Int J Theor PhysDOI 10.1007/s10773-014-2120-y

New Maximally Entangled States and PatternClassification in Two-Qubit System

Manu Pratap Singh ·B. S. Rajput

Received: 7 January 2014 / Accepted: 26 March 2014© Springer Science+Business Media New York 2014

Abstract Pattern classifications have been performed by employing the method of Grover’siteration on Bell’s MES and Singh-Rajput MES in two-qubit system and it has been demon-strated that for any pattern classification in a two-qubit system the maximally entangledstates of Singh-Rajput eigen basis provide the most suitable choice of search states while inno case any of Bell’s states is suitable for such pattern classifications.

Keywords Pattern classification · Grover’s iterations · Bell’s states: Singh-Rajput states ·Magic basis

1 Introduction

Quantum entanglement was already pointed out by Schrodinger [1] to be a crucial elementof quantum mechanics but the research has been refocused on it in the last fifteen yearsbecause the fields of quantum information theory [2, 3], quantum computers [4], universalquantum computing network [5], teleportation [6], dense coding [7, 8], geometric quantumcomputation [9, 10] and quantum cryptography [11–13] are being developed rather quickly.The physically allowed degree of entanglement and mixture is a timely issue given that theentangled mixed states could be advantageous for certain quantum information situation[14]. The simplest non-trivial multi-particle system that can be investigated theoretically,as well as experimentally, consists of two qubits which display many of the paradoxical

M. P. SinghDepartment of Computer Science, Institute of Engineering and Technology, Khandari Campus,DR. BR Ambedkar University, Agra, UP, Indiae-mail: manu p singh@hotmail.com

B. S. Rajput (�)I-11, Gamma-2, Greater Noida, UP, Indiae-mail: bsrajp@gmail.com

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features of quantum mechanics such as superposition and entanglement. Basis of entan-glement is the correlation that can exist between qubits. From physical point of view,entanglement is still little understood. What makes it too powerful is the fact that sincequantum states exist as superposition, these correlations exist in superposition as well andwhen superposition is destroyed, the proper correlation is somehow communicated betweenthe qubits. It is this communication that is the crux of entanglement. Entanglement is oneof the key resources required for quantum computation and hence the experimental creationand measurement of entangled states is of crucial importance for various physical imple-mentations of quantum computers. By quantum entanglement we mean quantum correlationamong the distinct subsystems of the entire composite system. For such correlated quantumsystems, it is not possible to specify the quantum state of any subsystem independently ofthe remaining subsystems. The generation of quantum entanglement among spatially sep-arated particles requires non-local interactions through which the quantum correlations aredynamically created [15] but our present knowledge of quantum entanglement is not at allsatisfactory [16]. However, the functional dependence of the entanglement measures likeconcurrence [17, 18], i-concurrence [19] and 3-tangle [20] on spin- correlation functionshave been established [21]. Protection of quantum states of open system from decoherenceis essential for robust quantum information processing and quantum control in quantumcomputers. The fact that the measurements can slow down decoherence is well known asquantum Zeno-effect [22, 23]. Recent papers concerning entanglement in quantum- spinsystems address the questions about the maximum entanglement of nearest neighbor qubitsbelonging to a ring of N qubits in a translational invariant quantum state [24], the depen-dence of entanglement between two qubits on temperature and external magnetic field[25–27], and three-qubits XYZ- model [28, 29] and XY-model [30]. In our very recent paper[31] a new set of maximally entangled states (MES) (Singh-Rajput states) constituting avery powerful and reliable eigen basis (Singh-Rajput Basis) (different from magic bases)of two-qubit systems has been constructed, the functional dependence of the entanglementmeasures on spin correlation functions has been established, correspondence between evo-lution of MES of two-qubit system and representation of SU(2) group has been worked outand the evolution of MES under a rotating magnetic field has been investigated.

In the present paper, the pattern classifications have been performed in straight forwardapproach employing the method of Grover’s iterate [3] on Bell’s MES and Singh-RajputMES in two-qubit system. It has been demonstrated that none of the maximally entangledBell’s state is suitable for correct pattern classification of the point ‘0?’ (where ? stands for 0or 1) upon measurement of two-qubit system on various iterations of Grover’s search whilethe first two maximally entangled states of Singh-Rajput basis are most suitable choice forthe desired pattern classification. It has also been demonstrated that any of the other twomaximally entangled states of Singh-Rajput basis is the most suitable choice as search statefor the desired pattern classification ‘1?’ based on Grover’s iterative search algorithm intwo-qubit system while the probability of correct desired pattern classification in this casealso never exceeds the limit of fifty percent when any of the Bell’s maximally entangled stateis chosen as search state. It has also been shown that the choice of phase inversion superpo-sition as the search state is also not suitable for pattern classification for a two-qubit systemsince in this case the probability of correct classification never exceeds the limit of 25 %.Performing pattern classifications of points ‘00’ and ‘11’ respectively, based on Grover’siterative search algorithm, it has been demonstrated that for any pattern classification in atwo-qubit system the maximally entangled states of Singh-Rajput Eigen basis provide themost suitable choice of search states and in no case any of Bell’s states is suitable for suchclassifications.

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2 Necessary and Sufficient Conditions for a Two-Qubit State to be MES

A general two-qubit state may be written as:∣∣∣� >= 1√

γ[a |00 > + b |01 > + c |10 > +d |11 > ]

= 1√γ

⎡

⎢⎢⎣

a

b

c

d

⎤

⎥⎥⎦

(2.1)

whereγ = |a|2 + |b|2 + |c|2 + |d|2 (2.2)

This state may also be written as:

|� > = 1√(2γ )

[i (a − d) |φ1 > + (a + d) |φ2 > +i (b+ c) |φ3 > + (b − c) |φ4 > ]

(2.3)where

|φ 1 >= − 1√2(|00 > − |11 >) ; |φ 2 >= − 1√

2(|00 > + |11 >)

|φ 3 >= − 1√2(|01 > + |10 >) ; |φ 4 >= − 1√

2(|01 > + |10 >) ,

(2.4)

are maximally entangled Bell’s states which satisfy the following conditions;∑4

μ=1|φ μ >< φμ |= 1 and < φμ |φ v >= δμv (2.5)

showing that these states constitute the orthonormal complete set and hence form the eigen-basis (magic basis) [17] of the space of two level Q-bits. Any two-qubit state may be writtenin magic basis as:

∣∣∣∣∣ψ >=

4∑

k=1

bk |φ k >

with its concurrence defined as [17, 18]

|C (|ψ >) =∣∣∣∣

∑4

k=1b2k

∣∣∣ (2.6)

If the concurrence C (|ψ >) = 1, the state is maximally entangled while for C (|ψ >) = 0,the state |ψ > is not entangled at all and for

0 < C (|ψ >) < 1, (2.7)

the state |ψ > is partially entangled. The concurrence of the state, given by (2.1) may bewritten as

C (|ψ >) = 2

γ|ad − bc| (2.8)

Thus, for non-entangled state (i. e. separable state), we have:

ad = bc (2.9)

and for partially entangled states,

0 <2 |ad − bc|

γ< 1 (2.10)

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For the state |ψ >, given by (2.1) to be maximally entangled state (MES), we have:

2 |ad − bc| = |a|2 + |b|2 + |c|2 + |d|2 (2.11)

or∣∣∣

(

a ∓ d∗)∣∣∣2 + ∣

∣(

b ± c∗)

∣∣∣2 = 0 (2.12)

This can be true either for;

d = a∗ and c = −b∗ (2.13)

or for;

d = −a∗ and c = −b∗ (2.14)

These are the necessary conditions for the state |ψ > of (2.1) to be maximally entangled.Thus, we get the following two sets of MES:

|�1 >= 1√

(2[|a|2 + |b|2}

[a|00 > +b|01 > −b ∗ |10 > +a ∗ |11 >] (2.15)

and

|�2 >= 1√

(2[|a|2 + |b|2}

[

a|00 > +b|01 > +b∗|10 > −a∗|11 >]

(2.16)

Bell states (i.e. magic bases), given by (2.4), may readily be obtained from the state |�1 >

of (2.15) on substituting:

(a = 1, b = 0) ; (a = −i, b = 0) ; (a = 0, b = 1) ; and (a = 0, b = −i) (2.17)

For these sets of values of a and b, the state |�2 > of (2.16) gives |∅1 > and |∅4 > withphase rotated by π

2 and |∅2 > and |∅3 > with phase rotated by −π2 .

Other maximally entangled two-qubit states which form the orthonormal complete set(i.e. eigen basis) may be obtained as follows by putting a = ±1 and b = 1 in state |�2 >

of (2.16) and a = 1, b = ±1 in state |�1 > of (2.15);

|ψ1 >= 1

2[−|00 > +|01 > +|10 > +|11 >] , (2.18)

|ψ2 >= 1

2[|00 > −|01 > +|10 > +|11 >] , (2.19)

|ψ3 >= 1

2[|00 > +|01 > −|10 > +|11 >] , (2.20)

|ψ4 >= 1

2[|00 > +|01 > +|10 > −|11 >] , (2.21)

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with their density matrices respectively given by:

ρψ1 = 14

⎡

⎢⎢⎣

1 −1 −1 −1−1 1 1 1−1 1 1 1−1 1 1 1

⎤

⎥⎥⎦,

ρψ2 = 14

⎡

⎢⎢⎣

1 −1 1 1−1 1 −1 −1

1 −1 1 1−1 −1 1 1

⎤

⎥⎥⎦,

ρψ3 = 14

⎡

⎢⎢⎣

1 1 −1 11 1 −1 1

−1 −1 1 −1−1 1 −1 1

⎤

⎥⎥⎦,

ρψ4 = 14

⎡

⎢⎢⎣

1 1 1 −11 1 1 −11 1 1 −1

−1 −1 −1 1

⎤

⎥⎥⎦

(2.22)

None of which can be factorized at all. The concurrence for each of these states is unity andthese states constitute the orthonormal set since

< ψμ|ψv >= δμv

and∑4

μ=1 |ψμ >< ψμ| = 1(2.23)

Thus the set of Bell states is not the only eigen basis (magic eigen basis) of the space of two-qubit system but the set of MES given by (2.18–2.21) also constitute a very powerful andreliable eigen basis of two-qubit systems. This is the new eigen basis and to differentiateit from the already known Bell’s basis we have designated it in our recent paper [31] asSingh-Rajput basis for its possible use in future in the literature. The MES constructed inthe form given by (2.18–2.21) have been correspondingly labeled as Singh-Rajput states. Inthis basis, various qubits of two-qubit states may be written as:

|00 >= 12 [|ψ2 > +|ψ3 > +|ψ4 > −|ψ1 >] ,

|01 >= 12 [|ψ1 > +|ψ3 > +|ψ4 > −|ψ2 >] ,

|10 >= 12 [|ψ1 > +|ψ2 > +|ψ4 > −|ψ3 >] ,

|11 >= 12 [|ψ1 > +|ψ2 > +|ψ3 > −|ψ4 >] ,

(2.24)

Substituting these relations in (2.4), Bell states may be constructed as follows in this newbasis;

|φ1 >= −i√2

[|ψ4 > −|ψ1 >] ; |φ2 >= 1√2

[|ψ2 > +|ψ3 >] ;|φ3 >= −i√

2[|ψ4 > +|ψ1 >] ; |φ4 >= 1√

2[|ψ3 > −|ψ2 >]

(2.25)

Concurrence of each of Bell states in this basis also is unity showing the invariance ofconcurrence in different bases.

Condition (2.10) for partial entanglement shows that if any coefficient of qubits in thestate |� >, given by (2.1), is vanishing, then the state is necessarily partially entangledand its concurrence is 2

3 if the sum of squares of moduli of non-zero coefficients is 3.For instance, let b = 0, and |a|2 + |c|2 + |d|2 = 3, then the concurrence given by (2.8)becomes 2

3 when a = ±1, c = ±1 and d = ±1. It may be readily shown that all the states1√3

[±|00 > ±|01 > ±|11 >] are partially entangled with concurrence = 23 .

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3 Method of Grover’s Iteration for Pattern Classification

The use of entanglement for constructing a quantum classification system has already beendiscussed [32] in a general way. Let us consider B = {0,1} and the set T = {(xi, yi)} be aset of m pairs of points, xi in Bn and yi in B. In pattern classification a quantum system isconstructed for correctly labeling points in T and for generalizing in a reasonable way tolabel other points belonging to Bn, which do not belong toT. Thus in pattern classificationsuch a quantum system is constructed that approximates the function f : Bn → B fromwhich the set T was drawn.There are three different approaches to state preparation, basedon information in set T of (n + 1) two states quantum systems (|0 > and |1 >);

i) Inclusion,ii) Exclusion,

iii) Phase Inversion

Inclusion is most intuitive where basis states not in T have zero coefficients and those in Thave non-zero coefficients in the superposition:

|�inc >= 1√m

∑

xiyi∈T |xiyi > (3.1)

Exclusion is an opposite approach, where basis states in T have zero coefficients and thosenot in T have non-zero coefficients in the superposition:

|�exc >= 1√2n −m

∑

xiyiACT |xiyi > (3.2)

In Phase Inversion, all basis states are included with coefficients of equal amplitudes butwith different phases based on membership in T:

|�ph >= 1√2n

(∑

xiyiACT |xiyi > −∑

xiyi∈T |xiyi >)

(3.3)

After state preparation, the pattern classification may be performed in straight forwardapproach employing the method of Grover’s iterate [3] which is described as a product ofunitary operators GR applied to quantum state iteratively and probability of desired resultmaximized by measuring the system after appropriate number of iterations. Here the opera-tor R is phase inversion of the state(s) that we wish to observe upon measuring the system. Itis represented by identity matrix I with diagonal elements corresponding to desired state(s)equal to -1 and the operator G is descried as an inversion about average:

If|� >= 1√2n+1

∑

(xiyi )∈Bn |xiyi > then G = 2 |� >< �| − 1 (3.4)

Let us apply this method of pattern classifications by using the maximally entangled statesfor two-qubit system in the following sections.

4 Pattern Classification using Bell’s States in Two-Qubit System

Let us choose T = {(00), (11)} for the two-qubit system with the set Bn given by

Bn = {(0, 0) , (0, 1) , (1, 0) , (1, 1)} (4.1)

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Then (3.4) become

|� >= 1

2[|00 > +| 01 > + |10 > +| 11 >] , (4.2)

and

G = 1

2

⎡

⎢⎢⎣

−1 1 1 11 −1 1 11 1 −1 11 1 1 −1

⎤

⎥⎥⎦

(4.3)

Let us find the probability of observing the correct classification of the point ‘0?’,where ? denotes 0 or 1, upon measurement on each iteration of Grover’s search applied tosuperposition given by (3.1), (3.2) and (3.3).

For the given search point the involved qubits are |00 > and |01 > and therefore thephase inversion operator R is given by

R =

⎡

⎢⎢⎣

−1 0 0 00 −1 0 00 0 1 00 0 0 1

⎤

⎥⎥⎦

(4.4)

Using relations (4.3) and (4.4), we have the following operator for Grover’s iterations;

D = GR = 1

2

⎡

⎢⎢⎣

1 −1 1 1−1 1 1 1−1 −1 −1 1−1 −1 1 −1

⎤

⎥⎥⎦

(4.5)

For an iteration of Grover’s search, let us apply this operator on the following state obtainedfrom (3.1),

|�inc >= 1√2

[|00 > +| 11 >] = 1√2

⎛

⎜⎜⎝

1001

⎞

⎟⎟⎠

(4.6)

This quantum state is the superposition of only the states |00 > and |11 > which havemaximum Hamming spread between two qubits. This state is maximally entangled state|ϕ2 >, second Bell’s state, which is an element of Bell’s magic basis [17]. We get

D|�inc >= 1√2

[|00 > −| 11 >] = 1√2

⎡

⎢⎢⎣

100

−1

⎤

⎥⎥⎦

= i|φ1 >, (4.7)

where |φ1 > is the first maximally entangled state of Bell, an element of magic basis. Itgives the probabilities PC and PW of correct classification ‘00’ and incorrect classification‘01’, respectively as:

PC = 0.5 and PW = 0 (4.7a)

with the total probability of the classification ‘0?’, P = PC + PW , given as 0.5, the prob-ability of irrelevant classification (other than that in which we are interested) given as

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PR = 0.5 and the conditional probability (if the desired pattern is classified then theprobability that the classification will be the correct one) is

Pcond = PC

PC + PW

= 1

After second iteration of the operator (4.5), we get:

D(

D |�inc >) = − 1√

2[|01 > +| 10 >] = −i |∅3 > (4.8)

where |∅3 > is the third Bell’s state, an element of magic basis. It gives the following valuesof different probabilities of classification after the second iteration;

PC = 0, PW = 0.5, P = 0.5, PR = 0.5, Pcond = 0 (4.8a)

Similarly, the third iteration gives the state

1√2= [|01 > −| 10 >] = |∅4 > (4.9)

which is the fourth Bell’s state, an element of magic basis. It also gives the same val-ues of various probabilities as obtained after second iteration. On fourth iteration the state|�inc >=|φ2 > is recovered. The same periodic behavior is repeated on further iterations.Thus the probability of observing the correct classification upon system measurement onvarious iterations of Grover’s search applied to second maximally entangled Bell’s state,i.e. inclusion superposition, never exceeds the value 0.5 and the probability of irrelevantclassification always remains 0.5.

Exclusion superposition, given by (3.2), has the following form in this case:

|�∈xc >= 1√2

[|01 > +| 10 >] = i |∅3 > (4.10)

which gives the following classification after the first iteration with the operator of (4.5);

D|�∈xc >= 1√2

[|01 > −| 10 >] =|∅4 > (4.11)

with

PC = 0;PW = 0.5;PR = 0.5;P = 0.5 and Pcond = 0 (4.12)

Second iteration classifies the state − |∅4 > with PC = 0.5;PW = 0;PR = 0.5;P =0.5, Pcond = 1. The third iteration classifies the state i |∅2 > and the fourth iterationrestores the original state |�∈xc > with inverted sign. The periodic behaviour is repeated onfurther iterations. Thus the probability of observing the correct classification upon systemmeasurement on various iterations of Grover’s search applied to third maximally entangledBell’s state, i.e. exclusion superposition, never exceeds the value 0.5 and the probabilityof irrelevant classification always remains 0.5. The similar results follow on choosing theprobability of observing correct classification upon system measurement on various itera-tions of Grover’s search applied to first and fourth maximally entangled Bell’s states also.Thus none of the maximally entangled Bell’s state is suitable for the pattern classificationin two-qubit system.

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Let us now choose the state prepared by phase inversion according to (3.3) which maybe written as follows for a two-qubit system;

|�ph >= 1

2[− |00 > +| 01 > + |10 > −|11 > ] (4.13)

which is a non-entangled state. On first iteration with the operator given by (4.5), this stateis transformed to the state

|� >= 1

2[− |00 > +| 01 > − |10 > +|11 > ] (4.14)

with the corresponding pattern classification ‘0?’ having the following probabilities;

PC = 0.25;PW = 0.25;PR = 0.5;P = 0.5 and Pcond. = 0.5 (4.15)

On the second iteration the original state∣∣�ph > is restored and the same periodicity is

repeated on further iterations. Thus this choice of search state is also not suitable for patternclassification for a two-qubit system.

5 Pattern Classification using New Maximally Entangled States in Two-Qubit System

Let us find the probability of observing the correct classification of the point ‘0?’, where ‘?’denotes 0 or 1, upon measurement on each iteration of Grover’s search applied to superposi-tion given by (2.18) i.e. the first maximally entangled Singh-Rajput state. The first iterationof this state by operator D, given by (4.5), leads to the pure state |01 > with the followingprobabilities of desired classification;

PC = 0;PW = 1;PR = 0;P = 1 and Pcond. = 0 (5.1)

The second iteration leads to the second MES |ψ2 >, with reversed sign, of Singh-Rajputbasis. It gives the following probabilities of desired classification;

PC = 0.25;PW = 0.25;PR = 0.5;P = 0.5 and Pcond. = 0.5 (5.2)

Third iteration leads to the pure state −|00 > with the following probabilities of desiredclassification;

PC = 1;PW = 0;PR = 0;P = 1 and Pcond. = 1 (5.3)

Fourth iteration restores the state |ψ2 > and the same periodicity is repeated in furtheriterations. Thus on the third iteration the first state of Singh-Rajput basis gives the 100 %probability of the correct pattern classification with the 0 % probability of irrelevant clas-sification (other than that in which we are interested).Though the probability of correctclassification on first iteration is zero but the total probability of classification of the desiredpattern ‘0?’, P = PC+ PW , is hundred percent and the probability of irrelevant classifica-tion is zero percent. The similar periodicity of probability of desired pattern classificationis observed by choosing the second maximally entangled Singh-Rajput state|ψ2 >, givenby (2.19), with the hundred percent probability of correct pattern classification after thefirst iteration and hundred percent total probability of classification of the desired patternand zero percent probability of irrelevant classification after the first and third iterations. Itshows that these states |ψ1 > and |ψ2 > of Singh-Rajput basis are most suitable choicefor desired pattern classification for a two-qubit system. On the other hand, if the third andfourth maximally entangled states, given by (2.20) and (2.21), are chosen as the search statesthen also the behavior of probabilities after third iterations is repeated periodically but theprobability of correct classification never exceeds beyond 25 % and the total probability of

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desired classification does not exceed beyond 50 % while the probability of irrelevant clas-sifications never falls below 50 %. Thus these states (third and fourth) are not good choiceas search state for the desired pattern classification ’0?’ in two-qubit system.

Let us now try to classify the point ‘00’ i.e. the pure state based on Grover’s iter-ative search algorithm by using the maximally entangled states of (2.18–2.21). For thisclassification the inversion operator is obtained as:

R =

⎡

⎢⎢⎣

−1 0 0 00 1 0 00 0 1 00 0 0 1

⎤

⎥⎥⎦

(5.4)

Multiplying the operator G with this operator, we get the operator D as:

D = 1

2

⎡

⎢⎢⎣

1 1 1 1−1 −1 1 1−1 1 −1 1−1 1 1 −1

⎤

⎥⎥⎦

(5.5)

The first iteration of this operator on the first state |ψ1 > of (2.18) leads to an un-entangledstate with only 25 % probability of the desired pattern classification while the second itera-tion leads to the pure state 100 > with hundred percent probability of the classification ofthe desired pattern. The third iteration of this state restores the state |ψ1 > with reversedsign and the fifth iteration again gives the hundred percent probability of the classificationof the desired pattern while the sixth iteration will restore the state |ψ1 > and the same peri-odicity is repeated on further iterations. Thus this state gives hundred percent probability ofcorrect classification on second and fifth iterations and hence the choice of the state |ψ1 >

as the search state is most suitable for the desired pattern classification. On the other handthe probability of the correct desired pattern classification is not better than 25 % with anyother state of Singh-Rajput basis, given by (2.19–2.21).

In our attempt to classify the point ‘00’ i.e. the pure state, based on Grover’s iterativesearch algorithm by using any of the Bell state, constituting the magic basis, we find thatthe probability of correct pattern classification does not exceed beyond fifty percent on anynumber of iterations and hence Bell states are not the suitable choice for the classificationof this pattern also.

Let us now classify the point ‘1?’ (where ? is 0 or 1) based on Grover’s iterative searchalgorithm by using our new maximally entangled states, given by (2.18–2.21). The inversionoperator R in this case may be written as:

R =

⎡

⎢⎢⎣

1 0 0 00 1 0 00 0 −1 00 0 0 −1

⎤

⎥⎥⎦

(5.6)

Using this relation and the operator G, given by relation (4.3), we get the search operator:

D = 1

2

⎡

⎢⎢⎣

−1 1 −1 −11 −1 −1 −11 1 1 −11 1 −1 1

⎤

⎥⎥⎦

(5.7)

The first iteration of this operator on the maximally entangled state |ψ3 >, given by (2.20),gives hundred percent probability of correct pattern classification ‘11’ and the zero percent

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probability of irrelevant classification while the third iteration gives the hundred percenttotal probability of the desired pattern classification. On the fourth iteration the state |ψ3 >

is restored and the same periodic behaviour is repeated in the further iterations. If we choosethe fourth maximally entangled state |ψ4 > as the search state then the first iteration byoperator D gives the hundred percent total probability of the desired classification and thethird iteration gives the hundred percent probability of correct pattern classification ‘11’ andthe zero percent probability of irrelevant classification. Thus these two maximally entangledstates |ψ3 > and |ψ4 > are the most suitable choice as search state for the desired patternclassification ‘1?’ based on Grover’s iterative search algorithm in two-qubit system. On theother hand if the first and second maximally entangled states, given by (2.18) and (2.19)are chosen as the search states then the probability of correct classification never exceedsbeyond 25 % and the total probability of desired classification does not exceed beyond50 % while the probability of irrelevant classifications never falls below 50 %. Thus thesetwo maximally entangled states (first and second) are not suitable for the desired patternclassification. Similarly the probability of correct pattern classification never exceeds thelimit of fifty percent when any of the Bell’s maximally entangled state is chosen as searchstate.

If the point’11’ is to be classified by the approach based on Grover’s iterative searchalgorithm in two-qubit system then the inversion operator may be written as:

R =

⎡

⎢⎢⎣

1 0 0 00 1 0 00 0 1 00 0 0 −1

⎤

⎥⎥⎦

(5.8)

and then the operator D gets the following form;

D = 1

2

⎡

⎢⎢⎣

−1 1 1 −11 −1 1 −11 1 −1 −11 1 1 1

⎤

⎥⎥⎦

(5.9)

Second and fifth iterations of the fourth state |ψ4 > of the new set of maximally entangledstates, given by (2.21), give the hundred percent probability of desired pattern classifica-tion and on sixth iteration the original state |ψ4 > is restored and same periodic behavioris repeated in the further iterations. On the other hand the probability of the correct desiredpattern classification is not better than 25 % with any of other three new maximally entan-gled states, given by (2.18–2.20). Similarly, the probability of correct pattern classificationdoes not exceed beyond fifty percent on any number of iterations on any of the Bell’s statesand hence Bell states are not the suitable choice for the classification of this pattern also.Thus the most suitable choice for the desired pattern classification ‘11’, based on Grover’siterative search algorithm, is the fourth maximally entangled state |ψ4 > of Singh-Rajputeigen basis for a two-qubit system.

6 Discussion

Results (4.7a) and (4.8a) show that the probability of observing the correct pattern clas-sification of the point ‘0?’, upon measurement of two-qubit system on various iterationsof Grover’s search applied to second maximally entangled Bell’s state, never exceeds thevalue 0.5 and the probability of irrelevant classification always remains 0.5. Similarly,

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result (4.12) shows that on applying various iterations of Grover’s search on the third max-imally entangled Bell’s state the probability of correct pattern classification never exceedbeyond the limit of fifty percent. The similar results follow on choosing the probability ofobserving correct classification upon system measurement on various iterations of Grover’ssearch applied to first and fourth maximally entangled Bell’s states also. Thus none of themaximally entangled Bell’s state is suitable for the pattern classification in two-qubit sys-tem. Result (4.15) shows that the choice of phase inversion superposition, given by (4.13),as the search state is also not suitable for pattern classification for a two-qubit system sincein this case the probability of correct classification never exceeds the limit of 25 %. It showsthat the probability of correct pattern classification based on any of Bell’s state does notexceed the limit of fifty percent on any number of iterations and that based on phase inver-sion superposition, remains fixed at twenty five percent on any number of iterations andhence these states (Bell’s states and

∣∣�ph > are not the suitable choice as search state for

the pattern classification based on the Grover’s iterative search algorithm for a two-qubitsystem.

Results (5.1–5.3) show the behaviors of probabilities of correct desired pattern classi-fication based on Grover’s iteration algorithm applied on the first and second maximallyentangled Singh-Rajput states |ψ1 > and |ψ2 > given by (2.18) and (2.19) respectively.It shows that on the third iteration the first state of Singh-Rajput basis gives the 100 %probability of the correct desired pattern classification with the 0 % probability of irrele-vant classification and on the first iteration of the second state of this basis gives 100 %probability of the correct desired pattern classification. Thus these states |ψ1 > and |ψ2 >

of Singh-Rajput basis are most suitable choice for the desired pattern classification for atwo-qubit system. It has also been shown that the total probability of desired pattern classi-fication is hundred percent after first, third and fifth iterations when any of these maximallyentangled state is used as the search state in Grover’s iteration algorithm. Starting with thesearch operator given by (5.7), it has been demonstrated that any of the other two maxi-mally entangled states |ψ3 > and |ψ4 > of Singh-Rajput basis are the most suitable choiceas search state for the desired pattern classification ‘1?’ based on Grover’s iterative searchalgorithm in two-qubit system. On the other hand the probability of correct desired patternclassification in this case also never exceeds the limit of fifty percent when any of the Bell’smaximally entangled state is chosen as search state.

Using the search operator, given by (5.5), to classify the point ‘00’ i.e. the pure statebased on Grover’s iterative search algorithm it has been shown that the first maximallyentangled state |ψ1 > of Singh-Rajput basis gives hundred percent probability of correctclassification on second and fifth iterations and hence the choice of this state as the searchstate is most suitable for the desired pattern classification. On the other hand the probabil-ity of desired pattern classification never exceeds the limit of fifty percent when any of theBell’s maximally entangled state is chosen as search state. Starting with the search oper-ator, given by (5.9), to classify the point’11’ by the approach based on Grover’s iterativesearch algorithm in two-qubit system, it has been shown that the most suitable choice for thedesired pattern classification is the fourth maximally entangled state |ψ4 > of Singh-Rajputeigen basis for a two-qubit system while the probability of correct pattern classificationdoes not exceed beyond fifty percent on any number of iterations on any of the Bell’sstates and hence Bell states are not the suitable choice for the classification of this patternalso. Thus for any pattern classification, based on Grover’s iterative search algorithm, in atwo-qubit system the maximally entangled states of Singh-Rajput eigen basis provide themost suitable choice of search states and in no case any of Bell’s states is suitable for suchclassifications.

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