ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

258 14. Gaussian entanglement sharing in non-inertial frames
results in the Dirac case where entanglement remains positive for all accelerations
(as a direct consequence of the restricted Hilbert space in that instance). Another
surprising result in this case is that we find that classical correlations are no longer
invariant to acceleration but are also degraded to some extent. We analyzed the entanglement
between the modes of the field detected by two non-inertial observers as
a function of the frequencies of their modes, and found that for a fixed acceleration
high frequency modes remain entangled while lower frequency modes disentangle.
In the limit of infinitely accelerated observers, the field modes are in a separable
state for any pair of frequencies.
The take-home message of this Chapter is the following.
➢ Continuous variable entanglement in non-inertial reference frames. The
degradation of entanglement due to the Unruh effect is analytically studied
for two parties sharing a two-mode squeezed state in an inertial frame, in the
cases of either one or both observers undergoing uniform acceleration. For
two non-inertial observers moving with finite acceleration, the entanglement
vanishes between the lowest frequency modes. The loss of entanglement
is precisely explained as a redistribution of the inertial entanglement into
multipartite quantum correlations among accessible and unaccessible modes
from a non-inertial perspective. Classical correlations are also lost for two
accelerated observers but conserved if one of the observers remains inertial.
The tools developed in this Chapter can be used to investigate the problem of
information loss in black holes [GA21]. There is a correspondence [233] between the
Rindler-Minkowski spacetime and the Schwarzschild-Kruskal spacetime, that allows
us to study the loss (and re-distribution) of quantum and classical correlations for
observers outside the black hole, extending and re-interpreting the results presented
in Sec. 14.3. In that case the degradation of correlations can be understood as
essentially being due to the Hawking effect [108, 109].
The next step concerns the study of classical and quantum correlations in the
most general particle states definable in a spacetime with at least two asymptotically
flat regions, represented by multi-mode squeezed states which involve all
modes being pair-wise entangled (like in the phase-space Schmidt decomposition,
see Sec. 2.4.2.1). The study of entanglement in this state, from a relativistic perspective,
will provide a deeper understanding of quantum information in quantum
field theory in curved spacetime [25].