ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

14.3. Distributed Gaussian entanglement due to both accelerated observers 249
partitions of the state σ ¯ LLN ¯ N is still possible in closed form thanks to the results
of Sec. 4.5.2 [GA7]. From Eqs. (4.74, 4.76, 6.13, 14.17), we find
m L| ¯ N = m N| ¯ L = m ¯ L| ¯ N = 1 , (14.18)
mL| L ¯ = cosh(2l) , mN| N ¯ = cosh(2n) ,
⎧
1 ,
⎪⎨ tanh(s) ≤ sinh(l) sinh(n) ;
(14.19)
mL|N =
⎪⎩
2 cosh(2l) cosh(2n) cosh 2 (s)+3 cosh(2s)−4 sinh(l) sinh(n) sinh(2s)−1
2[(cosh(2l)+cosh(2n)) cosh2 (s)−2 sinh2 (s)+2 sinh(l) sinh(n) sinh(2s)] ,
(14.20)
otherwise .
Let us first comment on the similarities with the setting of an inertial Alice
and a non-inertial Rob, discussed in the previous Section. In the present case of
two accelerated observers, Eq. (14.18) entails (we remind that m = 1 means separability)
that the mode detected by Leo (Nadia) never gets entangled with the
mode detected by anti-Nadia (anti-Leo). Naturally, there is no bipartite entanglement
generated between the modes detected by the two virtual observers ¯ L and ¯ N.
Another similarity found in Eq. (14.19), is that the reduced two-mode state σ X ¯ X
assigned to each observer X = {L, N} and her/his respective virtual counterpart
¯X, is exactly of the same form as σ R ¯ R, and therefore we find again that a bipartite
Gaussian contangle is created ex novo between each observer and her/his alter ego,
which is a function of the corresponding acceleration x = {l, n} only. The two
entanglements corresponding to each horizon are mutually independent, and for
each the X| ¯ X entanglement content is again the same as that of a pure, two-mode
squeezed state created with squeezing parameter x.
The only entanglement which is physically accessible to the non-inertial observers
is encoded in the two modes λI and νI corresponding to Rindler regions
I of Leo and Nadia. These two modes are left in the state σLN , which is not a
GMEMMS (like the state σAR in Sec. 14.2) but a nonsymmetric thermal squeezed
state (GMEMS [GA3]), for which the Gaussian entanglement measures are available
as well [see Eq. (4.76)]. The Gaussian contangle of such state is in fact given
by Eq. (14.20). Here we find a first significant qualitative difference with the case
of a single accelerated observer: a state entangled from an inertial perspective
can become disentangled for two non-inertial observers, both traveling with finite
acceleration. Eq. (14.20) shows that there is a trade-off between the amount of entanglement
(s) measured from an inertial perspective, and the accelerations of both
parties (l and n). If the observers are highly accelerated — namely, if sinh(l) sinh(n)
exceeds tanh(s) — the entanglement in the state σLN vanishes, or better said, becomes
physically unaccessible to the non-inertial observers. Even in the ideal case,
where the state contains infinite entanglement (corresponding to s → ∞) in the
inertial frame, the entanglement completely vanishes in the non-inertial frame if
sinh(l) sinh(n) ≥ 1. Conversely, for any nonzero, arbitrarily small accelerations
l and n, there is a threshold on the entanglement s such that, if the entanglement
is smaller than the threshold, it vanishes when observed in the non-inertial
frames. With only one horizon, instead (Sec. 14.2), any infinitesimal entanglement
will survive for arbitrarily large acceleration, vanishing only in the infinite acceleration
limit. The present feature is also at variance with the Dirac case, where
entanglement never vanishes for two non-inertial observers [6].