FOUNDATIONS OF QUANTUM MECHANICS

VIII. 3. MEASUREMENT ACCORDING TO QUANTUM MECHANICS 167
However, at a second look, the transition (VIII. 2) turns out to have peculiar consequences. The
formula (VIII. 2) assumed that the object system S was, before the measurement, in an eigenstate of
A. But what if S is in an arbitrary state |ψ⟩ ∈ H S ?
We can decompose this arbitrary state |ψ⟩ into the orthonormal eigenstates |a j ⟩ of A with coefficients
c j = ⟨a j | ψ⟩. Therefore, using |ψ⟩ = ∑ c j |a j ⟩ and the linearity of the evolution operator it
follows that
U ( |ψ⟩ ⊗ |r 0 ⟩ ) = U
N S ∑
j=1
c j |a j ⟩ ⊗ |r 0 ⟩ =
N S ∑
j=1
c j U ( |a j ⟩ ⊗ |r 0 ⟩ )
=
N S ∑
j=1
c j |a j ⟩ ⊗ |r j ⟩ =: |Φ⟩. (VIII. 4)
We see that the state |Φ⟩ of the composite system of object S and measuring apparatus M after the
measurement is no longer a product state, rather it is entangled. This implies that we cannot describe
S, nor M, with a pure state; the partial traces S and M yield mixed states, see section III. 4.
This aspect has no classical analogue. We will come back to this, but first we consider the question
whether this quantum mechanical description of the measurement process is compatible with the
measurement postulate. Or, more precisely, whether application of the measurement postulate to A
leads to the same result as its direct application to S. And we ask whether the desired correlation
between the values of A and R is achieved. We will show now that this is indeed the case.
The quantity R of the measuring apparatus M is represented on the Hilbert space H S ⊗ H M of
the composite system SM as 11⊗R. The probability to find for this quantity the value r k is, according
to the measurement postulate,
Prob |Φ⟩ (R : r k ) = ⟨Φ| ( 11 ⊗ |r k ⟩ ⟨r k | ) |Φ⟩. (VIII. 5)
With (VIII. 4) this yields
Prob |Φ⟩ (R : r k ) = |c k | 2 , (VIII. 6)
where we have used the orthonormality of the |r k ⟩ ∈ H M . This is the same result as yielded by
direct application of the measurement postulate to the arbitrary |ϕ⟩ from (VIII. 4). Apparently, the
probability to find an outcome r k when measuring R of M is always equal to the probability to find
the outcome a k of A on S. This former measurement can therefore be regarded as a substitute for the
latter.
The validity of (VIII. 6) itself does not show that a correlation between the value of A and R has
been established. To show that such a correlation exists, we have to know the probability of a certain
pair of outcomes (a i , r k ) for A ⊗ R, in the state |Φ⟩ of (VIII. 4). The joint probability to find this pair
of outcomes is
Prob |Φ⟩ (A : a i ∧ R : r k ) = ⟨Φ| ( |a i ⟩ ⟨a i | ⊗ |r k ⟩ ⟨r k | ) |Φ⟩
= ∣ ∣ ( ⟨a i | ⊗ ⟨r k | ) |Φ⟩ ∣ ∣ 2 = |c i | 2 δ ik . (VIII. 7)