The Born Rule and Projective Measurements

In the text we cited the Born rule

and according to Theorem 1.2, the states |(Subscript[a, i]) ⟩ are orthogonal for each state index (Subscript[a, i]). According to the collapse hypothesis, the act of measurement can be represented by the following

({{({{|ψ ⟩ ↦}}) |(Subscript[a, i])⟩ ,    where        ⟨(Subscript[a, j])|(Subscript[a, i])⟩ =(Subscript[δ, i j])       }})

if the result of the measurement leads to eigenvalue (Subscript[a, i]).  The above mapping is also called also called a projection. That is, defining the operator

({{({{(Subscript[M, i])}}) ≡ |(Subscript[a, i])⟩  ⟨ (Subscript[a, i])|}})

and using Dirac’s multiplication rule we find that

({{(Subscript[M, i])|ψ⟩= (|(Subscript[a, i])⟩ ⟨ (Subscript[a, i])|)|ψ ⟩ = |(Subscript[a, i])⟩  ⟨ (Subscript[a, i])|ψ ⟩ .}})

(Subscript[M, i]) projects the state along the direction |(Subscript[a, i]) ⟩.  Note that (Subscript[M, i])(Subscript[M, i])=(Subscript[M, i]) and so satisfies the definition for a projection operator. Thus we can equate the expectation values

({{⟨ψ| (Subscript[M, i]^†) (Subscript[M, i]) | ψ ⟩= ⟨ ψ |(Subscript[a, i])⟩ ⟨ (Subscript[a, i])|ψ ⟩ = |⟨ (Subscript[a, i])|ψ ⟩ ((|)^2).}})

which is simply  a re-statement of the Born rule for the measurement probabilities. In addition, using the fact that (Subscript[M, i]^†) (Subscript[M, i]) = (Subscript[M, i])(Subscript[M, i]) = (Subscript[M, i]) we construct the following state

({{|ψ' ⟩  =({{(Subscript[M, i]) |ψ ⟩}})/(Sqrt[({{⟨ψ| (Subscript[M, i]^†)}})(Subscript[M, i]) |ψ ⟩])}})

It is clear that ⟨ψ ’| ψ’ ⟩ = 1, and using definition (2) we can also express |ψ' ⟩  as

({{({{|ψ' ⟩ = |(Subscript[a, i])}}) ⟩   ⟨ (Subscript[a, i])|ψ ⟩ /|⟨ (Subscript[a, i])|ψ ⟩| = (e^(i (Subscript[α, i]))) |(Subscript[a, i]) ⟩}})

where we defined (e^( i (Subscript[α, i])))= ⟨ (Subscript[a, i])|ψ ⟩ /|⟨ (Subscript[a, i])|ψ ⟩|.  Thus, Eq. (3) is equivalent, up to an arbitrary phase factor, of the collapse mapping stated by postulate (1).

The projection approach is useful when considering a measurement that has degenerate eigenvalues. Suppose we take measurements with device A, whose eigenvalues a are  n-fold degenerate. That is,  the orthonormal states |(Subscript[a, 1]) ⟩, |(Subscript[a, 2]) ⟩ ... |(Subscript[a, i3]) ⟩,  are all eigenstates of A, with eigenvalue a.  We now define, as before, the set

({{(Subscript[({{M}}), i]) = |(Subscript[a, i])⟩  ⟨ (Subscript[a, i])|}})

but now construct

({{({{(Subscript[M, a])= }}) (∑_(i=1)^n)|(Subscript[a, i])⟩  ⟨ (Subscript[a, i])|}})

thus

({{({{({{⟨ψ| (SuperDagger[(Subscript[M, a])])}})(Subscript[M, a])|ψ ⟩ }})= (∑_(i=1)^n) ⟨ ψ |(Subscript[a, i])⟩ ⟨ (Subscript[a, i])|ψ ⟩= (∑_(i=1)^n)|⟨(Subscript[a, i])|ψ ⟩((|)^2)}})

which, again, conforms to the Born rule.

Also,

({{({{|(Subscript[ψ, a]) ⟩  =({{(Subscript[M, a]) |ψ ⟩ }})/(Sqrt[({{⟨ψ| (SuperDagger[M])}})M |ψ ⟩])}}) =     (∑_(i=1)^n) (Subscript[c, i])|(Subscript[a, i])⟩        (Subscript[c , i])≡((| ⟨(Subscript[a, i])|ψ ⟩  |)/(Sqrt[(∑_(i=1)^n)|⟨(Subscript[a, i])|ψ ⟩((|)^2)]))}})

Note that

({{(∑_(i=1)^n)({{|(Subscript[c, i])}})((|)^2) = (∑_(i=1)^n) | ⟨(Subscript[a, i])|ψ ⟩  ((|)^2)/(Sqrt[(∑_(k=1)^n)|⟨(Subscript[a, k])|ψ ⟩((|)^2)]) =1}})

and so   ⟨(Subscript[ψ, a]) | (Subscript[ψ, a]) ⟩ = 1.

Because, from our definition of measurement device A= Σ (Subscript[a, i]) (Subscript[M, i]) , the projection operators are orthogonal, i.e.

({{(Subscript[({{M}}), i]) (Subscript[M, j]) = (Subscript[δ, ij])}})

and complete

({{({{({{∑  (SuperDagger[(Subscript[M, i])])}})(Subscript[M, i])}}) = 1}})

These measurements are called projective, or von Neumann, measurements. However, measurement devices in which  relation (4) is relaxed can also be defined. The latter are called POVM (Positive Operator - Valued Measure ) 1

Generalized Born Rule2

Often,  we are concerned with a measurement that is restricted to a subspace of the system Hilbert space. For example, consider a 2-qubit system for which we measure only the second qubit with measurement operator

({{({{A= (Subscript[σ, Z]) ⊗ 1 }})
}})

Suppose the 2-qubit system is in the state

({{        ({{|ψ ⟩= (Subscript[c, 1])}})α |00⟩+ (Subscript[c, 1])β|01⟩ + ({{(Subscript[c, 2])}})a|10⟩ + (Subscript[c, 2])b|11⟩      

     |(Subscript[c, 1])((|)^2)+ |(Subscript[c, 2])((|)^2)=1         |a((|)^2)+ |b((|)^2)=1       |α((|)^2)+ |β((|)^2)=1 }})

We calculate the probability that measurement with A results in the value +1. Since the following states are eigenstates of A

({{         ({{       State            Eigenvalue}})}})
({{  ({{|00 ⟩        +1}, { |01 ⟩         +1}, { |10 ⟩          -1}, { |11 ⟩         -1}})}})

we construct the projection operator (Subscript[M,  +])= |00⟩⟨00|+|01⟩⟨10|  and so
        
        p(+1) = ⟨ψ|(SubPlus[M])|ψ ⟩ =  |⟨ψ  | 00 ⟩((|)^2)+ |⟨ψ|01⟩((|)^2)= |(Subscript[c, 1])((|)^2)(α^2)+ |(Subscript[c, 1])((|)^2)(β^2) = |(Subscript[c, 1])((|)^2)

and the system collapse into state (up to an overall phase factor)

({{({{(SubPlus[M])|ψ⟩/(Sqrt[⟨ψ|(SubPlus[M])|ψ ⟩])}})= α |00⟩+β |01⟩}})

Now, we can express the above state in the form

({{({{|ψ}})⟩= (Subscript[c, 1])|0⟩ ⊗|(Subscript[ψ, a])⟩+ (Subscript[c, 2])|1⟩⊗|(Subscript[ψ, b])⟩
}, {(Subscript[|ψ, a])⟩= α|0⟩ + β |1⟩,    |(Subscript[ψ, b])⟩= a|0⟩+ b|1⟩   }})

So measurement of the 2nd qubit leads to collapse into state |0 ⟩⊗(Subscript[|ψ, a])⟩,  with probability |(Subscript[c, 1])((|)^2), in agreement with the results obtained above. Likewise the probability for measurement value (-1) and collapse  into state |1⟩⊗|(Subscript[ψ, a])⟩ is given by |(Subscript[c, 2])((|)^2). 
Note that |(Subscript[ψ, a])⟩ ,  |(Subscript[ψ, b])⟩  are not orthogonal.

1	Michael Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge U Press, 2011

2	N. David Mermin, Quantum Computer Science, Cambridge U Press, 2007