In physics, a quantum instrument is a mathematical abstraction of a quantum measurement, capturing both the classical and quantum outputs. It combines the concepts of measurement and quantum operation.
Let
X
be the countable set describing the outcomes of a measurement and
{
E
x
}
x
∈
X
a collection of subnormalized completely positive maps, given in such a way that the sum of all
E
x
is trace preserving, i.e.
tr
(
∑
x
E
x
(
ρ
)
)
=
tr
(
ρ
)
for all positive operators
ρ
.
Now for describing a quantum measurement by an instrument
I
, the maps
E
x
are used to model the mapping from an input state
ρ
to the outputstate of a measurement conditioned on an classical measurement outcome
x
. Thereby the probability of measuring a specific outcome
x
on a state
ρ
is given by
The state after a measurent with the specific outcome
x
is given by
If the measurement outcomes are recorded in a classical register, i.e. this can be modelled by a set of orthonormal projections
|
x
⟩
⟨
x
|
∈
B
(
C
|
x
|
)
, the action of an instrument
I
is given by an channel
I
:
B
(
H
1
)
→
B
(
H
2
)
⊗
B
(
C
|
x
|
)
with
Here
H
1
and
H
2
are the Hilbert spaces corresponding to the input and the output quantum system of a measurement.
A quantum instrument is an example of a quantum operation in which an "outcome"
x
of which operator acted on the state is recorded in a classical register. An expanded development of quantum instruments is given in quantum channel.
