ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

0
entanglement
condition
14.3. Distributed Gaussian entanglement due to both accelerated observers 251
2000
0
1
2
Ν
(a)
3
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0
2
1
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Λ
entanglement 1
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condition
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Ν
(b)
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Figure 14.6. Entanglement condition, Ineq. (14.22), for different frequency
modes assuming that Leo and Nadia have the same acceleration (a) ā = 2π and
(b) ā = 10π. Entanglement is only present in the frequency range where the
plotted surfaces assume negative values, and vanishes for frequencies where the
plots become positive; the threshold [saturation of Ineq. (14.22)] is highlighted
with a black line. Only modes whose frequencies are sufficiently high exhibit
bipartite entanglement. For higher accelerations of the observers, the range of
entangled frequency modes gets narrower, and in the infinite acceleration limit
the bipartite entanglement between all frequency modes vanish.
2.5
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Λ
1
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Ν
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mLN
Figure 14.7. Entanglement between different frequency modes assuming
that Leo and Nadia have the same acceleration ā = 2π.
Considering once more equally accelerated observers, aL = aN ≡ ā with finite
ā, it is straightforward to compute the Gaussian contangle of the modes that do
remain entangled, in the case of a maximally entangled state in the inertial frame.
From Eq. (14.20), we have
m L|N(s → ∞) =
2
1
5
4
3
Λ
cosh(2l) cosh(2n) − 4 sinh(l) sinh(n) + 3
2[sinh(l) + sinh(n)] 2 . (14.23)
In Fig. 14.7 we plot the entanglement between the modes, Eq. (14.23), as a function
of their frequency λ and ν [using Eq. (14.2)] when Leo and Nadia travel with the
same acceleration ā = 2π. We see that, consistently with the previous analysis, at
fixed acceleration, the entanglement is larger for higher frequencies. In the infinite
acceleration limit, as already remarked, entanglement vanishes for all frequency
modes.