ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

248 14. Gaussian entanglement sharing in non-inertial frames
with uniform accelerations aL and aN, respectively. They have single-mode detectors
sensitive to modes λ and ν, respectively. We consider, that in the inertial
frame all the field modes are in the vacuum except for modes λ and ν which are in
the pure two-mode squeezed state σ p
LN
(s) of the form Eq. (2.22), with squeezing
parameter s as before. Due to their acceleration, two horizons are created so the
entanglement is redistributed among four parties: Leo, anti-Leo (living respectively
in Rindler region I and II of Leo’s horizon), Nadia, anti-Nadia (living respectively
in Rindler region I and II of Nadia’s horizon). These four (some real and some
virtual) parties will detect modes λI, λII, νI, νII, respectively. By the same argument
of Sec. 14.2, the four observers will share a pure, four-mode Gaussian state,
with CM given by
σLLN ¯ N ¯ (s, l, n) = SλI ,λII (l)SνI ,νII (n)SλI ,νI (s) · 8
· S T λI ,νI (s)ST νI ,νII (n)ST λI ,λII (l) ,
(14.16)
where the symplectic transformations S are given by Eq. (2.24), 8 is the CM of
the vacuum |0〉λII ⊗|0〉λI ⊗|0〉νI ⊗|0〉νII , while l and n are the squeezing parameters
associated with the respective accelerations aL and aN of Leo and Nadia [see
Eq. (14.2)]. Explicitly,
⎛
σL¯ εLL ¯ εLN ¯ εL¯ N¯
⎜
σLLN ¯ N ¯ = ⎜ ε
⎝
T LL ¯ σL εLN εLN¯ εT LN ¯ εT LN σN εNN¯ εT L¯ N¯ εT L¯n εT N ¯ ⎞
⎟
⎠ , (14.17)
σ N N¯
where:
σ ¯ X = [cosh 2 (x) + cosh(2s) sinh 2 (x)]2 ,
σX = [cosh 2 (x) cosh(2s) + sinh 2 (x)]2 ,
ε ¯ XX = ε X ¯ X = [cosh 2 (s) sinh(2x)]Z2 ,
ε ¯ XY = ε Y ¯ X = [cosh(y) sinh(2s) sinh(x)]2 ,
ε ¯ X ¯ Y = [sinh(2s) sinh(x) sinh(y)]Z2 ,
εXY = [cosh(x) cosh(y) sinh(2s)]Z2 ,
with Z2 = 1 0
0 −1 ; X, Y = {L, N} with X = Y , and accordingly for the lower-case
symbols x, y = {l, n}.
The infinite acceleration limit (l, n → ∞) can now be interpreted as Leo and
Nadia both escaping the fall into a black hole by accelerating away from it with
acceleration aL and aN, respectively. Their entanglement will be degraded since
part of the information is lost through the horizon into the black hole [GA21]. Their
acceleration makes part of the information unavailable to them. We will show that
this loss involves both quantum and classical information.
14.3.1. Bipartite entanglement
We first recall that the original pure-state contangle Gτ (σ p
L|N ) = 4s2 detected by
two inertial observers is preserved under the form of bipartite four-mode entanglement
Gτ (σ ( ¯ LL)|(N ¯ N)) between the two horizons, as the two Rindler change of
coordinates amount to local unitary operations with respect to the ( ¯ LL)|(N ¯ N) bipartition.
The computation of the bipartite Gaussian contangle in the various 1×1