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in terms of the projectors P . Now the only (real) numbers that are present in
i
thiscontextaretheeigenvaluesλ sowemightsuspectthattheoutcomesofthe
i
measurement must be related to these numbers. Measurement is then defined
by the following postulate
Measuringtheoperator inthestate ψ givestheresultλ withprobability
i
O | i
p(λ )= ψ P ψ . (5.69)
i i
h | | i
If the outcome of the measurement was λ , the state of the system immedi-
i
ately after the measurement is
P ψ
i
| | i (5.70)
p(λ )
i
4Howeveritmightbethattheproperunpderstandingoftheinterpretationandmeasurement
issuesinquantum mechanics couldhavebearingsoncomputationtheory.
118
Withoutlossofgeneralitywecanthinkofthestateintermsofanexpansion
in the eigenstates of the operator
O
n
ψ = α j .
j
| i | i
Xj=1
Note, however, that the coefficients in this expansion are unknown, unless
we have deliberately prepared the system in a certain superposition. Now the
probability can be calculated explicitly
n n n
p(λ )=( α k )P ( α j )=( α k )α i =
i ∗kh i j ∗kh i
| | i | | i
Xk=1 Xj=1 Xk=1
α α = α 2.
∗i i i
| |
The state of the system after the measurement becomes
P ψ α i
i i
| | i = | i
p(λ ) α 2
i i
| |
p p
5.6.2 General measurement
Itispossibletodefineaslightlymoregeneralconceptofquantummeasurement.
Insteadofhavinganobservablewithaspectralresolutionasinequation(5.68),
it is sufficient to have a set of measurement operators M acting on the state
i
{ }
space of the system. The index i refers to the outcome of the measurement.
These operators are subject to a completeness requirement
M i†M =I (5.71)
i
Xi
expressing the fact that the sum of probabilities must be 1. This follows
since the probability for outcome k is defined as
p(k)= hψ |M k†M |ψ i. (5.72)
k
In this case, the state of the system after the measurement is
M ψ
k
| i (5.73)
p(k).
p
Itis thus easyto seethatprojectivemeasurementsarea specialcaseofgen-
eralmeasurements. Togofromageneralmeasurementtoaprojectivemeasure-
ment,wedemandthatthemeasurementoperatorsM areorthogonalprojectors,
i
that is they are hermitean and satisfy M M =δ M .
i j ij i
119
5.6.3 POVM measurement
Yet another special case of generalmeasurements is the so calledPOVM5 mea-
surements. The idea is the following. If one is only interested in the prob-
abilities, and not the resulting states, it is enough to know the combination
M k†M k, the operators M themselves are not needed. If we define new opera-
k
tors E = M k†M k, then E is a positive operator such that iE = I and the
k k i
probabilities are given by p(k)= ψ E k ψ . P
h | | i
Turning this argumentaround,a POVMmeasurementis defined by any set
ofoperators E suchthateachoperatorE ispositiveandthecompletenessre-
i i
{ }
lation E =I holds. Theprobabilityofoutcomekbecomesp(k)= ψ E ψ .
i i h | k | i
Of couPrse, not having the ”square roots” of the E i, the new states cannot be
computed.
5PositiveOperatorValuedMeasurement
120
Chapter 6
Abstract quantum