quantum mechanics

Qubit Measurement

Let us revisit some of the properties of quantum measurement, already dis- cussed in Sect. 1.2.4, in their application to qubits. In discussing classical bits, we have tacitly assumed that one can easily measure the bit as many times as one wishes, without disturbing its state, and the result of the mea- surement is 0 if the bit state is 0, and 1 if the state is 1. This is indeed the case in classical information processing devises. In quantum mechanics the situation is drastically di?erent: If one has a single copy of a quantum system—a qubit in an arbitrary state |?1) = ? |0) + ? |1), the measurement yields either |0) or |1). A single measurement does not allow one to infer the amplitudes ? and ? of the quantum state. Moreover, the no-cloning the- orem, discussed in the following section, forbids one to clone an unknown quantum state so as to produce many copies of the system. In addition, the measuring process itself modi?es the state of the system: If during the measurement one ?nds the system, e.g., in |0), the post-measurement state
of the system will be |?pm) = ?0 |?1)/?P0 = |0), where ?0 = |0)(0| is
the projection operator and P0 stands for the renormalization of the wave- function. Similarly, ?nding the system in |1) yields the post-measurement
state |?pm) = ?1 |?1)/?P1 = |1), where ?1 = |1)(1| . The probabili-
1) are given, respectively, by
P0 = (?1| ?0 |?1) = |?|2 and P1 = (?1| ?1 |?1) = |?|2, but these proba-
bilities can experimentally be determined only after many measurements on the ensemble of identical systems all being in state |?).
Consider now the measurement performed on a subset of a composite quantum system. The simplest example is a pair of qubits in state (8.2). If we measure the state of the ?rst qubit with the result |0), the post-measurement state of the system collapses to
1

|?pm

?0 |?2)

c00 |00) + c01 |01) c00 |0) + c01 |1)

2 ) = , 1
0

=
|c00|2 + |c00|2

= |0) ,

.
|c00|2 + |c00|2

This shows, in particular, that even if the two qubits were initially entangled, after measuring the ?rst qubit the state of the system becomes factorisable and, in general, the second qubit is left in a superposition state.


?1

Fig. 8.5. Symbol designating qubit measurement in the computational basis.


The last, but not least, important ingredient of quantum circuits is the projective measurement, which is denoted by the symbol shown in Fig. 8.5.


It is applied at the end of the circuit to yield the result of computation, but often the measurement is also performed in the middle of computation to control the subsequent quantum gates, as is the case in, e.g., quantum error correction discussed in Sect. 9.4.
It is su?cient to perform the measurement on individual qubits only in the computational basis { |0), |1)}. If we need to perform a measurement in some other orthogonal basis, we can apply a suitable unitary operator to transform the basis to the computational one and then measure. For example, if we want to perform a measurement that distinguishes the states |±) = 1 ( |0)± |1)),
the required unitary transformation is the Hadamard gate H which transforms the basis { |+), |?)} to { |0), |1)}.


?1 U 1
0 H H
Fig. 8.6. Circuit for measuring the observable U .


Finally, suppose we want to measure an observable associated with an operator U acting on a qubit in state |?1) and having the eigenvalues ±1 and the corresponding eigenstates |?±). The circuit for doing this is shown in Fig. 8.6. Recall that we can represent state |?1) in terms of the eigenstates
|?±) as |?1) = c+ |?+) + c? |??). Following the steps of the circuit, we have

H 1


WCU

1

|?1) |0) ??

?2 |?1)( |0) + |1)) ??
1

?2 ( |?1) |0) + U |?1) |1))

= ?2 [(c+ |?+) + c? |??)) |0) + (c+ |?+)? c? |??)) |1)]
H
?? c+ |?+) |0) + c? |??) |1) . (8.14)

Consequently, detecting the lower (ancilla) qubit in state |0) or |1) indi- cates, respectively, the eigenvalue +1 or ?1 of operator U , while the post- measurement state of the interrogated qubit collapses to the corresponding eigenstate of U ,
1 ) = |?±) . (8.15)
Note that in (8.14) we have only used the fact that U has just two eigenval- ues ±1; we did not explicitly rely on whether U corresponds to a single or many qubit observable. This measurement scheme can therefore be applied equally well to observables U acting on any number of qubits but having only two eigenvalues ±1.