ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

6
s
4
14.2. Distributed Gaussian entanglement due to one accelerated observer 245
2
0
0
1
r
2
50
0
3
100
GΤΣ ARR
Figure 14.4. Genuine tripartite entanglement, as quantified by the residual
Gaussian contangle Eq. (14.12), among the inertial Alice, Rob in Rindler region
I, and anti-Rob in Rindler region II, plotted as a function of the initial
squeezing s and of Rob’s acceleration r. The tripartite entanglement increases
with r, and for r → ∞ it approaches the original entanglement content 4s 2
shared by Alice and Rob in the Minkowski modes.
while m R|(A ¯ R) is always bigger than the other two quantities. Therefore, by using
Eqs. (6.13, 7.36, 14.7, 14.8, 14.9) together with Gτ (σ A| ¯ R) = 0, we find that the
residual Gaussian contangle is given by
Gτ (σ A|R| ¯ R) =
=
2 g[m ¯R|(AR) ] − g[m2 R| ¯ R ], r < r∗ ;
g[m2 A|(R ¯ R) ] − g[m2 (14.12)
A|R ], otherwise.
⎧
−4r2 + arcsinh 2
cosh 2 2 2
(r) + cosh(2s) sinh (r) − 1,
⎪⎨
⎪⎩
r < r ∗ ;
[2 sinh 2 (r)+(cosh(2r)+3) cosh(2s)] 2
[2 cosh(2s) sinh 2 (r)+cosh(2r)+3] 2 − 1,
otherwise.
4s 2 − arcsinh 2
The tripartite entanglement is plotted in Fig. 14.4 as a function of r and s. Very
remarkably, for any initial squeezing s it increases with increasing acceleration r.
In the limit of infinite acceleration, the bipartite entanglement between Alice and
Rob vanishes so we have that
lim
r→∞ Gτ (σA|R| R) ¯ = Gτ (σA|(RR)) ¯ = Gτ (σ p
A|R ) = 4s2 . (14.13)
Precisely, the genuine tripartite entanglement tends asymptotically to the two-mode
squeezed entanglement measured by Alice and Rob in the inertial frame.
We have now all the elements necessary to fully understand the Unruh effect on
CV entanglement of bosonic particles, when a single observer is accelerated. The
acceleration of Rob, produces basically the following effects:
• a bipartite entanglement is created ex novo between the two Rindler regions
in the non-inertial frame. This entanglement is only function of the
acceleration and increases unboundedly with it.