ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

238 14. Gaussian entanglement sharing in non-inertial frames
of an uniformly accelerated observer. Two different sets of Rindler coordinates,
which differ from each other by an overall change in sign, are necessary for covering
Minkowski space. These sets of coordinates define two Rindler regions (I and II)
that are causally disconnected from each other:
at = e aζ sinh(aτ), az = e aζ cosh(aτ);
at = −e aζ sinh(aτ), az = −e aζ cosh(aτ).
A particle undergoing eternal uniform acceleration remains constrained to either
Rindler region I or II and has no access to the opposite region, since these two
regions are causally disconnected.
Now consider a free quantum scalar field in a flat background. The quantization
of a scalar field in the Minkowski coordinates is not equivalent to its quantization
in Rindler coordinates. However, the Minkowski vacuum state can be expressed in
terms of a product of two-mode squeezed states of the Rindler vacuum [259]
where
|0〉 ρM
= 1
cosh r
∞
tanh n r |n〉 ρI |n〉 ρII = U(r) |n〉 ρI |n〉 , (14.1)
ρII
n=0
cosh r =
1 − e
− 2π|ωρ|
− a
1
2
, (14.2)
and U(r) is the two-mode squeezing operator introduced in Eq. (2.21). Each
Minkowski mode of frequency |ωρ| has a Rindler mode expansion given by Eq. (14.1).
The relation between higher energy states can be found using Eq. (14.1) and
the Bogoliubov transformation between the creation and annihilation operators,
âρ = cosh rˆbρI − sinh rˆb † ρII , where âρ is the annihilation operator in Minkowski
space for mode ρ and ˆbρI and ˆbρII are the annihilation operators for the same mode
in the two Rindler regions [63, 233]. A Rindler observer moving in region I needs
to trace over the modes in region II since he has no access to the information in
this causally disconnected region. Therefore, while a Minkowski observer concludes
that the field mode ρ is in the vacuum |0〉ρM , an accelerated observer constrained
to region I, detects the state
|0〉〈0|ρM
→ 1
cosh 2 r
∞
tanh 2n r|n〉〈n|ρI
n=0
(14.3)
which is a thermal state with temperature T = a
2πkB where kB is Boltzmann’s
constant.
14.2. Distributed Gaussian entanglement due to one accelerated observer
We consider two inertial observers with detectors sensitive to modes α and ρ, respectively.
All field modes are in the vacuum state except for modes α and ρ which
are originally in a two-mode squeezed state with squeezing parameter s, as in [4].
This Gaussian state, which is the simplest multi-mode squeezed state (of central
importance in quantum field theory [25]), clearly allows for the exact quantification
of entanglement in all partitions of the system in inertial and non-inertial frames.