ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

246 14. Gaussian entanglement sharing in non-inertial frames
• the bipartite entanglement measured by two inertial observers is redistributed
into a genuine tripartite entanglement shared by Alice, Rob and
anti-Rob. Therefore, as a consequence of the monogamy of entanglement,
the entanglement between Alice and Rob is degraded and eventually disappears
for infinite acceleration.
In fact, bipartite entanglement is never created between the modes measured by
Alice and anti-Rob. This is very different to the distribution of entanglement of
Dirac fields in non-inertial frames [6], where the fermionic statistics does not allow
the creation of maximal entanglement between the two Rindler regions, implying
that the entanglement between Alice and Rob is never fully degraded; as a result of
the monogamy constraints on entanglement sharing, the mode measured by Alice
becomes entangled with the mode measured by anti-Rob and the entanglement in
the resulting three-qubit system is distributed in couplewise correlations, and a
genuine tripartite entanglement is never created in that case [6].
In the next Section, we will show how in the bosonic case the picture radically
changes when both observers undergo uniform acceleration, in which case the
relativistic effects are even more surprising.
14.2.3. Mutual information
It is interesting to compute the total (classical and quantum) correlations between
Alice and the non-inertial Rob, encoded in the reduced (mixed) two-mode state
σ A|R of Eq. (14.6), using the mutual information I(σ A|R), Eq. (2.40). The symplectic
spectrum of such state is constituted by ν−(σ A|R) = 1 and ν+(σ A|R) =
Det σ ¯R: since it belongs to the class of GMEMMS, it is in particular a mixed
state of partial minimum uncertainty (GLEMS), which saturates Ineq. (2.19) (see
Sec. 4.3.3.1). Therefore, the mutual information reads
I(σ A|R) = f( Det σA) + f( Det σR) − f( Det σ ¯ R) , (14.14)
with f(x) given by Eq. (2.39)
Explicitly:
I(σ A|R)
= log[cosh 2 (s) sinh 2 (r)] sinh 2 (r) cosh 2 (s) + log[cosh 2 (s)] cosh 2 (s)
+ log[cosh 2 (r) cosh 2 (s)] cosh 2 (r) cosh 2 (s) − log[sinh 2 (s)] sinh 2 (s)
− 1
2
− 1
2
log{ 1
2 [cosh(2s) cosh2 (r) + sinh 2 (r) − 1]}[cosh(2s) cosh 2 (r) + sinh 2 (r) − 1]
log{ 1
2 [cosh2 (r) + cosh(2s) sinh 2 (r) + 1]}[cosh 2 (r) + cosh(2s) sinh 2 (r) + 1].
The mutual information of Eq. (14.14) is plotted in Fig. 14.5(a) as a function
of the squeezing degrees s (corresponding to the entanglement in the inertial
frame) and r (reflecting Rob’s acceleration). It is interesting to compare the
mutual information with the original two-mode squeezed entanglement measured
between the inertial observers. In this case, it is more appropriate to quantify
the entanglement in terms of the entropy of entanglement, EV (σ p
A|R ), defined
as the Von Neumann entropy of each reduced single-mode CM [see Eq. (1.25)],
EV (σ p
A|R ) ≡ SV (σ p
A ) ≡ SV (σ p
B ). Namely,
EV (σ p
A|R ) = f(cosh 2s) , (14.15)
with f(x) given by Eq. (2.39). In the inertial frame (r = 0), the observers share a
pure state, σA|R ≡ σ p
A|R and the mutual information is equal to twice the entropy