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8/2/2019 Anin Mimos Math

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Quantum Algebra-1

A. Pathak

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What are we going to discuss?

It is not in the scope of this 50 minutes lecture to provide all themathematical knowledge required for quantum computing. But we willgive some basic ideas which are essential for us to proceed further. In

particular we will discuss the following: Vector space (only introduction and elementary definitions)

Dirac notation

Inner product and outer product

Gram-Schmidt procedure

Eigen values and eigen operators

Normal, unitary and positive operators

Tensor product

Partial trace

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Vector Space: Hilbert space is a finite dimensional complex vector space

Recall the vectors in conventional three dimensional space.

In a 3 dimensional coordinate space you need 3 numbers to describe

any vector. These three numbers are essentially projection along three directions

(or three axes) specified by three orthonormal unit vectors.

In three dimensional space we need a set of three orthonormal unit

vectors i, j and k to describe any other vectorP (in terms of these

unit vectors) asP = xi + yj + zk.

If we generalize this idea and do not restrict ourself to threedimension and allow the scaler projections (x, y, z etc.) to becomplex then we obtain a vector space in general. Note: All thequantum computing literature refers to a finite-dimensional complexvector space by the name Hilbert Space . We will use H to denotesuch a space.

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Large Vector Space

A vector space consists of a set of vectors ( , , .......), together witha set of scalars (a, b, c.........), which is closed under vector addition and

scaler multiplication. Thus a set of objects constitutes a vector space, ifit obeys following rules:

1. Closure: Addition of two objects of the set gives another object of

the same set (addition of two vectors gives a vector). Therefore a

well defined operation will never take you outside the vector space.

2. Has a zero: for every object V there exist another object 0 suchV + 0 = V .

3. Scalar Multiplication : If c is a scaler then cV is also a vector

4. Inverse: For every V there exists a V such that V + ( V) = 05. Associative: (V + W) + X = V + ( W + X)

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The Cn Vector Space

In the conventional cartesian coordinate system where the unit vectors, i, j and k forms a complete set of

orthonormal bases. Under this bases we can describe an arbitrary vector P as (x , y , z) or as x

yz. Thus

the elements of the column matrix is essentially the coefficients of the bases.

To describe an arbitrary n-dimensional state/vector we need a complete set of n orthonormal bases and

the state can be described by a set of n complex numbers which are the coefficients of the n bases. In

general the coefficients of the bases can be complex number and the space spanned by the set of n

orthonormal bases is called Cn space. A very special case of Cn space is one in which all the coefficients

are real and that space is known as Rn space.

In Cn vector space you need n complex numbers to describe any vector of this space. In other words any

object of this set is n tuples of complex numbers. For example

a =

z1

z2

z3

.

.

.

.

zn

is a vector in Cn space.

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Therefore we should have

z1

z2

z3

.

.

.

.

zn

+

z1

z2

z3

.

.

.

.

zn

=

z1 + z1

z2 + z2

z3 + z3

.

.

.

.

zn + zn

,

and

c

z1

z2

z3

.

.

.

.

zn

=

cz1

cz2

cz3

.

.

.

.

czn

.

For a discrete quantum system with n possible states, we will be interested in Cnspace.

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Inner Products

An inner product is a generalization of the dot product. In a vectorspace, it is a way to multiply two vectors to yield a complex number(

|

,

|

|

Cathat obeys the following rules:

1. ( , ) is linear in its second argument

(|vk, kak|wk) = kak (|vk, |wk) ,2.

(|v, |w) = (|w, |v) ,3.

(

|v

,

|v

)

0.

a Actually is bracket now we can divide it into two parts as | . Thus thebra-ket is divided in two parts and the left part (i.e |) is known as bra andthe right part (i.e. | is known as ket. Thus the bra and ket in Dirac notation isessentially division of conventional bracket in to two parts.

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Bases and Linear Independence:

If there exist a set of vectors {|v1, |v2, ..., |vn} such that any vector |v in the space can be written as a

linear combination of the vectors in the set (i.e. |v =

n

j=1|vj). Then the set is called spanning set.

A spanning set does not always forms a basis but bases forms a spanning seta

. Actually, a linearlyindependent spanning set forms a basis. The condition of linear independence of a set of vectors

{|v1, |v2, ..., |vn} is mathematically stated as

nj=1

aj |vj = 0 if and only if all aj = 0.

In other words if there does not exist any linear combination of the set of vectors {|v1, |v2, ..., |vn} which

adds to zero nontrivially then it forms a set of linearly independent vectors.

The elements of the basis set are linearly independent of each other so they satisfy the following

condition of orthogonalityb

vi|vj = 0 fo r alli = j.

aA set of vectors that spans the space is also called complete.bIn general any two vectors |v and |w are orthogonal to each other if they satisfy v|w = 0.

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Qubit: A vector in 2 dimensional space

If we have a vector in a particular direction we can easily obtain an unit vector in the same direction by

dividing the vector by its norm. Since in the direction of|v.

unit vector = |vv|v

, (1)

where

normof |v = ||v| =

v|v. (2)

It is usual practice to use a basis set whose elements are of unit magnitude and orthogonal to each other.

Such a basis set is known as orthonormal basis set and the elements of the set satisfy

vi|vj = ij,

where ij is Kronecker delta function:

In two dimensional quantum space we use a basis {|0 and|1} to describe an arbitrary state (qubit)

| = |0 + |1 and |0 and |1 satisfy

0|0 = 1|1 = 1 and 0|1 = 1|0 = 0. (3)

Thus {|0, |1} forms an orthonormal basis in 2 dimensional quantum vector space.

Problem:Assume that (4) is satisfied and then show that the

|00+|112

,|00|11

2,|01+|10

2,|01|10

2

forms a

complete set of orthonormal basis in C4 space.

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Outer Products

Let |v be a vector in the vector space V and |w be a vector in thevector space W. Then the outer product of |w and |v (which is denoted

as |wv| in Dirac notation) is a linear map from V into W defined by|wv|(|v) = |wv|v. (6)

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Linear Operators

An operator A is said to be linear if and only if it satisfies the following

properties:

A[f(x) + g(x)] = Af(x) + Ag(x) (7)and

A[cf(x)] = cAf(x) (8)

where f and g are arbitrary functions and c is an arbitrary constant.Examples: d

dxis a linear operator since,

d

dx[f(x) + g(x)] =

d

dxf(x) +

d

dxg(x)

andd

dx[cf(x)] = cd

dxf(x).

But ( )2 is not a linear operators since,

[f(x) + g(x)]2 = [f(x)]2 + [g(x)]2.Similarly, we can show that x2, d

2

dx2etc. are linear operators and

, log()

are nonlinear operators. It is an outstanding curiosity to note that theoperators occurring in quantum mechanics are essentially linear.

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Pauli Matrices

Following set of four useful matrices which acts on a 2-dimensional

vector space (or on single qubit) are known as Pauli matrices:

0 = I =

1 00 1

, (9)

1 = x = X =

0 11 0

, (10)

2 = y = Y =

0 ii 0

(11)

and

3 = z = Z =

1 00 1

. (12) The identity matrix is not always included in the set of Pauli

matrices but we have included it to make a complete set oforthogonal basis. Thus any 2 2 complex Hermitian matrices can beexpressed in terms of the the Pauli matrices.

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Properties of Pauli matrices

All single-qubit quantum gates are 2 2 unitary matrices. The Pauli matrices are some of the most

important single-qubit operations, for example, 1 represents the NOT gate. As this matrices are very

useful in quantum computing here we will note some of the properties of these matrices.

For i, j = 1, 2, 3 Pauli matrices satisfy

2i = I, (13)

ij = ji f o r i = j, (14)

det(1) = 1, (15)

T r(i) = 0 (16)

and

[i, j ] = iijkk (17)

a where ijk is Levi-Civita symbolb

a[A,B]=AB-BA defines commutation of A and B. If [A,B]=0, then we say A and B commutes.b The value of Levi-Civita symbol is 1 if (i, j, k) is an even permutation of (1,2,3), -1 for odd permutations and 0 for all other cases, to

be more precise

ijk =

1 if(i,j,k) is (1, 2, 3) or (2, 3, 1) or (3, 1, 2)

1 if(i,j,k) or (1, 3, 2) or (3, 2, 1) or (2, 1, 3)

0 otherwise

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Gram-Schmidt procedure:

Suppose we start with a non orthonormal basis (|e1, |e2, .....|en).Gram-Schmidt procedure provides us a simple prescription to generatean orthonormal basis (

|e1

,

|e2

, .....

|en

) from the non orthonormal basis

(|e1, |e2, .....|en). The prescription is as follows:1. Normalize the first basis vector:

|e1 =|e1||e1|| .

2. Find the projection of the second vector along the first and subtract

it off:

|e2

e1

|e2

|e1

.

This is orthogonal to |e1. Normalize it to obtain |e23. Subtract from |e3, its projection along |e1 and |e2:

|e3 e1|e3|e1 e2|e3|e2and normalize the resultant vector to get |e3. Continue the processtill we obtain |en.

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Example

Let us construct an orthonormal basis from the non orthonormal set {|e1, |e2:|e1 =

2

1

and |e2 =

1

3

}

Step 1: Normalize the first vector

|e1 =

21

2 1

2

1

= 15 21 .Step 2.1: construct the second vector orthogonal to the normalized first vector

|e2 e1|e2|e

1 =

1

3

.4 .2

13

2

1

=

1

3

2

1

=

1

2

Step 2.2: Normalize the vector constructed in step 2.1

|e2 =

12

1 2 12 =

1

5

12 .

Thus we have the required orthonormal set of basis as

15

2

1

, 1

5

1

2

.

Problem: Use Gram-Schmidt procedure to orthonormalize the basis |e1 = (1 i)i + 2j + ik , |e2 = (2i)i + 3j + 2kand |e3 = (1 i)i +j + (1 + i)k.

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Eigenvalues and Eigenvectors

If an operator is operated on a vector (state) and yield a scaler

multiplied by the vector (state), then the equation is called the eigen

value equation. The operator is called the eigen operator, the vector(state) is called eigen state (eigen vector) and the scaler is called eigen

value. Therefore, if we have

A|v = |v, (18)then A is the eigen operator, |v is the eigen vector (eigen state) and is the eigen value. For example, if we consider d

dxas the eigen operator

then exp(nx) is a valid eigen state having eigen value n but sin(x) is not

an eigen function of the operator

d

dx. Problem:1. Find out the eigenvalues of the Pauli matrices [hints: find roots to

the equation (i i = 0)].

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Hermitian Operators

To define Hermitian operator first we have to define adjoint operators.

Ais the adjoint operator of the operator A if A and A satisfies,

(A|v, |w) = (|v, A|w) (19)for all vectors |v and |w in the vector space V. The adjoint operatorssatisfies following properties:

A

= A, A = (A)T , (AB) = BA. (20)

Now we can loosely define Hermitian operator as an operator which

satisfies,

A = A. (21)This is essentially the definition of a self adjoint operator. In a strictsense all the self adjoint operators defined by (21) is Hermitian but allHermitian operators are not self adjoint. But the Hermitian operatorswhich are not self adjoint are not very important in the context ofquantum computing.

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Normal, unitary and positive operators:

Following definitions are also important for the understanding of thetext:

1. Normal operator:any operator that satisfies AA = AA is known asnormal operator

2. Unitary operator: If the operator A satisfies AA = AA = I, then A iscalled unitary. So it is easy to note that the unitary operators areessentially normal and an unitary operator A must satisfy A1 = A.

3. Positive Operators: operator B is called positive operator if itsatisfies, (|v, B|v) 0 for all |v in the vector space V. That means apositive operator does not have any negative eigen value. But apositive operator can have eigen value 0. Now if we exclude thispossibility and demand that all the eigenvalues B are positive then

the B is called positive definite operator. Thus a positive definiteoperator B satisfies, (|v, B|v) > 0 for all |v in the vector space V.

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Tensor Products:

A tensor product space is a larger vector space formed from two smaller ones simply by combining elements

from each in all possible ways that preserve both linearity and scalar multiplication. If V is a vector space of

dimension n and W is a vector space of dimension m. Then V W is a vector space of dimension nm. The idea

of tensor product can be clarified from the following examples: Consider Pauli matrices X = 0 11 0 andY =

0 i

i 0

tensor product of these matrices is

X Y =

0.Y 1.Y

1.Y 0.Y

=

0 0 0 i

0 0 i 0

0 i 0 0

i 0 0 0

.Similarly we have

|0 |1 = |01 = 10

01

= 0

1

0

0

.

The fact that tensor product preserves linearity and scalar multiplication can be mathematically stated as

z(|v |w) = (|v z|w) = (z|v |w) scalar multiplication

and

|v (|w1 + |w2) = |v |w1 + |v |w2)

(|w1 + |w2) |v = |w1 |v + |w2 |v

linearity.

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Trace and Partial trace:

The traceaof a n n square matrix A is defined as the sum of the elements on the main diagonal. Thus

T r(A) =

n

j Ajj . (22)Example:

The trace operation has some interesting properties, let us list few of them here:

1.

T r(AT) = TrA.

Since the diagonal elements of a square matrix does not change on transposition therefore

2.

T r(A) = T r(AB) = T r(BA)

3. The trace is a linear map i.eT r(A + B) = T r(A) + T r(B)

and

T r(rA) = r T r(A).

a The use of the term trace arises from a German word spur which is anonymous to English word spoor.

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Partial Trace

Partial trace is a generalized version of trace. It is easy to understand if

we consider state |AB HA HB. Even if the states are entangled thestate of the first qubit can in general be described by a density operatorAon HA. Popularly,

A is called reduced density operator. The

mathematical operation that calculates the reduced density operator is

the partial trace. The reduced density operator A can be defined in

terms of the density operator of the composite system as

A T rB(AB), (23)where T rB is the partial trace over system B, which is defined as

T rB (|a1a2| |b1b2|) |a1a2|T r (|b1b2|) . (24)Using the cyclic property of the trace

T r (|b1b2|) = T r (b2|b1) = b2|b1 (25)we can simplify (24) as

T rB (|a1a2| |b1b2|) |a1a2|b2|b1. (26)This operation (TrB) is often called tracing out system B.

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Example

Trace out the second qubit of the two qubit entangled state

| = 12

(|00 + |11) .The density matrix for this state is

= || = 12

(|0000| + |0011| + |1100| + |1111|) . (27)Now we can compute the partial trace as

A = T rB()

= 12

T rB (|0000| + |0011| + |1100| + |1111|)= 1

2(|00|T r(|00|) + |01|T r(|01|) + |10|T r(|10|) + |11|T r(|11|))

=1

2 (|00|0|0 + |01|1|0 + |10|0|1 + |11|1|1)= 1

2(|00| + |11|) .

(28)Problem: Trace out the first qubit of all four Bell states.

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What happens in quantum mechanics?

In quantum mechanics there exists an operator corresponding to each dynamical observable of classical

physics. Examples: Momentum and energy are classical observable, in quantum mechanics ih ddx

represents

momentum operator pxalong X direction, ihddt

represents energy.

These operators may be represented by matrices and these operators satisfy eigen value equations of the form

Aop| = | (29)

where, Aop is the eigen operator, | is the eigen-state and is the eigen value. As it is expected the eigen

values are discrete and this means all values of the observable are not possible. The meaning of the word

quantum is discrete and this discrete nature of the quantum mechanics intrinsically lies in the inherent

operator algebra.

Again as a measurement of the observable can only yield one of the eigen values of the operator that

represents the particular observable and we cannot measure anything imaginary, so the eigenvalues of

the meaningful physical operators will be real. And this requirements make it essential that the

quantum mechanical operators has to be Hermitian. Other physical conditions makes it unitary too.

Therefore, most of the time we will deal with Hermitian unitary operators.

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