0198506961.pdf

284 Quantum computing

Fig. 13.3The state | 0 〉 lies at the
north pole of thisBloch sphereand
| 1 〉at the south pole; all other posi-
tion vectors are superpositions of these
two basis states. The Bloch sphere
lies in the Hilbert space spanned by
the two eigenvectors| 0 〉and| 1 〉.The
Hadamard transformation defined in
eqn 13.3 takes| 0 〉 →| 0 〉+| 1 〉(cf.
Fig. 7.2).

spin-1/2 particle determines whether its orientation is up or down along

a given axis. After that measurement the particle will be in one of those

states since the act of measurement puts the system into an eigenstate

of the corresponding operator. In the same way, a qubit will give either

0 or 1, and the read-out process destroys the superposition.

The Bloch sphere is very useful for describing how individual qubits

transform under unitary operations. For example, theHadamardtrans-

formation that occurs frequently in quantum computation (see the ex-

(^2) In this chapter wavefunctions are writ- ercises at the end of this chapter) has the operator 2
ten without normalisation, which is the
common convention in quantum com-
putation.

UˆH| 0 〉→| 0 〉+| 1 〉,

UˆH| 1 〉→| 0 〉−| 1 〉.

(13.3)

This is equivalent to the matrix

UˆH=√^1

2

(

11

1 − 1

)

The effect of this unitary transformation of the state is illustrated in

Fig. 13.3—it corresponds to a rotation in the Hilbert space containing

the state vectors. This transformation changes| 0 〉, at the north pole,

into the superposition given in eqn 13.3 that lies on the equator of the

sphere.

13.1.1 Entanglement

We have already encountered some aspects of the non-intuitive behaviour

of multi-particle quantum systems in the detailed description of the two

electrons in the helium atom (Chapter 3), where the antisymmetric spin

state [|↓↑〉 − |↑↓〉]/

√

2 corresponds to the wavefunction

Ψ=| 01 〉−| 10 〉 (13.4)

in the notation used in this chapter (without normalisation). This does

not factorise into a product of single-particle wavefunctions:

Ψ
=ψ 1 ψ 2 , (13.5)