2f,Ê) also
S T , S −1 , −S ∈ Sp(2f,Ê), where the inverse of S is given by S −1 = σ S T σ −1 . Symplectic
transformations have determinant detS = +1. In addition, the symplectic
matrix σ itself is a symplectic transformation, as can be seen from one of its standard
forms (2.3) or (2.4). The inverse is σ −1 = σ T = −σ. For the special case of a single
mode, the symplectic group consist of all real 2×2 matrices with determinant one,
i.e. Sp(2,Ê) = SL(2,Ê). Extensive discussions of the symplectic group, including
the topics of this section, can be found e.g. in [11,12,13].
By (2.2), symplectic transformations of the vector of field operators, R ′ = S R,
do not change the canonical commutation relations; they do not alter the physics of
a continuous-variable system but merely present a change of the symplectic basis.
Since σ is itself a symplectic transformation, this argument justifies neglecting the
distinction between R ′ k and Rk in the computation of the moments above. Under a
symplectic transformation S, displacement vector and covariance matrix are modified
according to d ↦→ S·d and γ ↦→ S T ·γ·S. Weyl operators are mapped to Weyl operators
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