Quantum Information Theory with Gaussian Systems

2 Basics of Gaussian systems
unitary operator US implements this transformation on a density operator ρ such
that US ρ U ∗ S decomposes into a tensor product of one-mode Gaussian states:
ρ =
f
ρj , (2.32a)
j=1
where the ρj are thermal states with covariance matrix γj and γj as the symplectic
eigenvalues of γ. Computing 〈mj| ρj |nj〉 from (2.30) and (2.31) for the eigenvectors
|nj〉 of the occupation number operator ˆ Nj for mode j yields the spectral decompo-
sition
ρj = 2
γj + 1
∞
nj=0
nj γj − 1
|nj〉〈nj| . (2.32b)
γj + 1
The eigenvalues νn1,n2,...,nf of the full state ρ with f modes can be labeled by the
occupation number of each of its normal modes and are given by
νn1,n2,...,nf =
f
j=1
2
γj + 1
γj − 1
. (2.33)
γj + 1
The occupation number expectation value Nj of a single mode (undisplaced) is
obtained as
Nj = tr ρj a ∗ 2
jaj =
γj + 1
∞
nj=0
nj γj − 1
nj =
γj + 1
γj − 1
.
2
Note that Nj ≥ 0 corresponds to the condition on symplectic eigenvalues, γj ≥ 1,
induced by the state condition (2.22). If the expectation value N of the occupation
number follows a Bose distribution, N = (e −β − 1) −1 with inverse temperature β,
the resulting single-mode state ρ is a Gibbs state, ρ = e −β ˆ N / tr[e −β ˆ N ].
The above spectral decomposition (2.32b) directly gives rise to an exponential
form for the Gaussian state ρj of a single mode j [d]:
ρj = exp log 2 − log(γj + 1) + log(γj − 1) − log(γj + 1) a ∗ ja
j ,
where a∗ j and aj with this mode. Since a∗ j aj = (Q2j + P 2 j
are the creation and annihilation operators, respectively, associated
− )/2, the above can be recast as
1
ρj = exp 2 log(γj − 1) − log(γj + 1) (Q 2 j + P 2 1
j ) − 2 log(γ2
j − 1) + log 2 .
Generalizing this to the case of f modes and denoting the symplectic basis where
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