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L = = = . (4.96)
z |− 2i 2(cid:18)0 1(cid:19)(cid:18)1(cid:19) −2(cid:18)1(cid:19) −2|− 2i
−
95
Thus we havea concreterealizationofthe angularmomentum theoryofthe
preceding section in the particular case of l = 1.
2
Connection to quantum computation
Spin1/2isanexampleofatwo-statequantumsystem. Inquantumcomputation
such systems are referred to as qubits. In that context, a slightly different
notation is used. Instead of 1 and 1 , the states are denoted by 0 and
|2i |− 2i | i
1 respectively, and of course, they are to be thought of as quantum versions
| i
of the classical bit values 0 and 1.
Pauli matrices
In the context of describing spin 1/2 is customary to express the angular mo-
mentum operators in terms of the Pauli matrices. As is seen below, it is really
atrivialrescalingbyafactor1/2involved,buttheoreticalphysicistsseemtobe
fond of shuffling factors of 1/2 around.11
0 1
σ = (4.97)
x (cid:18)1 0(cid:19)
0 i
σ y = (cid:18)i − 0 (cid:19) (4.98)
1 0
σ = . (4.99)
z (cid:18)0 1(cid:19)
−
We now leave the theory of angular momentum at this stage.
11Aswellasfactorsof2,√2,π etcetera!
96
Chapter 5
General quantum theory
In this chapter,quantumtheory will be outlined in anabstractandformalway
suitablefordiscussingquantumcomputationalmodelsandquantumcomplexity
theory. Theconceptsoftheprecedingchapterwillreappearbutinamuchmore
formal setting. A modern general reference to quantum mechanics suitable for
studies into quantum computation is the book by A. Peres [36].
5.1 State spaces
The state spaces of quantum mechanics are modeled on Hilbert spaces. These
spaces can be either finite dimensional or infinite dimensional. The infinite di-
mensionalcaserequiresaratherelaboratemathematicaltreatmentifonewants
to be stringent. For quantum computation it suffices to consider finite dimen-
sionalHilbert spaces, as any realquantum computer would have a finite size in
terms of number of states.
In the first section, the theory of vector spaces will reviewed. The intuition
behind the theory ofvectorspacesis the familiarrealvectorsofordinarythree-
dimensionalspace. Thus if v andv are twovectorsin the space,so is α v +
1 2 1 1
α v , where α and α are real numbers. These notions will be made precise
2 2 1 2
below.
Aconciseandreadablephysics-stylereferenceto vectorspacesis C.Isham’s
book [37].
5.1.1 Vector spaces
LetVdenoteann-dimensionalvectorspaceoverthe complexnumbersC. This
meansthatforanyvectorsx1, x2 andxinV andanycomplexnumbersα 1, α
2
and α the following equations expressing linearity hold
α(x +x )=αx +αx (5.1)
1 2 1 2
(α +α )x=α x+α x (5.2)
1 2 1 2
97
α (α )x=(α α )x (5.3)
1 2 1 2
1x=x (5.4)
0x=0 (5.5)
The space V can be thoughtof as n-tuples of complex numbers arrangedas
column vectors so that we can write V = Cn.1 An inner product (or a scalar
product)overVisdefinedasacomplexvaluedfunction x,y definedonV V,
h i ×
subject to the conditions
x,x 0 and x,x =0 if and only if x=0 (5.6)
h i≥ h i
x,α y +α y =α x,y +α x,y (5.7)
1 1 2 2 1 1 2 2
h i h i h i
x,y = y,x ∗ (5.8)
h i h i
forall x,y,y ,y V andall α ,α C. The complex conjugateofacomplex
1 2 1 2
∈ ∈
number α is denoted by α . Note that x,x is a realnumber by condition (c).
∗
h i
Two vectors x and y are said to orthogonal if x,y =0.
h i
A vector space supplied with an inner product is called an inner-product
vector space.
It can be shown that the inner product satisfies the Schwarz inequality
x,y x,x y,y . (5.9)
|h i|≤ h i h i
p p
Norm
From the inner product, a norm can be defined by x = x,x . From this
k k h i
definition and the properties of the inner product, it followspthat
αx = α x for all complex numbers α (5.10)