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2024年4月16日发(作者：fopen函数说法正确的是)
Mostofthislecturewillbedevotedtoestablishingcharacterizationsofquantumoperations,based
,though,wewilldiscussonemoreimpor-
tantfactaboutmeasurements,knownasNaimark’sTheorem,thatrelatesgeneralmeasurementsto
projectivemeasurements.
5.1Naimark’sTheorem
Ameasurement
{
P
a
:a
∈
Γ
}
onacomplexEuclideanspace
X
issaidtobeaprojective(orvonNeu-
mann)measurementifitisthecasethateachmeasurementoperatorP
a
isanorthogonalprojection
operatoron
X
.Forsuchameasurement,itnecessarilyholdsthatP
a
P
b
=
0fora
=
refertoameasurementwithrespecttosomeorthonormalbasis
{
x
a
:a
∈
Γ
}
of
X
,itismeantthat
themeasurementisgivenby
{
P
a
:a
∈
Γ
}
,whereP
a
=
x
a
x
∗
a
foreacha
∈
Γ.
Thereisasenseinwhichnogeneralityislostinconsideringonlyprojectivemeasurements:
every(general)measurementonagivenregisterXcanberealizedasaprojectivemeasurementon
apairofregisters
(
X,Y
)
,providedthatYislargeenoughandinitializedtoaknownpurestate.
ThisfactwillbeeasilyestablishedoncewehaveprovedNaimark’sTheorem,whichisasfollows.
Theorem5.1(Naimark’sTheorem).Let
X
beacomplexEuclideanspace,let
{
P
a
:a
∈
Γ
}⊂
Pos
(
X
)
beameasurement,andlet
Y=
C
Γ
.ThenthereexistsalinearisometryA
∈
U
(
X
,
X⊗Y
)
suchthat
P
a
=
A
∗
(
1
X
⊗
E
a,a
)
A
foreverya
∈
Γ.
fineA
∈
L
(
X
,
X⊗Y
)
as
A
=
∑
a
∈
Γ
√
k’sTheorem;Characterizationofquantumoperations42
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