In quantum mechanics, especially in the study of open quantum systems, reduced dynamics refers to the time evolution of a density matrix for a system coupled to an environment. Consider a system and environment initially in the state
ρ
S
E
(
0
)
(which in general may be entangled) and undergoing unitary evolution given by
U
t
. Then the reduced dynamics of the system alone is simply
ρ
S
(
t
)
=
T
r
E
[
U
t
ρ
S
E
(
0
)
U
t
†
]
If we assume that the mapping
ρ
S
(
0
)
↦
ρ
S
(
t
)
is linear and completely positive, then the reduced dynamics can be represented by a quantum operation. This mean we can express it in the operator-sum form
ρ
S
=
∑
i
F
i
ρ
S
(
0
)
F
i
†
where the
F
i
are operators on the Hilbert space of the system alone, and no reference is made to the environment. In particular, if the system and environment are initially in a product state
ρ
S
E
(
0
)
=
ρ
S
(
0
)
⊗
ρ
E
(
0
)
, it can be shown that the reduced dynamics are completely positive. However, the most general possible reduced dynamics are not completely positive.
