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Quantum Observable

Wafa Elmannai, Varun Pande, Ajay Shreshta, Khaled Elleithy

Department of Computer Science and Engineering

University of Bridgeport

Bridgeport, CT 06604, USA

{welmanna, vpande, sajay, elleithy} @bridgeport.edu

Abstract— Since the last decade, many researchers have

started focusing on the implementation of quantum

systems and describing the states of the quantum systems.

The most of focus has been on the quantum states

strategy. Researchers considered it as outstanding

strategy as compared to classical strategies.

Complementary quantum Observables interaction and the

information are the most important features that refereed

by Observables. This shows us how uniquely these

features can define a packet of information in

mathematical term as a state. Also, it provides

technological information that indeed captures the

properties of quantum systems. In this paper, we prove

how to implement certain logical methods in quantum

computing to obtain more accurate information as

classical computing systems provide. This new

methodology can make the work more efficient. First, we

are examining the photons and consider the various spin

states, and then we consider the direction of the spinner

photons to handle more information. Second, we use the

unique property of photons to behave both as a wave and

a particle when we exploit a new property by passing the

photon through mirrors and beam splitters to

demonstrate and achieve results that are not possible in

the classical world.!

Keywords- state of Quantum systems, quantum

Observables, physical systems, Photons Function, quantum

bit.

I. INTRODUCTION

Quantum systems have been studied since 1960s

[1]. Moving from classical systems to quantum systems

is like updating program from assembly language to

high level programming languages. The motivation of

quantum computers is always to provide faster and

better computation ability. Most important, quantum

computing will help improve the communication and

cryptography servers.

Quantum computation has been accepted to overtake

the classical computation Systems for some unsolvable

problem, since Shor’s algorithm presented the quantum

algorithm of polynomial time [2].

Lately, Quantum systems proved that it has more

influence than solving only the computation issues. It

could demonstrate how to discover better solution for

even natural problems. By comparing the quantum and

classical computers, we can find that the classical

computers apply some rules on its inputs with help of

the basic algorithms and wait for an output result. With

the quantum computers and its unconventional way of

computing, these problems could be solved faster and

more efficiently. These developments are a huge

motivation to the developers to introduce quantum

computer software [3].

In addition, some computation problems that we need

to solve require more than the expected computation

power or calculations. However, those problems on

quantum systems can be done in less time using

quantum bits (qubit). Therefore, in this paper we are

going to demonstrate that the use of photons as bits in

quantum systems can change certain fixed rules in

classical systems.

II. OVERVIEW OF EXISTING QUANTUM

OBSERVABLES

We focused on time, space, motion and curving of light

as quantum observables with photons as the quantum

object to represent the qubits. Just as other sub-atomic

quantum entities, photons exhibit both particle and

wave like properties. In this section we are discussing

these properties [4]

1. Time is a constant and unalterable underlining

element in our lives. Yet when we look closely we

realize it is relative as proposed by Einstein’s

Theory of Universal Relativity. In the quantum

world, time makes an important observable.

2. Space is another variable or observable, which

when coupled with time allows us to place events

in the time-space coordinates. Space can be defined

as a continuous three dimensional variable where

objects can be placed. In the time-space

construction, time is considered to be the 4th

dimension.

3. The gravitational field (attraction) of an object

changes the shape of space time and as a result the

path of light (photons) bends slightly towards the

object (mass). Therefore, the deflection or the

curving of the path of light (photon particle) can

also be considered as an observable.

4. Motion of uneven acceleration or momentum

through cavity can create the interesting

phenomenon of entanglement, in which quantum

particles separated by extremely large distance

behave or exhibit same state properties such as

position, spin or polarization. Thus both motion

and momentum qualify as observables.

III. PROBLEM STATMENT

The quantum observable is an object of quantum

system that can be observed. It can be considered in

classical systems as an angle, a length or even

temperature. Researchers found that the quantum

observable include the structure information of

multiverse entities.!!

Some problems were considered as unsolvable

problems in classical systems; but they can be solved in

quantum systems for example Shor’s algorithm [5]. In

classical systems to factor 100 digit numbers, you

would take more than 1024 years. But shore’s proved

that it would take only 20 minutes in quantum systems.

Therefore, we will use the photon function to prove

algebraically that the NOT function of two equal bits

can be EQUAL to one and not zero as defined in

classical systems. So, our implementation can explore a

new type of NOT gate with two inputs. In our case, the

gate inputs would be the photons states in two

directions. Hence, our experiment will be based on two

terms.

Such an experiment will show that conventional way of

thinking today is not complete, which will bring new

dimensions to the development of quantum systems.

IV. RELATED WORK

In [6], the authors postulate the idea that an uneven

acceleration of cavity in space-time generates an

entanglement of greater degree in the quantum gates

between the cavity field modes. It highlights the

relationship between motion and gravity to the

generation of entanglement. Quantum computers

promise exponential growth in computing power by

storing and processing information manifold greater

than classical computers. The movement of cavities to

achieve this goal is possible as these cavity fields can

be manifested in a controlled way and observed directly

too. The paper demonstrates that the degree of

entanglements is directly proportional to the repetition

of travel segments.

Vladimir Dzhunshalev introduced the idea of non-

associative operator in non-associative quantum theory

based on observable and unobservable [7]. This paper

makes the assumption that non associative field theory

may be applied to quantization that is one of the

promising interacting fields to journalize the quantum

mechanism. Hence, the paper discusses the application

of non-associative in quantum theory that leads to

interesting characteristics The paper also investigates

the extended particles that are similar to strings in string

theory. is the author concludes that if non associative

algebra of quantum field operator consisted of

associative algebra then the operator of extended

particles can be displayed as similar to string

representation of primary particles sting theory.

An open quantum system was introduced to control the

landscape in [8]. This paper introduces the idea of how

to control the environment and influences on it by open

quantum systems. The paper focuses on critical support

for measuring maximum environmental landscape in

the liberty. The authors talk about the possibility of

developing well-organized quantum control algorithm.

Also, it introduced Kraus operator-sum representation

based on heuristic approach. This approach combines

the system and the environment model. Based on their

results, we can conclude that on basis of theoretical

knowledge and landscape analysis can be handy for

controlling external environment. It can lead to

introduce a new optimized quantum system.

In [9], the authors propose a method to investigate time

dependent correlations of non-trivial observables in

many-body ultra-cold lattice gases. The scheme uses a

quantum non-demolition matter-light interface. First, it

maps the observable of interest on the many body

systems into the light and, then, it coherently stores

such information into an external system acting as a

quantum memory. Correlations of the observable at two

(or more) instances of time are retrieved with a single

final measurement that includes the readout of the

quantum memory. Such method brings at reach the

study of dynamics of many-body systems in and out of

equilibrium by means of quantum memories in the field

of quantum simulators.

Duncan et. al., describe the interaction of

complementary quantum observables such as position

and momentum [10]. Furthermore, they demonstrate

how they play a role in the information processing in

their paper. Furthermore, they delve into data

store/encoded in complex phases and how quantum

gates logics can be facilitated simulate algorithms and

achieve transformation and complex computations. The

authors leverage the principles from classical systems

and use those same rules to provide a series of

equations to validate the concepts of quantum

mechanics.

V. PROPOSED WORK

A. System Architecture:

In this section we are describing our system architecture

by considering that a quantum simulation is complete to

give us values of a variable. Based on the logic of

information processing and quantum mechanics, we

would know the end state values. So, we can observe

the various states of the final variable before it arrives

to this state. Figure1 describes how our simulation

allows us to do followings:

• We define the logic like regular computing but in

terms of individual states rather than statements.

These states can be anywhere from 2 to infinity

because of the limited computing capabilities in

today’s world. However, we are showing our

simulation only up to 4 states.

• Once the simulation starts, we can choose the state

to run our calculations then the unique and

individual events are logged in an event logger.

• The event logger allows us to choose the unique

values at the different states when we perform

reverse information processing.

• Yet we can see the output of the individual states

or the final state as the user wishes to be.

!

Figure1: Flow chart of the proposed system

Finally, in this section we are just showing how

multiple states can be chosen to define a unique output

of quantum observable. We are not trying to give any

logical code to show input equal to the output rather we

are performing information processing by observing the

various quantum states in terms of time and in terms of

space.

B. Mathematical Model:

The idea of our experiment comes from the experiment

that Galileo performed in the 17th century. He

measured the distance by which the ball falls down,

with respect to the time. The results at t= 1, 2, 3 units

the distance fallen, were 1, 4, 9 times the distance fallen

in unit time. Later, Newton demonstrated with the Laws

of Motion that the governing equation is:

!!"#$%&'(!!"#$%&&%'! = !! ∗ !! − !!. ! ∗ ! ∗ !" !!! = !!,

All dynamical variables such as: photons, orbital

angular momentum, translational momentum, spin, total

angular momentum, energy are related to a Hermitian

operator that performances on the state of the quantum

system. However, those eigenvalues match with the

conceivable values of dynamical variables. Let’s

assume |! > and named eigen ket of the observable!U!,

per eigen value!! which subsists of a dimensional

Hilbert space. Based on that, we got

!! ! >!= ! ! >.

This equation demonstrates that if a measurement of the

observable ! is complete while the interest system was

in |! >!state, then the eigen value (!) with certainty has

to be returned using a certain measurement. Yet, if our

interest system was generally in state!|∅ >!∈ ! , then

the above absorbed eigenvalue ! will be returned with

possibility of!| < ∅|! > |

!

. However, to observe the

system in the original state, we assume A and B are

vector spaces of finite-dimensional over a field, the

dimensions are m and n. For any space U let L (U)

signify the linear operators space on U. The fractional

trace over B, Tr

B

, is a plotting:!

! ∈ ! !!!"#$%&!!"#$%&'!!"#$!! → !"!! ! ! ∈ !!(!).

Which can be defined as e

1

….e

m

and F

1

… f

n

.

Furthermore, T will have a matrix which represents:

!

!",!"!

!"#$!! ≤ !, ! ≤ !!, !! ≤ !, ! ≤ !.

That is qualified to the basis A tensor product with B.

To specify K and I in term of 1 to m, we are going to

consider the following summation:

!

!,!

= ! !

!",!".

!

!!!

This summation will be expressed in matrix which

gives a linear operator of A. That is liberated of the

bases choice and is considered as partial trace.

This can be called "tracing out" or "tracing over" A to

consent only an operator on B in the setting. Both A

and B are considered as Hilbert spaces related to

quantum systems.

The restricted trace machinist can be invariantly

demarcated as follows:

!"

!

∶ ! !!!"# $%&!!"#$%&'!!"#$!! → !(!).

The state control the restricted trace, assume a

1

,….a

m

and b

1

,….b

n

. Based on that we allow a

i

send to a

j

and

other elements go t zeros. Same thing we allow b

i

send

to b

j

from basis ! !!!!"#$%!!"#$%&'!!"#ℎ!! .! Based

on this mapping, we can get:

!"

!

!

!!!"#$%&!!"#$%&'!!"#$!!

= !"# !!!

!!

!

This is mismatch of observables in quantum mechanics:

!" − !"! ≠ !. Q and R are matrices.

This difference can show the need to extent results on

the order in which measurements of observables !! and

!!are achieved. Observables compliant to non-

commutative machinists that are named incompatible.

However, altering the states of observation can show

!!!!"#!!"! = ! −! (Imaginary state).

That can lead to a world of all new potentials.

C. Proposed Experiment:

Our proposed work is done based on two experiments.

In the first experiment, we observing the state of the

photon based on its rotation. Figure 2 shows first part of

the experiment.

Figure 2: The photons state observation experiment based on the

spinning

Assume we have eight photon’s guns (G) plus six

electronics to rotate the photon in different direction.

Hence, based on the photon’s direction, we can observe

the state. Also, we have four photon detectors. Based on

our assumed parameters which are X-X’=1; that means

if the photons was shot from x direction. So, the first

observed value which is the direction will be equal to 1.

But if it was shot from y direction then the value will be

0. Other observed value (photon state) will be based on

the photon’s spinning. If it is in the left side then the

value will be equal to 1, but if it spins to right side the

value will be 0.

As we demonstrated above we are observing two values

that are the state of the photon based on the spinning

and the photon’s direction.

So, assume at time T1 the G1 has shot the first photons

to Y direction and in the same time S1 rotates the

photon. Then N1 will detect two values based on the

assumed values. The detector will get 0, 0 value which

means the photons came from x to y direction with

negative rotation.

Based on above explanation, we can conclude our first

part that gives us two outputs out of one input. That

means in classical computing as half of computation.

By applying this experiment basis on the complexity

time with having four inputs we could get eight outputs.

So, if we want to have (00000000) output we can shoot

the photons at all times to same detector which is N1.

By that, we could observe the desired state based on our

inputs.

In the second experiment, we considered one photon in

part of two directions (X, Y) as showing in figure3; that

is using the beam to shot the photon and the reflection

of the mirror to reflect the photons on the two

directions.

Figure 3: Proposed experiment based on the photon reflection in

different direction

X"X’=1

Y"y’=0

S+1/2=1

S"1/2=0

G

3

G

4

G1

G2

S

2

S

1

S4

S3

N1 N2

N

3

N

4

X X

’

Y

Y

’

D

S

T1

0

0

0

1

1

0

0

1

T

2

T

3

T

4

G5

S

5

G6

S

6

G

8

S8

G

7

S7

t=1

t=2

t=3

t=4

Beam+

Splitter+B1

Beam+

Splitter+B2

MirrorM1

MirrorM2

(P)Photon

Photon+

Detector

Photon+

Detector

B

B

not

X+direction+1

Y+direction+0

Based on this reflection, we are going to observe the

photon’s state in each direction and in same time by

helping of the photon detector in direction x and y.

We have assumed Q (photon state) is equal to 1 if the

photon moves in the screen X direction, i.e., left to

right. As well as Q is 0 if the photon moves in the

screen Y direction, i.e. bottom to top. Figure 3 shows

the behavior of the photon after been shot.

First of all, we have directed the photon on beam

splitter (B1) from left to right when Q will be equal to

1. Then the mirror will start reflecting the direction of

the photons. So, first direction will be changed from 1

to 0 by mirror1 (M1), when based on next mirror (M2)

we will have the direction from 0 to 1. This can be

expressed in classical computing as NOT function.

The second beam splitter B2 will join both directions.

Based on our experiment, the right side of the photon

detector will be fired up by considering the results of

the photons from right side. Using quantum algebra, we

can prove this equation as well as the expectation value

functions and Hermitian matrices. By that, we can say

Beam operation !. !! is 1. So, if we conclude this

experiment, we can observe that Q remains the same.

Our experiment shows the following results:

B·NOT·B = I.

I is the identity matrix (similar to 1 in classical

algebra). It will always remain the same.

D. Results and Analysis of Our Simulation:

Based on the quantum concepts, all Quantum states at

the same nesting level have the same color. This makes

a number of things to be defined easily:

• It is easy to observe peer Quantum states.

• In case of nesting any state within other, this

can take next level of color.

• If the state’s transition moves from one state to

other, then the transition will be in color from

the front state to the target state.

A state’s edge becomes slightly heavier at every nesting

level. That can be helpful in case of differentiation. A

heavier border capitals the state is more extremely

nested. First state at any level would be designated with

a horizontal line in the top left angle of that state.

Therefore, there must be only one stat at each level.

However, in other nesting level there is no more than

one initial state that is allowed within an immediate

paternal state. We designate the final state by using a

diagonal line in the bottom right angle of that state.

In our experiment, the first state will be the initial state

as an external one when it arrived. Initial state will be

able to get a transition and moved to next state on

receiving an event T1 to T2.

In previous paragraphs we provided an explanation of

our experiment in terms of quantum concepts. Based on

this analysis, we can now easily explain our proposed

simulation that is based on Quantum State change

Logger use as showing in Figure 4.

Figure 4 shows with the use of Quantum State change

Logger, how we could implement our experiments. Our

implementation is based on creating a quantum timer

function, and then we create an event manager to log

each state. Based on this scenario, we create a runner

which shows transition from one state to many states.

Figure 4: Quantum State change Logger to create time quantum

machine

However, by passing the event manger function in our

system, we can create the quantum state machine. In

this work we provide a complete and fully implemented

quantum system to observe the system states at the

same time. In the beginning we always need to initialize

the quantum state machine to allow us observing and

controlling the states of timing quantum system. Hence,

based on Figure 4, our implementation could display

multiple events of the quantum state machine plus

changing in any state without affecting the other state.

VI. CONCLUSION

Quantum computing is yet in early stages of its

development. Considering the various advancements in

the field of computers, we have tried to show a different

perspective to computing. In this paper, we have

demonstrated that observing a particle as different

variable in terms of space gives us unique outputs and

considering the same particle in terms of its own

orientation gives us a new variable as an output. We

have successfully simulated the idea of a quantum

based gate while we observe a particle over space and

time. Considering the unique properties of the same

photon where it can behave as a wave or a particle, we

have shown that new logical gates are possible to be

built.

We have presented a simulation that enables us to log

states in a quantum state machine. Thus, providing us

with enough information to do information processing

which is an significant issue with quantum computing

as there is no way of properly measuring the quantum

states.

In the future we are planning to extend this work to

more practical applications with the objective to

improve the time complexity that might help making

quantum computing a reality.!

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