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Elements of Quantum Computation and Quantum Communication by Anirban Pathak

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Chapter 3
Mathematical tools and
simple quantum mechanics
required for quantum
computing
It is not in the scope of this elementary textbook to introduce quantum
mechanics in detail or to provide an elaborate mathematical background re-
quired for quantum computing. However, we will provide some basic ideas
that are essential for us to proceed further. To be precise, here we will
briefly describe vector space, different types of operators, Gram-Schmidt
procedure, postulates of quantum mechanics, POVM, density operator, en-
tanglement and its measures, partial trace, Schmidt decomposition, Bloch
sphere, nocloning theorem, etc. Let us start our journey toward the world
of quantum computation by reviewing some simple ideas of algebra.
3.1 A little bit of algebra required for quan-
tum computing
As far as the algebra of quantum computing is concerned, the most im-
portant concept is Hilbert space, which is a complex vector space. To
understand that, we need to understand, vector space first. In order to
understand vector space, in general, we may recall vectors in conventional
three dimensional space. In a three dimensional coordinate space we need
three numbers to describe a vector. These three numbers are essentially
projections along three directions (or three axes) specified by three or-
thonormal unit vectors. So in three dimensional space we need a set of
57
58 Mathematical tools and quantum mechanics for quantum computing
three orthonormal unit vectors
ˆ
i,
ˆ
j and
ˆ
k to describe any other vector
P
(in terms of these unit vectors) as
P = x
ˆ
i + y
ˆ
j + z
ˆ
k. Now, if we generalize
this idea and do not restrict ourselves to three dimensions and allow the
scalar projections (x, y, z, etc.) to be complex, then we obtain a vector
space. Thus we may lucidly describe a vector space in general as a gener-
alization of familiar three dimensional space into an n-dimensional space
where complex projections are also allowed. A formal definition of the
vector space is provided in the following subsection.
3.1.1 Vector space
A vector space is a set of elements called vectors (
α,
β,
γ,···), together
with a set of scalars (a, b, c, ···), which is closed under vector addition and
scalar multiplication. Thus a set of objects constitutes a vector space if it
obeys the following rules:
1. Closure: Addition of two objects of the set gives another object of
the same set (i.e., addition of two vectors gives a vector). Therefore,
a well defined operation will never take us outside the vector space.
2. Has a zero: For every object
V there exists another object
0 such
that
V +
0=
V.
3. Scalar multiplication: If c is a scalar and
V is a vector then c
V is
also a vector.
4. Inverse: For every
V there exists a
V such that
V +(
V )=
0.
5. Associative: The addition operation in a linear vector space is asso-
ciative, i.e., (
V +
W )+
X =
V +(
W +
X).
Example 3.1: Show that a set of vectors (
α,
β,
γ,···),
together with a
set of positive scalars (a, b, c, ···) cannot form a vector space.
Solution: If (a, b, c, ···) are only positive then the inverse of a vector
cannot exist and consequently a necessary property of vector space cannot
be satisfied.
3.1.1.1 The C
n
vector space
At this stage, we would like to draw the attention of the reader toward the
conventional Cartesian coordinate system, where the unit vectors,
ˆ
i,
ˆ
j and
ˆ
k form a complete set of orthonormal basis vectors. Under this basis we
can describe an arbitrary vector
P = x
ˆ
i+y
ˆ
j +z
ˆ
k as (x, y, z) or as
x
y
z
.
Thus the elements of the column matrix are essentially the coefficients
of the basis vectors. Analogously to describe an arbitrary n-dimensional

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