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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