Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
Since σ(Aξ, η) = (Aξ) T · σ · η, the symplectic transpose is explicitly obtained as
A +
= σ−1 · A T · σ.
The symplectic form governs the abstract description of a quantum system via
the canonical commutation relations (ccr) between canonical or field operators Rk
for k = 1, 2, . . .,2f:
[Rk, Rl] = iσk,l . (2.2)
For a system of f harmonic oscillators, the field operators correspond to position
and momentum operators Qj and Pj of each mode j = 1, 2, . . .,f. In quantum optics
Q and P are replaced by the quadrature components of the electromagnetic field.
By fixing a particular harmonic oscillator as a reference for Q, P of each mode and
choosing a modewise ordering of the field operators,
R2j−1 = Qj , R2j = Pj ,
the symplectic matrix takes on a standard form:
σ =
f
j=1
0 1
= f ⊗ σ0 for σ0 =
−1 0
0 1
−1 0
(2.3)
(where f indicates the f × f identity matrix). In a different ordering, where all
position operators are grouped together and followed by all momentum operators,
i.e.
Rj = Qj , Rf+j = Pj ,
the symplectic matrix has a different block structure:
0
σ =
− f
f
.
0
(2.4)
We refer to this ordering as blockwise or (Q, P)-block ordering. Depending on the
situation, one form for σ or the other might be advantageous. In either case, the set of
field operators can be compactly written as a vector R = (Q1, P1, Q2, P2, . . .,Qf, Pf)
or R = (Q1, Q2, . . .,Qf, P1, P2, . . .,Pf).
An equivalent description of a continuous-variable quantum system does not use
the field operators Q and P, but builds upon the annihilation and creation operators
ak and a∗ k , respectively, which are defined by
ak = (Qk + iPk)/ √ 2
and because of (2.2), (2.3) obey the bosonic commutation relations
The operator
10
[ak , a ∗ l ] = δk,l , [ak , al ] = [a ∗ k , a∗l ] = 0 .
ˆN k = a ∗ k a k = (Q 2 k + P 2 k − )/2