ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

7.4. Promiscuous entanglement versus noise and asymmetry 139
evolution of the residual Gaussian contangle as a measure of tripartite correlations.
The results here obtained will be recovered in Sec. 12.2.4, and applied to the study
of the effect of decoherence on multiparty protocols of CV quantum communication
with the classes of states we are addressing, thus completing the present analysis
by investigating its precise operational consequences.
In the most customary and relevant instances, the bath interacting with a set of
N modes can be modeled by N independent continua of oscillators, coupled to the
bath through a quadratic Hamiltonian Hint in the rotating wave approximation,
reading
N
Hint = vi(ω)[a †
i bi(ω) + aib †
i (ω)] dω , (7.46)
i=1
where bi(ω) stands for the annihilation operator of the i-th continuum’s mode
labeled by the frequency ω, whereas vi(ω) represents the coupling of such a mode
to the mode i of the system (assumed, for simplicity, to be real). The state of the
bath is assumed to be stationary. Under the Born-Markov approximation, 19 the
Hamiltonian Hint leads, upon partial tracing over the bath, to the following master
equation for the N modes of the system (in interaction picture) [47]
˙ϱ =
N
i=1
γi
ni L[a
2
†
i ]ϱ + (ni
+ 1) L[ai]ϱ , (7.47)
where the dot stands for time-derivative, the Lindblad superoperators are defined
as L[ô]ϱ ≡ 2ôϱô † − ô † ôϱ − ϱô † ô, the couplings are γi = 2πv 2 i (ωi), whereas the
coefficients ni are defined in terms of the correlation functions 〈b †
i (ωi)bi(ωi)〉 = ni,
where averages are computed over the state of the bath and ωi is the frequency of
mode i. Notice that ni is the number of thermal photons present in the reservoir
associated to mode i, related to the temperature Ti of the reservoir by the Bose
statistics at null chemical potential:
1
ni =
exp( ωi . (7.48)
) − 1 kTi
In the derivation, we have also assumed 〈bi(ωi)bi(ωi)〉 = 0, holding for a bath at
thermal equilibrium. We will henceforth refer to a “homogeneous” bath in the case
ni = n and γi = γ for all i.
Now, the master equation (7.47) admits a simple and physically transparent
representation as a diffusion equation for the time-dependent characteristic function
of the system χ(ξ, t) [47],
N
γi ∂xi
˙χ(ξ, t) = − (xi pi) + (xi pi)ω
2 ∂pi
T
xi
σi∞ω
pi
χ(ξ, t) , (7.49)
i=1
where ξ ≡ (x1, p1, . . . , xN , pN ) is a phase-space vector and σi∞ = diag (2ni +
1, 2ni + 1) (for a homogeneous bath), while ω is the symplectic form, Eq. (2.8).
The right hand side of the previous equation contains a deterministic drift term,
which has the effect of damping the first moments to zero on a time scale of γ/2,
and a diffusion term with diffusion matrix σ∞ ≡ ⊕ N i=1 σi∞. The essential point
19 Let us recall that such an approximation requires small couplings (so that the effect of
Hint can be truncated to the first order in the Dyson series) and no memory effects, in that the
‘future state’ of the system depends only on its ‘present state’.