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n n
e ,x = e , α e = e ,α e
j j i i j i i
h i h i h i
Xi=1 Xi=1
n n
= α e ,e = α δ =α .
i j i i ij i
h i
Xi=1 Xi=1
Thus we get the simple formula for the expansion coefficients
α = e ,x . (5.17)
i j
h i
Dual vector space
To each vectorx in V there is a dual vectorx . If x is representedas a column
†
vector, then x is represented by a row vector whose elements are the com-
†
plex conjugates of the elements of x. In this representation, the inner product
between two vectors x and y can be defined as
x,y =x y +x y +...+x y . (5.18)
∗1 1 ∗2 2 ∗n n
h i
This form of the inner product satisfies all the conditions (5.6), (5.7) and
(5.8).
Another, more abstract point of view, is to consider x as linear functional
†
from V to C defined by x (y) = x,y . However, in practical calculations the
†
h i
explicit vector representation is useful.
5.1.2 Hilbert spaces
In order for a general vector space V to be a Hilbert space, two requirements
mustbemeat: (i)theremustexistanormdefinedintermsofaninnerproduct,
and (ii) the set of vectors must be complete with respect to the norm. Com-
pletenessmeansthatallCauchysequencesofvectorsconvergestovectorsinthe
space [38]. This last point is somewhat elaborated in the next section.
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Completeness
A Hilbert space is complete in the sense that if x is a sequence in
n
H { } H
with lim x x = 0 then there is an x in to which the sequence
n,m n m
→∞k − k H
x converges, i.e lim x = x. Thus, in a Hilbert space, every Cauchy
n n n
{ } →∞k k
sequence is convergent. (The converse, that every convergent sequence is a
Cauchy sequence, is true in every normed vector space).
For infinite-dimensional vector spaces, this condition is not automatically
true. In certain cases an intricate process of completing the space with all
limits of Cauchy sequences can be preformed (in much the same way as the
set of rational numbers are completed to form the set of real numbers). The
requirement of completeness is needed in order that the usual tools of analysis;
performing limits, differentiating et cetera, can be used.
Finite dimensional inner-product vector spaces over the complex numbers
are Hilbert spaces. This is follows from the fact that the complex numbers are
complete with respect to their usual absolute value norm, and this is enough
to ensure completeness of finite-dimensional vector spaces over the complex
numbers. No process of completion is needed in this case.
Inthe contextofquantumcomputation,this isactuallymorethanwewant,
sincecompletenessinthissensemeansthattherearevectorsintheHilbertspace
whose components are non-computable numbers. Non-computable functions
could therefore be hidden within the specification of the quantum computer.
We will return to this point later on.
5.1.3 Dirac notation
Afterthisgeneralintroductiontovectorspaces,wewillnowchangethenotation
somewhat in order to be in conformity with mainstream quantum theory.
Dirac inventedanotationalsystemthat is veryusefulboth forformaltreat-
ment of quantum mechanics and explicit calculations. A vector in the vector
space is denoted by a ket . Inside the ket, one places any symbol or symbols
|i
that characterizesthe state, so for example the vector x can be denoted by x .
| i
A generic quantum state is often denoted by ψ .
| i
This system is very versatile. If we are working with a specific orthonormal
basis for a certain vector space, the basis vectors can be denoted concisely by
i , i.e. we just record the index within the ket symbol. The expansion of a
| i
vector x in terms of the basis i is written as
| i | i
n
x = α i , (5.19)
i
| i | i
Xi=1
in analogy to equation (5.13).
In this notational system, the dual to x is denoted by x and we have
| i h |
x = x. A vector ψ in the dual space is called a bra. The inner product
†
| i h | h |
is written x,y = xy . Thus, the orthonormality requirement for a basis
h i h | i
becomes
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ij =δ , (5.20)