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Multiple choice
Stinespring's dilation theorem writes any CPTP channel
E
E
E
as
E
(
ρ
)
=
t
r
E
(
V
ρ
V
†
)
E(\rho) = \mathrm{tr}_E(V \rho V^\dagger)
E
(
ρ
)
=
tr
E
(
V
ρ
V
†
)
for a linear map
V
V
V
into the system-plus-environment space. What condition must
V
V
V
satisfy, and how large can the environment be taken?
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V
V
V
is Hermitian (
V
=
V
†
V = V^\dagger
V
=
V
†
), and
dim
(
H
E
)
\dim(H_E)
dim
(
H
E
)
must be infinite
V
V
V
is unitary on the system alone (
V
†
V
=
V
V
†
=
V^\dagger V = V V^\dagger =
V
†
V
=
V
V
†
=
identity), and
dim
(
H
E
)
\dim(H_E)
dim
(
H
E
)
must equal
dim
(
H
A
)
\dim(H_A)
dim
(
H
A
)
V
V
V
is an isometry (
V
†
V
=
V^\dagger V =
V
†
V
=
identity), and
dim
(
H
E
)
\dim(H_E)
dim
(
H
E
)
can be taken equal to the Kraus rank of
E
E
E
V
V
V
is a projector (
V
2
=
V
V^2 = V
V
2
=
V
), and
dim
(
H
E
)
\dim(H_E)
dim
(
H
E
)
equals the output dimension
dim
(
H
B
)
\dim(H_B)
dim
(
H
B
)
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