Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
DYNAMICS
OF
QUANTUM
ENTANGLEMENT
IN
RELATIVISTIC
OPEN
SYSTEMS
∗
M˘
ad˘
alin
Calamanciuc
†
Aurelian
Isar
‡
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.205
Abstract
We
review
the
results
obtained
by
investigating,
in
the
framework
of
the
theory
of
open
systems
based
on
completely
positive
quantum
dynamical
semigroups,
the
time
evolution
of
quantum
entanglement
of
two
different
bimodal
systems,
each
one
interacting
with
a
thermal
environment.
The
first
system
is
formed
of
the
free
massless
bosonic
mode
of
a
stationary
observer
in
a
region
that
is
asymptotically
flat
(Kruskal
observer)
or
freely
falling
in
Schwarzschild
black
hole,
and
the
mode
of
a
uniformly
accelerated
observer
hovering
outside
the
event
horizon
of
black
hole.
For
initial
bimodal
Gaussian
squeezed
thermal
states
of
the
system,
we
show
that
under
the
influence
of
Hawking
ra-
diation
the
quantum
entanglement
of
the
two
modes
is
destroyed
in
a
finite
time,
for
non-zero
values
of
the
temperature
of
the
thermal
en-
vironment,
i.e.
the
phenomenon
of
entanglement
sudden
death
takes
place.
The
second
system
consists
of
two
bosonic
modes
associated
with
a
scalar
quantum
field
in
de
Sitter
space,
characterized
by
two
re-
gions
that
are
causally
disconnected
and
described
using
open
universe
coordinates
on
two
open
charts.
The
survival
time
of
the
entanglement
of
this
bimodal
system
strongly
depends
on
the
competition
between
the
contrary
effects
provided
by
the
squeezing
of
the
initial
bimodal
state,
the
curvature
of
de
Sitter
space,
the
mass
parameter
and
the
∗
Accepted
for
publication
on
July
30,
2026
†
calamanciuc.madalin@theory.nipne.ro
,
Faculty
of
Physics,
University
of
Bucharest;
National
Institute
of
Physics
and
Nuclear
Engineering,
Magurele,
Romania
‡
isar@theory.nipne.ro
,
Faculty
of
Physics,
University
of
Bucharest;
National
Insti-
tute
of
Physics
and
Nuclear
Engineering,
Magurele,
Romania;
Academy
of
Romanian
Scientists
205
On
quantum
entanglement
212
mode
B
.
By
performing
partial
trace
over
¯
B
in
the
tripartite
covariance
matrix
(
17
)
we
will
get
rid
of
the
mode
¯
B
,
obtaining
the
initial
covariance
matrix
for
the
modes
A
and
B
,
associated
with
region
R
:
σ
AB
(
s,
γ
p
)
=
A
C
C
T
B
,
(18)
with
A
=
cosh
2
s
I,
B
=
|
γ
p
|
2
+
cosh
2
s
1
−
|
γ
p
|
2
I,
C
=
sinh
2
s
1
−
|
γ
p
|
2
Z.
(19)
3
Time
evolution
of
quantum
entanglement
In
this
Section
we
will
investigate
the
time
evolution
of
the
entanglement
be-
tween
the
two
bosonic
modes
of
the
considered
systems,
each
one
immersed
in
its
common
thermal
bath.
To
study
the
dynamics
of
the
bimodal
sys-
tems
we
use
the
axiomatic
formalism
of
open
systems,
based
on
completely
positive
quantum
dynamical
semigroups.
In
this
framework
the
Markovian
irreversible
time
evolution
of
an
open
system
is
described
by
the
following
Gorini-Kossakowski-Sudarshan-Lindblad
master
equation
for
the
density
op-
erator
ρ
(
t
)
(we
set
=
1)
[
17
–
19
]
dρ
(
t
)
dt
=
−
i[
H
,
ρ
(
t
)]
+
1
2
k
2
L
k
ρ
(
t
)
L
†
k
−
ρ
(
t
)
,
L
†
k
L
k
+
,
(20)
where
H
=
1
2
(
x
2
+
p
2
x
+
y
2
+
p
2
y
)
(21)
is
the
Hamiltonian
of
the
two
bosonic
modes
(we
take
identical
frequencies
ω
=
1
for
the
two
modes)
and
the
operators
L
k
,
L
†
k
describe
the
interaction
of
the
system
with
a
general
environment.
If
the
initial
states
are
Gaussian
and
the
operators
L
k
are
chosen
poly-
nomials
of
first
degree
in
the
canonically
conjugated
quadrature
operators
x,
p
x
,
y,
p
y
of
the
two
bosonic
modes,
then,
due
to
the
linear
character
of
the
dynamics,
the
Gaussianity
is
preserved
in
time
[
14
,
28
].
