Quantum Observable

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.

Full text

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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