A Study on the basics of Quantum Computing
Prashant
Department d’Informatique et de recherché operationnelle,
Universite de Montreal, Montreal. Canada.
{prashant}@iro.umontreal.ca
http://www-etud.iro.umontreal.ca/~prashant/
Abstract
Quantum theory is one of the most successful theories that have influenced the course of
scientific progress during the twentieth century. It has presented a new line of scientific
thought, predicted entirely inconceivable situations and influenced several domains of
modern technologies. There are many different ways for expressing laws of science in
general and laws of physics in particular. Similar to physical laws of nature, information
can also be expressed in different ways. The fact that information can be expressed in
different ways without losing its essential nature, leads for the possibility of the automatic
manipulation of information.
All ways of expressing information use physical system, spoken words are
conveyed by air pressure fluctuations: “No information without physical representation”.
The fact that information is insensitive to exactly how it is expressed and can be freely
translated from one form to another, makes it an obvious candidate for fundamentally
important role in physics, like interaction, energy, momentum and other such abstractors.
This is a project report on the general attributes of Quantum Computing and Information
Processing from a layman’s point of view.
Keywords: computation, EPR,
transformation, decoherence.
quantum
mechanics,
superposition,
unitary
TABLE OF CONTENTS
Abstract
Chapter 1: Introduction
Chapter 2: Literature Survey
2.1 A Brief History of Quantum Computing
2.2
Limitations of Classical Computers and birth of art of Quantum Computing
2.2.1 Public key Cryptography and Classical factoring of big integers.
2.2.2 Quantum Factoring
2.2.3 Searching of an item with desired property.
2.2.4 Simulation of quantum system by classical computer.
2.3
Quantum Computing: A whole new concept in Parallelism
2.4
Quantum Superposition and Quantum Interference: Conceptual
visualization of Quantum Computer.
2.5
Quantum Entanglement
2.5.1 Bertleman’s Socks
2.5.2 EPR situation, Hidden Variables and Bell Theorem
2.5.2.1 An EPR situation
2.5.2.2 Bell Inequalities.
2.6
Quantum Teleportation and Quantum Theory of Information
2.7
Thermodynamics of Quantum Computation
2.8
Experimental Realization of Quantum Computer
2.8.1 Heteropolymers
2.8.2 Ion Traps
2.8.3 Quantum Electrodynamics Cavity
2.8.4 Nuclear Magnetic Resonance
2.8.5 Quantum Dots
2.8.5.1 Josephson Junctions
2.8.5.2 The Kane Computer
2.8.6
2.9
Topological Quantum Computer
Future Directions of Quantum Computing
2
Chapter 3: A New outlook to the Principle of Linear Superposition
3.1 Modification of Wave function as a requirement of Quantum
Teleportation
3.2 Introduction of EPR correlation term in Expansion Theorem
3.2.1 Suitability of Quantum bits for Quantum Computation
3.3 An alternative interpretation of Quantum No Cloning Theorem.
Chapter 4: Experimental realization of Quantum Computers
4.1 Materials of Low dimensionality-Quantum Dot a promising
candidate.
4.2 Need for Modified Coulomb Potential and its analysis
4.3 Analysis of Quantum Dots using Modified Coulomb Potential.
4.4 Study of Quantum Wires using Modified Coulomb Potential
4.5 Visit to Nano Technology Lab in Barkatullah University, Bhopal
Chapter 5: Future Directions of Research
5.1 Explanation of Measurement Problem by Symmetry breaking
5.2 EPR Correlation: Interacting Hamiltonian Vs Non linear wave
function
5.3 Possibility of third paradigm in Quantum Mechanics
5.4 Conclusion and Future Scope
References
3
List of Figures:
Fig 1: Showing number of dopant impurities involved in logic in bipolar transistors with year.
Fig 2: Beam splitting of light
Fig3: Example to show wave particle duality of light
Fig 4: EPR paradox description using He atom
Fig 5: Graphical description of the EPR situation
Fig 6: Quantum Dots
Fig 7: Quantum Teleportation using Entanglement
Fig 8: The graphical representation of the Modified Coulomb potential
Fig 9: Plot of Wave Function Vs. distance in a Quantum Dot
Fig 10: Quantum Wire as a 1-D system
Fig 11: Quantum Dots as seen from an Atomic Force Microscope
Fig 12: The tip of the Scanning Tunneling Microscope
Fig 13: AFM Plots of Quantum Dots prepared in laboratory.
4
CHAPTER 1
1.0 INTRODUCTION
With the development of science and technology, leading to the advancement of
civilization, new ways were discovered exploiting various physical resources such as
materials, forces and energies. The history of computer development represents the
culmination of years of technological advancements beginning with the early ideas of
Charles Babbage and eventual creation of the first computer by German engineer Konard
Zeise in 1941. The whole process involved a sequence of changes from one type of
physical realization to another from gears to relays to valves to transistors to integrated
circuits to chip and so on. Surprisingly however, the high speed modern computer is
fundamentally no different from its gargantuan 30 ton ancestors which were equipped
with some 18000 vacuum tubes and 500 miles of wiring. Although computers have
become more compact and considerably faster in performing their task, the task remains
the same: to manipulate and interpret an encoding of binary bits into a useful
computational result.
The number of atoms needed to represent a bit of memory has been decreasing
exponentially since 1950. An observation by Gordon Moore in 1965 laid the foundations
for what came to be known as “Moore’s Law” – that computer processing power doubles
every eighteen months. If Moore’s Law is extrapolated naively to the future, it is learnt
that sooner or later, each bit of information should be encoded by a physical system of
subatomic size. As a matter of fact this point is substantiated by the survey made by
Keyes in 1988 as shown in fig. 1. This plot shows the number of electrons required to
store a single bit of information. An extrapolation of the plot suggests that we might be
within the reach of atomic scale computations with in a decade or so at the atomic scale
however.
1014
1012
1010
No. Of
Impurities
108
106
104
102
1
1950
1970
1980 1990 2000
Year
2010
Fig 1: Showing number of dopant impurities in logic in bipolar transistors with year.
5
Matter obeys the rules of quantum mechanics, which are quite different from the
classical rules that determine the properties of conventional logic gates. So if computers
are to become smaller in future, new, quantum technology must replace or supplement
what we have now. Not withstanding, the quantum technology can offer much more than
just cramming more and more bits to silicon and multiplying the clock speed of
microprocessors. It can support entirely a new kind of computation with quantitatively as
well as qualitatively new algorithms based on the principles of quantum mechanics.
With the size of components in classical computers shrinking to where the behaviour
of the components, is practically dominated by quantum theory than classical theory,
researchers have begun investigating the potential of these quantum behaviours for
computation. Surprisingly it seems that a computer whose components are all to function
in a quantum way are more powerful than any classical computer can be. It is the
physical limitations of the classical computer and the possibilities for the quantum
computer to perform certain useful tasks more rapidly than any classical computer, which
drive the study of quantum computing.
A computer whose memory is exponentially larger than its apparent physical size, a
computer that can manipulate an exponential set of inputs simultaneously – a whole new
concept in parallelism; a computer that computes in the twilight (space like) zone of
Hilbert Space (or possibly a higher space – Grassman Space & so on), is a quantum
computer. Relatively few and simple concepts from quantum mechanics are needed to
make quantum computers a possibility. The subtlety has been in learning to manipulate
these concepts. If such a computer is inevitability or will it be too difficult to build on, is
a million dollars question.
6
CHAPTER 2
LITERATURE SURVEY
2.1 A BRIEF HISTORY OF QUANTUM COMPUTING:
The idea of computational device based on quantum mechanics was first explored
in the 1970’s and early 1980’s by physicists and computer scientists such as Charles H.
Bennet of the IBM Thomas J. Watson Research Centre, Paul A. Beniof of Arogonne
National Laboratory in Illinois, David Deustch of the University of Oxford and Richard
P. Feynman of Caltech. The idea emerged when scientists were pondering on the
fundamental limits of computation. In 1982 Feynman was among the fewer to attempt to
provide conceptually a new kind of computers which could be devised based on the
principles of quantum physics. He constructed an abstract model to show how a quantum
system could be used to do computations and also explain how such a machine would be
able to act as a simulator for physical problems pertaining to quantum physics. In other
words, a physicist would have the ability to carry out experiments in quantum physics
inside a quantum mechanical computer. Feynman further analysed that quantum
computers can solve quantum mechanical many body problems that are impractical to
solve on a classical computer. This is due to the fact that solutions on a classical
computer would require exponentially growing time where as the whole calculations on
quantum computer can be done in polynomial time.
Later, in 1985, Deutsch realized that Feynman assertion could eventually lead to a
general-purpose quantum computer. He showed that any physical process, in principle
could be modelled perfectly by a quantum computer. Thus, a quantum computer would
have capabilities far beyond those of any traditional classical computer. Consequently
efforts were made to find interesting applications for such a machine. This did not lead to
much success except continuing few mathematical problems. Peter Shor in 1994 set out a
method for using quantum computers to crack an important problem in number theory
which was namely factorisation. He showed how an ensemble of mathematical
operations, designed specifically for a quantum computer could be organized to make
such a machine to factor huge numbers extremely rapidly, much faster than is possible on
conventional computers. With this breakthrough, quantum computing transformed from a
mere academic curiosity directly to an interest world over.
Perhaps the most astonishing fact about quantum computing is that it took
exceedingly large time to take off. Physicists have known since 1920’s that the world of
subatomic particles is a realm apart, but it took computer scientists another half century
to begin wondering whether quantum effects might be harnessed for computation. The
answer was far from obvious.
7
2.2 LIMITATIONS
COMPUTING
OF
CLASSICAL COMPUTER
AND BIRTH OF ART OF
QUANTUM
2.2.1: Public Key Cryptography and Classical factoring of big integers:
In 1970 a clever mathematical discovery in the shape of “public key” systems
provided a solution to key distribution problem. In these systems users do not need to
agree on a secret key before they send the message. The principle of a safe with two keys,
one public key to lock it, and another private one to open it, is employed. Everyone has a
key to lock the safe but one person has a key that will open it again, so anyone can put a
message in the safe but only one person can take it out. In practice the two keys are two
large integer numbers. One can easily derive a public key from a private key but not vice
versa. The system exploits the fact that certain mathematical operations are easier to
perform in one direction that the other e.g. multiplication of numbers can be performed
much faster than factorising a large number. What really counts for a “fast” algorithm is
not the actual time taken to multiply a particular pairs of numbers but the fact that the
time does not increase too sharply when we apply the same method to ever-large
numbers. We know that multiplication requires little extra time when we switch from two
three digit numbers to two thirty digit numbers using the simpler trial division method
about 10∧13 times more time or memory consuming than factoring a three digit number.
In case of factorisation the use of computational resources is enormous when we keep on
increasing the number of digits. As a matter of fact public key cryptosystems could avoid
key distribution problem. However their security depends upon unproven mathematical
assumptions such as the difficulty of factoring large integers. Nevertheless one such
protocol is RSA, which maps electronic banking possible by assuming banks and their
customers that a bogus transfer of funds or a successful forgery would take the world’s
fastest computer millions of years to carry out. Another is the under spread Data
Encryption Standard (DES) which remains secure far most ordinary business
transactions.
The procedure of factorising a large integer can be quantified as follows. Consider
a number N with L decimal digits (N ~ 10 to power L). The number is factored using trial
division method. On conventional computers one of well known factoring algorithm runs
for number of operations of the order of
or explicitly,
s ~ O (exp ( (64/9)1/3 (lnN)1/3 (ln lnN)2/3 ))
s ~ A exp ( 1.9 L1/3 ( lnL)2/3 )
This algorithm therefore, scales exponentially with the input size log N (log N determines
the length of the input. The base of the logarithm is determined by our numbering system.
Thus base 2 gives the length in binary, a base 2 gives the length in binary, a base of 10 in
decimal and so on) e.g. in 1994 a 129 digit number (known as RSA 129) was successfully
factored using this algorithm on approximately 1600 workstations scattered around the
world, the entire factorisation took eight months. Using this to estimate the per factor of
the above exponential scaling, it is found that it would take roughly 800,000 years to
factor a 250 digit number with the same computer power, similarly a 1000 digit number
would require 10 to the power 25 years (much longer than the age of universe). The
difficulty of factoring large numbers is crucial for public key cryptography such as used
8
in banks around 250 digits. Using the trial division method for factorisation 10L/2 (=√N)
divisions are needed to solve the problem on exponential increase as function of L.
Suppose a computer performs 1010 decisions per second. Thus the computer can factor
any number N, in about e.g. a 100 digit number will be factored in 1040 seconds, much
longer than 3.8 X 1017 second (12 billion years), the currently estimated age of the
universe!
2.2.2 Quantum Factoring
From the analysis of classical factoring of big integers it seems that factoring big
numbers will remain beyond the capabilities of any realistic computing devices and
unless mathematicians or computer scientists come up with an efficient factoring
algorithm, the public key crypto systems will remain secure. However it turns out that
this is not the case. The Classical Theory of Computation is not complete simply because
it does not describe all physically possible computations. In particular, it does not
describe computations, which can be performed by quantum devices. Indeed, recent work
in quantum computation shows that a quantum computer can factor much faster than any
classical computer. According to an algorithm developed by Peter Shor factoring an
integer using quantum computer runs in O((lnN) 2+ε) steps, where ε is small. This is
roughly quadratic in the input size, so factoring a 1000 digit number with such an
algorithm would require only few million steps. The implication is that public key
cryptosystems based on factoring may be breakable.
2.2.3 Searching of an item with desired property:
Searching of an item with desired property from a collection of N items is another
problem that admits tremendous speed up using quantum logic based algorithm. Suppose
we pick up an item at random from a collection of N items likelihood of correct selection
is the same as that of right one, the probability of right selection is half. Hence on an
average we require N/2 operations for getting the right item. However Grover invented
quantum logic based algorithm, which accomplishes the same task in an average of √ N
number of operations.
2.2.4 Simulation of quantum system by classical computer:
Richard P. Feynman, in 1982 proposed that a quantum physical system of N
particles with its quantum probabilities can not be simulated by the usual computer
without an exponential slowdown in the efficiency of simulation. However, a system of
N particles in classical physics can be simulated with a polynomial slowdown. The main
reason for this is that the description size of a particle system is linear in N in classical
physics but exponential in N according to quantum computer (computer based on the
laws of quantum mechanics) can avoid the slowdown encountered in the simulation
process of quantum systems. Feynman also addressed the problem of simulating a
quantum physical system with a probabilistic computer but due to interference
phenomena, it appears to be a difficult problem.
9
2.3 QUANTUM COMPUTING: A WHOLE NEW CONCEPT IN PARALLELISM
Performing mathematical calculations, searching the internet, modelling the
national economy, forecasting the weather and so on puts a constraint on the capacity of
even the fastest and most powerful computers. The difficulty is not so much that
microprocessors are too slow; it is that computers are inherently inefficient. Modern
(classical) computers operate according to programs that divide a task into elementary
operations, which are then carried out serially, one operation at a time. Efforts have been
made to coax two or more computers (or at least two or more microprocessors) to work
on different aspects of a problem at the same time, but the progress in such parallel
computing has been slow and fitful. The reason to a large extent is that the logic built into
microprocessors is inherently serial (normal computers sometimes appear to be doing
many tasks at once, such as running both a word processor and spreadsheet programme,
but in reality, the central processor is simply cycling rapidly from one task to the next).
In a true sense, parallel computer would have simultaneity built into its very nature.
It would be able to carry out many operations at once, to search instantly through a long
list of possibilities and point out the one that solves the problem. Such computers do
exist. They are called quantum computers. In reality, the more exciting feature of
quantum computing is quantum parallelism. A quantum system in general is not in one
“classical state” but in a “quantum state” consisting (broadly speaking) in a superposition
of many classical or classical like states. This is called principle of linear superposition
used to construct quantum states. If the superposition can be protected from unwanted
entanglement with its environment (known as decoherence) a quantum computer can
output results depending on details of all its classical like states. This is quantum
parallelism- parallelism on a serial machine.
2.4 QUANTUM SUPERPOSITION AND
visualization of quantum computer.
QUANTUM
INTERFERENCE:
Conceptual
In a quantum computer the fundamental unit of information (is called a quantum bit
or “qubit”, analogous to classical bit used in ordinary computer), is not binary but rather
more quaternary in nature. This qubit property arises as direct consequence of its
adherence to the laws of quantum motions. A qubit can exist not only in a state
corresponding to the logical state 0 or 1 as in a classical state bit but also in states
corresponding to a blend of superposition of those classical states. In other words a qubit
can exist as a zero, a one, or simultaneously as both 0 and 1, with numerical coefficient
representing the probability for each state. This concept may appear to be counterintuitive
because every day phenomenon is governed by classical physics, not quantum
mechanics, which taps out at the atomic level. Physically qubit can be visualized by the
spin s=1/2 of one electron system, the two state +1/2 and –1/2 being two eigenstates of Sz
(z component direction of an external magnetic field of spin ½.). Alternatively a beam of
single photon can also be used, the total states being the state of polarization (horizontal
or vertical) with respect to some chosen axis. Thus qubit can take 2 values, 0 or 1, which
are associated with two eigenstates of a spin of a single electron (say):
|1> = |↑>
|0> = |↓>
|0> + |1> = |↑> + |↓>
10
And further qubit can be a superposition of these two states with complex coefficient and
this property distinguishes them form classical bits used in conventional computers. In
mathematical terms, we say that since the general state of a qubit can be superposition of
two pure states with arbitrary complex coefficients, then the state is described as a vector
in the two dimensional complex space c2 and the two pure states form the basis of the
representation.
Experimentally the phenomenon of quantum superposition can be demonstrated as
follows. Consider figure 2.
Fig 2: Beam splitting of light
Here a light source emits a photon along a path towards a half-silvered mirror. This
mirror splits the light, reflecting half vertically toward detector A and transmitting half
toward detector B. a photon however represents a single quantised energy state (E=hν)
and hence cannot be split, so it is detected with equal probability at either A or B. this is
verified by observation that if the detector A registers the signal than B does not and viceversa. With this piece of information one would like to think that any given photon
travels either vertically or horizontally, randomly choosing between the two paths.
Quantum mechanics however predicts that the photon actually travels both paths
simultaneously, collapsing down to one path only upon measurement (collapse of the
wave function). This effect is known as single-particle interference resulting from the
linear superposition of the possible photon states of potential paths. The phenomenon of
single particle interference can be better illustrated in a slightly more elaborate
experiment, outlined in fig 3. In this experiment the photon first encounters a half
polished mirror (beam splitter) thus getting split into two parts: reflected beam and
transmitted beam. The two beams are recombined with the half of the fully silvered
mirrors and finally another half silvered mirror splits before reaching a detector. Each
half silvered mirror introduces the probability of the photon following down one path or
the other. Once a photon strikes the mirror along either of two paths after the first beam
splitter, the arrangement is identical to that in fig 2 thus a single photon travelling
vertically, when strikes the mirror, then by the experiment in fig 2, there should be an
equal probability (50%) that the photon will strike either detector A or detector B. the
same goes for a photon travelling down the horizontal path. However, the actual result is
drastically different. If the two possible paths are exactly equal in length, then it turns out
that there is 100% probability that photon reaches the detector A & 0% probability that it
reaches the detector B thus the photon is certain to strike the detector A it seems
inescapable that the photon must, in some sense, have actually travelled both routes
simultaneously.
11
B
Fully Silvered
V
H
Half Silvered
(Beam splitter)
Fig 3: Example to show wave nature and particle nature of light
This can be demonstrated by placing an absorbing screen in the way of either of the
routes, then it becomes equally probable that detector A or B is reached. Block of one of
the paths actually allows detector B to be reached; with both routes open, the photon
somehow knows that it is not permitted to reach detector B, so it must have actually felt
out both routes. It is therefore perfectly legitimate to say that between the two half
silvered mirrors the photon took both transmitted and reflected paths or using more
technical language, we can say that photon is in a coherent superposition of being in the
transmitted beam and in the reflected beam. This quantum interference is resulting due to
linear superposition principle. This is one of those unique characteristics that make
current research in quantum computing not merely a continuation of today’s idea of
computer but rather an entirely new branch of thought and underlying concept and it is
because quantum computers harness those special characteristics that gives them the
potential to be incredibly powerful computational device.
2.5 QUANTUM ENTANGLEMENT
The observation of correlation among various events is day to day phenomena.
These correlations are well described with the help of laws of classical physics. Let us
consider the following example: Imagine a scene of bank robbery is picturised. The bank
robber is pointing a gun at the terrified teller. By looking at the teller one can tell whether
the gun has gone off or not if the teller is alive and unharmed, one can be sure the gun has
not been fired. If the teller is lying dead of a gunshot wound on the floor, one knows the
gun has been fired. This is a simple detective case. Thus there is a direct correlation
between the state of gun and the state of the teller ‘gun fired’ means “teller alive”. In the
event it is presumed the robber only shots to kill and he never misses.
In the world of microscopic objects described by quantum mechanics, correlation
among the events is not so simple. Consider a nucleus which might undergo a radioactive
decay in a certain time or it might not. Thus with respect to decay the nucleus exist in two
12
possible states only: ‘decayed’ and ‘not decayed’, just as we had two states, ‘fired’ and
‘not fired’ for the gun or ‘alive’ and ‘dead’ for the teller. However, in the quantum
mechanical world, it is also possible for the atom to be in a combined state decayed-not
decayed in which it is neither one nor the other but somewhere in between. This is due to
the principle of linear superposition of two quantum mechanical states of the atom, and is
not something we normally expect of classical objects like guns or tellers. Further let us
consider a system consisting of two nuclei. Two nuclei may be correlated so that if one
has decayed, the other will also have decayed. And if one has not decayed, neither has the
other. This is 100% correlation. However, the nuclei may also be correlated so that if one
is in the superposition state, ‘decayed-not decayed’, the other will also be. Thus quantum
mechanically, then one more correlation between nuclei than we would expect
classically. This kind of quantum ‘super correlation’ is called ‘Entanglement’.
Entanglement was in fact originally named German, ‘Verschrankung’, by Erwin
Schrodinger, a Nobel laureate in physics, for his basic contribution in quantum
mechanics. Schrodinger was the first to realize the strange character of entanglement.
Imagine it is not the robber but the nucleus, which determines whether the gun fires. If
the nucleus decays, it sets off a hair trigger, which fires the gun. If it does not decay, the
gun does not fire. But if the nucleus is in the superposition it can be correlated to the gun
in a superposition state fired-not fired. However such a correlation leads to a catastrophic
situation. In the present case teller is dead or alive at the same time! Schrodinger was
worried about the similar situation where the victim of the quantum entanglement was a
cat in a box (Schrodinger cat: A paradox). For Schrodinger cat in a box decay nucleus
could trigger the release of lethal chemical. The basic problem is that in the everyday
world we are not used to see anything like dead-alive cat or dead-alive teller. However, in
principle, if quantum mechanics is to be a complete theory describing every level of our
experience, such strange states should be possible. Where does the quantum world stop
and the classical world begins? Do we really have an interface separating quantum
phenomenon from the classical one? And so on. These and allied problems have been
described since long and in the process a number of different interpretations of quantum
theory have been suggested.
The problem was brought into focus by a famous paper in 1935 by Einstein,
Podolsky and Rosen, who argued that the strange behaviour of entanglement that
quantum mechanics is an incomplete theory not wrong. This is widely known as EPR
paradox. The concept of EPR paradox can be understood with the help of following
example: consider a Helium atom in the ground state. It has two electrons having
following quantum numbers: n=1, l=0, s=1/2, sz = +1/2 for one and sz = -1/2 for another.
Thus we have j = 0 & 1. But jz = (sz )1 + (sz )2 = 0. Hence only j=0 state is allowed. Thus
in a Helium atom, two electrons are antiparallel to each other and hence form entangled
pair of system. The atom is provided sufficient energy (equal to the binding energy of the
atom). So that it disintegrates at rest. Consequently two electrons fly away in opposite
direction.
13
Fig 4: EPR paradox description using He atom
Two electrons are taken apart. With the application of magnetic field when the spin of
one electron is flipped, the spin of other electron is also flipped instantaneously
(communication with speed faster than speed of light). This is a real phenomenon
Einstein called it spooky action at a distance; the mechanism of which cannot, as yet be
explained by any theory- it simply must be taken as given and this was Einstein’s
objection about the completeness of quantum theory. However we know that further
developments (Bell inequality and its experimental verification) proved that quantum
considerations are correct even if it means communication between space like events.
Even more amazing is the knowledge about the state of spin of another electron without
making a measurement on it.
Quantum Entanglement allows qubits that are separated by incredible distances to
interact with each other instantaneously (not limited to the speed of light). No matter how
large the distance between the correlated particles, they will remain entangled as long as
they are isolated. Taken together, quantum superposition and entanglement creates an
enormously enhanced computing power.
2.5.1 Bertleman’s socks:
In the first instant one may be inclined to ask: What is so special about quantum
entanglement? One does encounter similar situations (phenomenon of correlation
between two events) in areas other that quantum world. Let us consider the case of Mr.
Bertleman who has the peculiar habit of wearing socks of different colours in left and
right foot. If he wears red coloured sock in one foot, it should be green in the other, or if
it is yellow in one then it should be blue in the other. Presumably Mr. Bertleman never
breaks the rule. Therefore looking at the colour of one sock one can tell the colour of the
other sock which he is wearing. However on deeper scrutiny, the kind of objection raised
above, does not stand. As a matter of fact in quantum entanglement the choice of
measurement also plays a crucial role. One may decide to measure x-component of spin,
or its y-component or a compound along s direction inclined at an arbitrary angle to xaxis. The other particle arranges its spin accordingly. In case of Bertleman’s example the
onlooker has a role to play. The onlooker once decides to see the yellow-blue
combination of colours for Bertleman socks, looks at accordingly the intention of
onlooker in deciding the colour is interesting and equally interesting is the instant
communication of this intention.
14
2.5.2 EPR SITUATION, HIDDEN VARIABLES AND BELL THEOREM:
The critical examination of the paper “Is Quantum Mechanics Complete?” by
Einstein, Podolsky and Rosen (EPR) carried by John Bell lead to following contradictory
conclusion:
1. EPR correlations (usually referred as quantum entanglement) predicted by
quantum mechanics are so strong that one can hardly avoid the conclusion that
quantum mechanics should be completed by some supplementary parameters
(those so called hidden variables.)
2. The elaboration of the above result, demonstrates that the hidden variables
description in fact contradicts some predictions of quantum mechanics.
In the face of these two perfectly convincing and contradictory results, there is only
one way out: ask Nature how it works. Till the end of 1970 there was no experimental
result to answer this question. The contradiction discovered by Bell in EPR paper, is
so subtle that it appears only in a very peculiar situations that had not been
investigated. And require design and build specific experiments.
2.5.2.1 An EPR situation:
Consider a source emits a pair of photons v1 & v2 travelling in opposite
direction (Fig 4.). Each photon impinges onto polarizer, which measure the linear
polarization along both of two directions (a & b) determined by orientation of the
corresponding polarizer.
+1
+1
ν1
ν2
S
-
-1
a
b
Fig 5: Graphical description of the EPR situation
There are two possible outcomes for each measurement, which are +1 and –1. Quantum
mechanics allows for the existence of a two-photon state (EPR state) for which the
polarization measurements taken separately appear random but which are strongly
correlated. More precisely, denoting P+(a) & P-(a) as the probabilities that the
polarization of ν1 along a is found equal to + or -, these probabilities are predicted to be
equal to 0.5, similarly the probabilities P+(b) & P-(b) for photon ν2 are equal to 0.5 and
independent of the orientation b.
On the other hand, the joint probability P++(a) for observing + for both the
photons is equal to 0.5 Cos2(a.b). In case of parallel polarisers [(a.b)=0], this joint
probability is P++(0)=0.5, Similarly P--(0). For cross polariser [(a.b)= π/2], the joint
probability is P-+( π/2)=P+-(π/2)=0. The results for the two photons of the same pair are
thus always identical, both + or – i.e. they are completely correlated. Such correlations
15
between events each of which appears to be random, may arise outside the physics world
as well. Consider for instance, the occurrence of some defined disease (say G) and let us
assume that biologists have observed its development in 50% of the population aged 20,
and its absence in the remaining half. Now on investigating specific pair of (true) twin
brothers, a perfect correlation is found between the outcomes: if one brother is affected,
the other is also found to be inflicted with the disease; but if one member of the pair has
not developed the disease, then the other is also unaffected. In face of such perfect
correlation for twin brothers, the biologists will certainly conclude that the disease has a
genetic origin. A simple scenario may be invoked: at the first step of conception of the
embryo, a genetic process which is random in nature, produced a pair of chromosome
sequence-one which is responsible for the occurrence or absence of the disease that has
been duplicated and given to both brothers.
Thus the two situations the case of correlation between the polarized states of two
photons and the case of twin brothers (a number of such situations can be exemplified)
are exactly analogous. It seems therefore, natural to link this correlation between the pairs
of photons to some common property analogous to the common genome of the two twin
brothers. This common property changes form pair to pair, which accounts for the
random character of the single event. This is the basic conclusion drawn by John Bell
regarding EPR states. A natural generalization of the EPR reasoning leads to the
conclusion that quantum mechanics is not a complete description of physical reality. As a
matter of fact, introduction of “some common property” which changes from pair to pair,
invokes the idea that complete description of a pair must include “something” in addition
to the state vector, which is the same for all pairs. This “something” can be called
supplementary parameter or hidden variables. Inclusion of hidden variables sends an
account of the polarized states of two photons, for any set (a, b) of orientations.
2.5.2.2 Bell Inequalities
Bell examined critically the requirement for hidden variables to explain the expected
correlation between the two polarized states of photons. It was shown that the expected
correlations, for the joint measurements of polarized states of photons as mentioned
above, cannot take any set of values, but they are subjected to certain constraints. More
precisely, if we consider four possible sets of orientations [(a.b), (a.b’), (a’.b), (a’.b’)],
the corresponding correlation coefficients (which measure the amount of correlation) are
restricted by Bell inequalities which states that a given combination of these four
coefficients ‘s’ is between –2 and +2 for any reasonable hidden variable theory. Thus
Bell inequalities prescribe a test for the validity of hidden variable theory. However,
quantum mechanics predicts the value of s as 2.8 i.e., it violates Bell inequalities, and the
same is tested by experiments. Thus the hidden variable theories envisaged above are
unable to send an account of the EPR correlation (quantum entanglement) predicted by
quantum mechanics. As a matter of fact quantum mechanical correlations are more
intricate to understand as compared to mutual correlations between twin brothers.
Bell inequality is based on the assumption of local hidden variable models. The
assumption of locality states that the result of a measurement by a polariser cannot be
directly influenced by the choice of the orientation of the other remotely located
polariser. Actually this is nothing but the consequence of Einstein causality (No signal
can move with a speed greater than speed of light in vacuum). Nevertheless, Bell
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inequalities apply to wide class of theories than local hidden variable theories. Any
theory, in which each photon has a “physical reality” localized in space-time, determining
the outcome of the corresponding measurement, will lead to inequalities that (sometimes)
conflict with quantum mechanics. Bell’s theorem can thus be phrased in the following
way: some quantum mechanical predictions (EPR correlations-quantum entanglement)
cannot be mimicked by any local realistic model in the spirit of Einstein ideas of theory
of hidden variables.
2.6 QUANTUM TELEPORTATION AND QUANTUM THEORY OF INFORMATION
Information is physical and any processing of information is always performed by
physical means-an innocent sounding statement, but its consequences are anything but
trivial. In the last few years there has been an explosion of theoretical and experimental
innovations leading to the creation of a fundamental new discipline: a distinctly Quantum
Theory of Information. Quantum physics allows the construction of qualitatively new
types of logic gates, absolutely secure cryptosystems (systems that combine
communications and cryptography), the cramming of two bits of information into one
physical bit and, has far just been proposed, a son of “teleportation”.
Until recently teleportation was not taken seriously by scientists. Usually
teleportation is the name given by science fiction writers to the feat of making an object
or person disintegrate in one place while a perfect replica appears somewhere else.
Normally this is done by scanning the object in such a way as to extract all the
information from it, then this information is transmitted to the information from it, then
this information is transmitted to the receiving location and used to construct the replica,
not necessarily from the actual material of the original, but probably from atoms of the
same kinds, arranged in exactly the same pattern as the original. A teleportation machine
would be like a fax machine, except that it would work on 3-dimensional objects as well
as documents, it would produce an exact copy rather an approximate facsimile and it
would destroy the original in the process of scanning it.
In classical physics, an object can be teleported, in principle by performing a
measurement to completely characterize the properties of the object that information can
then be sent to another location, and this object reconstructed. Moreover classical
information theory agrees with everyday intuition: if a message is to be sent using an
object which can be put into one of N distinguishable states, the maximum number of
different messages which can be sent is N. For example, single photon can have only two
distinguishable polarization states: left handed and right handed. Thus a single photon
can not transmit more than two distinguishable messages i.e. one bit of information.
Notwithstanding the basic question: is it possible to provide a complete reconstruction of
the original object? The answer is no. All the physical systems are ultimately quantum
mechanical and quantum mechanics tells us that it is impossible to completely determine
the state of an unknown quantum system, making it impossible to use the classical
measurement procedure to move a quantum system from one location to another. This is
due to Heisenberg Uncertainty Principle which states that more accurately an object is
scanned, the more it is disturbed by the scanning process, until one reaches a point where
the objects original state has been completely disrupted, still without having extracted
enough information to make a perfect replica. This sounds like a solid argument against
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teleportation: if one cannot extract enough information from an object to make a perfect
copy, it would be seen that a perfect copy cannot be made.
Charles H Bennet with his group and Stephen Wiesner have suggested a remarkable
procedure for teleporting quantum states using EPR states (entangled states). Quantum
teleportation may be described abstractly in terms of two particles, A & B. A has in its
possession an unknown state |ψ> represented as:
|ψ> = α|0> + β|1>
This is a single quantum bit (qubit)-a two level quantum system. The aim of teleportation
is to transport the state |ψ> from A to B. This is achieved by employing entangled states.
A & B each posses one qubit of a two qubit entangled state;
|ψ>(|0>A |0>B) + |1>A|1>B)
The above state can be rewritten in the Bell basis (|00>±|11>)), (|01>±|10>) for the first
two qubit and a conditional unitary transformation of the state |ψ> for the last one, that is
(|00>+|11>)|ψ>+(|00>-|11>)σZ|ψ> + (|01>+|10>)σX |ψ> + (|01>-|10>)(-iσY|ψ>)
Where σX, σY, σZ are Pauli matrices in the |0>, |1> basis. A measurement is preformed on
A’s qubits in the Bell basis. Depending on the outcomes of these measurements, B’s
respective states are |ψ>, σZ |ψ>, σX |ψ>, -iσZ |ψ> A sends the outcome of its
measurement to B, who can then recover the original state |ψ> by applying the
appropriate unitary transformation I, σZ, σ Y or iσY depending on A’s measurement
outcome. It may be noted that quantum state transmission has not been accomplished
faster than light because B must wait for A’s measurement result to arrive before he can
recover the quantum state.
Fig 6: Quantum Teleportation using Entanglement
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2.7 THERMODYNAMICS OF QUANTUM COMPUTATION
Computers are machines and like all machines they are subject to thermodynamics
constraints, based on the laws of thermodynamics. Analogous to any physical system, the
modern computers based on digital devices produce heat in operation. The basic question
is: Can digital computers be improved so as to minimize production of heat. It turns out
that it is possible to think (consistent with the laws of physics) of an ideal computer
capable of shaping, maintaining and moving around digital signals, maintaining and
moving around digital signals without any heat generation. Nevertheless there is one
place where heat must be produced. Whenever information is erased the phase space
associated with the system that stores information shrinks. Erasing a single bit of
information reduces entropy of the system that stored that information by at least
∆S=klog2. This reduction of entropy results in heat transfer to the environment.
Thus if a computer could be constructed that does not erase any information, such a
computer could work without generating any heat at all. This is precisely the situation in
quantum computers. Quantum Computation is reversible (but not the read out of the
result of that computation). It is therefore possible, at least in principle, to carryout
quantum computation without generating heat. Of course, in reality the computer would
still generate a lot of heat. Electric pulses moving along copper wires would have to work
their way against resistance. Electrons diffusing from a source would still collide with
crystal imperfections and with electrons in the drain again, generating heat. But at least in
ideal situation, copper wires could be replaced with superconductors, imperfect crystals
with prefect ones.
The reversibility of quantum computers is being realised by conceiving and
fabricating special gates. In digital computers gate like NOR, AND, NAND and XOR are
employed. All these gates are irreversible: they must generate heat. Amount of
information on the right hand side of
(a,b)
(a∧b)
is less than the amount of information on the left hand side. Using Toffoli gates Charles
Bennett has demonstrated that quantum computers are capable of performing any
computation utilizing only reversible steps. These special gates maintain all information
that is passed to them, so that the computation can be run forward and backward.
Consequently the computation results in a very large amount of data, because every
intermediate step is remembered, but heat generation is eliminated which the computation
goes on. After the computation is over the computation can be run backwards in order to
restore the initial state of the computer and avoid its spontaneous combustion.
2.8 Experimental Realisation of Quantum Computer
The architectural simplicity makes the quantum computer faster, smaller and cheaper
but its conceptual intricacies are posing difficult problems for its experimental
realization. Nevertheless a number of attempts have been made in this direction with an
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encouraging success. It is envisaged that it may not be too far when the quantum
computer would replace digital computer with its full prospects. Some of the attempts for
the experimental realization of quantum computer are summarized as follows:
2.8.1 Heteropolymers:
The first heteropolymer based quantum computer was designed and built in 1988 by
Teich and then improved by Lloyd in 1993. In a heteropolymer computer a linear array of
atoms is used as memory cells. Information is stored on a cell by pumping the
corresponding atom into an excited state. Instructions are transmitted to the
heteropolymer by laser pulses of appropriately tuned frequencies. The nature of the
computation that is performed on selected atoms is determined by the shape and the
duration of the pulse.
2.8.2 Ion Traps:
An ion trap quantum computer was first proposed by Cirac and Zoller in 1995 and
implemented first by Monroe and collaborators in 1995 and then by Schwarzchild in
1996. The ion trap computer encodes data in energy states of ions and in vibrational
modes between the ions. Conceptually each ion is operated by a separate laser. A
preliminary analysis demonstrated that Fourier transforms can be evaluated with the
ion trap computer. This, in turn, leads to Shor’s factoring algorithm, which is based
on Fourier transforms.
2.8.3 Quantum Electrodynamics Cavity:
A quantum electrodynamics (QED) cavity computer was demonstrated by
Turchette and collaborators in 1995. The computer consists of a QED cavity filled
with some cesium atoms and an arrangement of lasers, phase shift detectors, polarizer
and mirrors. The setup is a true quantum computer because it can create, manipulate
and preserve superposition and entanglements.
2.8.4 Nuclear Magnetic Resonance:
A Nuclear Magnetic Resonance (NMR) computer consists of a capsule filled with
a liquid and an NMR machine. Each molecule in the liquid is an independent
quantum memory register. Computation proceeds by sending radio pulses to the
sample and reading its response. Qubits are implemented as spin states of the nuclei
of atoms comprising the molecules. In an NMR computer the readout of the memory
register is accomplished by a measurement performed on a statistical ensemble of
say, 2.7x1019 molecules. This is in contrast to QED cavity computer ion-trap
computer, in which a single isolated quantum system was used for memory register.
NMR computer can solve NP (Non-polynomial) complete problems in polynomial
time. Most practical accomplishments in quantum computing so far have been
achieved using NMR computers.
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2.8.5 Quantum Dots
Quantum computers based on quantum dot technology use simpler architecture and
less sophisticated experimental, theoretical and mathematical skills as compared to the
four quantum computer implementations discussed so far. An array of quantum dots, in
which the dots are connected with their nearest neighbors by the means of gated
tunneling barriers are used for fabricating quantum gates using split gate technique. This
scheme has one of the basic advantages: the qubits are controlled electrically. The
disadvantage of this architecture is that quantum dots can communicate with their nearest
neighbors only resulting data read out is quite difficult.
2.8.5.1 Josephson Junctions
The Josephson junction quantum computer was demonstrated in 1999 by Nakamura
and the coworkers. In this computer a Cooper pair box, which is a small superconducting
island electrode is weakly coupled to a bulk superconductor. Weak coupling between the
superconductors create a Josephson junction between them which behaves as a capacitor.
If the Cooper box is small as a quantum dot, the charging current breaks into discrete
transfer of individual Cooper pairs, so that ultimately it is possible to just transfer a single
Cooper pair across the junction. Like quantum dot, computers in Josephson junction
computers, qubits are controlled electrically. Josephson junction’s quantum computers
are one of the promising candidates for future developments.
2.8.5.2 The Kane Computer
This computer looks a little like a quantum dot computer but in other ways it is more
like an NMR computer. It consists of a single magnetically active nucleus of p31 in a
crystal of isotropically clean magnetically inactive Si28. The sample is then placed in a
very strong magnetic field in order to set the spin of p31 parallel or antiparallel with the
direction of the field. The spin of the p31 nucleus can then be manipulated by applying a
radio frequency pulse to a control electrode, called A-gate, adjacent to the nucleus.
Electron mediated interaction between spins could in turn be manipulated by applying
voltage to electrodes called J-gates, placed between the p31 nuclei.
2.8.6 Topological Quantum Computer
In this computer qubits are encoded in a system of anyons. “Anyons” are quasiparticles in 2-dimensional media obeying parastatistics (neither Fermi Dirac nor Bose
Einstein). But in a way anyons are still closer to fermions, because a fermion like
repulsion exists between them. The respective movement of anyons is described by braid
group. The idea behind the topological quantum computer is to make use of the braid
group properties that describe the motion of anyons in order to carry out quantum
computations. It is claimed that such a computer should be invulnerable to quantum
errors of the topological stability of anyons.
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2.9 Future Directions of Quantum Computing
The foundation of the subject of quantum computation has become well established,
but everything else required for its future growth is under exploration. That covers
quantum algorithms, understanding dynamics and control of decoherence, atomic scale
technology and worthwhile applications. Reversibility of quantum computation may help
in solving NP problems, which are easy in one direction but hard in the opposite sense.
Global minimization problems may benefit from interference (as seen in Fermat’s
principle in wave mechanics). Simulated annealing methods may improve due to
quantum tunneling through barriers. Powerful properties of complex numbers(analytic
functions, conformal mappings) may provide new algorithms.
Quantum field theory can extend quantum computation to allow for creation and
destruction of quanta. The natural setting for such operations is in quantum optics. For
example, the traditional double slit experiment (or beam splitter) can be viewed as the
copy operation. It is permitted in quantum theory because the intensity of the two copies
is half the previous value. Theoretical tools for handling many-body quantum
entanglement are not well developed. Its improved characterization may produce better
implementation of quantum logic gates and possibilities to correct correlated errors.
Though decoherence can be described as an effective process, its dynamics is not
understood. To be able to control decoherence, one should be able to figure out the eigen
states favored by the environment in a given setup. The dynamics of measurement
process is not understood either, even after several decades of quantum mechanics.
Measurement is just described as a non-unitary projection operator in an otherwise
unitary quantum theory. Ultimately both the system and the observer are made up of
quantum building blocks, and a unified quantum description of both measurement and
decoherence must be developed. Apart from theoretical gain, it would help in improving
the detectors that operate close to the quantum limit of observation. For physicist, it is of
great interest to study the transition from classical to quantum regime. Enlargement of the
system from microscopic to mesoscopic levels, and reduction of the environment from
macroscopic to mesoscopic levels, can take us there. If there is something beyond
quantum theory lurking there, it would be noticed in the struggle for making quantum
devices. We may discover new limitations of quantum theory in trying to conquer
decoherence.
Theoretical developments alone will be no good without a matching technology.
Nowadays, the race for miniaturization of electronic circuits is not far away from the
quantum reality of nature. To devise new types of instruments we must change our viewpoint from scientific to technological-quantum effects are not for only observation, we
should learn to control them from practical use. The future is not foreseen yet, but it is
definitely promising.
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Chapter 3
A New Outlook to the Principle of Linear Superposition
3.1 Modification of Wave function as a requirement of Quantum Teleportation
Here we introduce a possibility of modification of Expansion Theorem as
applicable to an EPR situation for correlated terms. In this section we attempt to
introduce a correlation term in the definition of wave function (ψ) as given by Linear
Superposition Principle. A general expression of wave function (ψ) can be expressed
in word equation as:
Wave Function (ψ
ψ) = Term of Correlated states (δ
δ) + Sum of Un-correlated
terms.
This word equation can be expressed numerically as:
ψ = δ + Σ an ψn
where n ε allowed un-correlated states & ψ(n) are eigen states.
Hence in general with the introduction for our formalism of wave function, we can
now divide wave function into 3 categories which are:
1. Quantum Systems with Corr. = 0
ψ = Σ an ψn where n ε (1,2,3,4..) & ψ(n) are
eigen states.
2. Quantum Systems with 0 < Corr. < 1
ψ = δ + Σ an ψn where n ε allowed uncorrelated states & ψ(n) are eigen states.
3.Quantum Systems with Corr. = 1 ψ = δ and Σ an ψn = 0 ψ(n) are eigen states.
Thus the above treatment of Expansion Theorem in Quantum Systems suggests that
the definition of wave function should be modified to take into account the representation
of EPR states and further investigation should be done to determine the value of δ in the
wave function definition. To further support and validate the above formalism I have
applied the Schrodinger Wave Eq. both time dependent and time independent to check if
the form of Expansion Theorem changes or not.
In the standard formulation of Quantum Mechanics we have Schrodinger Wave Eq.
given by :
∇2 ψ+ 2m/h’ (E-V) ψ = 0 ------(1) Schrodinger time independent Eq.
in this eq. ψ : wave function, m is effective mass of the system, E is total energy, V is
potential energy of the system and h’ = h2 / 4π2 (h: Plank Constant) and Schrodinger Time
dependent wave eq. is given by:
ih/2π
π (∂
∂/∂
∂t) ψ = H ψ ---------(2)
where ψ : wave function of the system, H is Hamiltonian of the system.
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Now, by putting all the three definitions of ψ into these fundamental wave equations of
Quantum Mechanics that the modified definition of ψ is correct as we get the expected
results.
3.1.1 Suitability of Quantum Bits for Quantum Computation
Since there are three types of wave function systems as we have discussed in
previous sections, thus how do they relate to qubits is given in following lines:
1. For purely unentangled states in a Quantum system i.e. Corr.= 0, in such type of
systems wave functions is given by Linear Superposition Principle:
ψ = Σ an ψn & Σ |an|2 =1 where n ε (1,2,3,4……) and ψ(n) are eigen states.
In these type of system of qubits quantum algorithms can be made “efficiently” as in
this case the underlying parallelism of computation and vast storage of information is
possible according to the conception of Bloch Sphere or otherwise since every state
ψn is independent of each other and hence can be used for computation and storing of
information.
2. For mixed entangled states in a Quantum System i.e. 0< Corr. < 1, in such type of
systems wave function is given by:
ψ = δ + Σ an ψn & Σ |an|2 ≠ 1 where n ε allowed un-correlated states & ψ(n) are Eigen
states. In such a case since entangled and unentangled states cannot be separated as that
would amount to an interaction with the system leading to information loss and wave
function collapse. Hence such type of a state is not fit for computational purposes as it
may lead to spurious results.
3. For purely entangled states in a Quantum system i.e. Corr. = 1, in such type of
systems wave function is given by:
ψ = δ and Σ an ψn = 0 ψ(n) are Eigen states.
These states are suitable for teleportation of information using EPR states and not for
information storage or computational purposes. Thus case 3 is well suited only for
information communication keeping the validity of Quantum No Cloning Theorem.
3.2 An alternative interpretation of Quantum No Cloning Theorem.
The definition of Quantum No Cloning Theorem states that:
“It is not possible to copy quantum bits of an unknown quantum state.”
Or “Replication of information of arbitrary qubits is not possible.”
This fundamental limit is brought by Heisenberg’s Uncertainty Principle given by:
∆x. ∆p >= h/2π
π
∆x is the uncertainty in the measurement of position.
∆p is the uncertainty in the measurement of momentum.
24
Thus Uncertainty Principle brings about the problem of Measurement in Quantum
Mechanics which says that act of observation leads to wave function collapse and
information carried by a Quantum System is lost i.e.:
ψ = α|0> + β|1> observation
either |0> or |1>
Now, another explanation of information loss associated with measurement of a qubit can
be given by energy exchange leading to irreversibility brought into the system according
to the 2nd Law of Thermodynamics. The very act of measurement or observation has
brought an element of irreversibility into the system which can also be appreciated by the
fact that the there is a concept of canonically conjugate variables in Quantum Mechanics.
That is if we have two observables say A and B in a quantum system which are
canonically conjugate, they follow the inequality AB≠ BA and it is represented in
Quantum Mechanics as commutator brackets [A, B] = h/2π AB-BA=h/2π
Now the vary fact that AB≠
≠ BA suggests that there is some irreversibility brought in the
process of measurement of a quantum system and hence the uncertainty creeps in which
is given by Heisenberg Uncertainty Principle. This leads to some important questions
like:
1. Does the notion of quantum of action h (h, 2h, 3h…) lead to irreversibility in the
measurement process since ½ h or any fraction value of h cannot take place?
2. Irreversibility is brought in due to entropy change in the system, since entropy is a
statistical phenomenon, does this anyway relate to statistical behaviour of
quantum probabilities?
3. Is irreversibility in a Quantum measurement more fundamental that uncertainty in
the measurement process and leads to uncertainty in the system, thus explaining
the need for Uncertainty Principle?
Thus in Classical Mechanics entropy (S) of a system is associated with the loss of
information in the system due to the energy interaction and 2nd Law of Thermodynamics.
But in Quantum Mechanics we have the concept of choice of information due to wave
function collapse, so we can define a new parameter analogous to entropy in case of
Quantum systems related to the choice of information. Thus we introduce here a
possibility of the need of a new parameter for depicting the concept of choice of
information analogous to entropy. The new parameter can be an area of further research.
The concept of irreversibility in the system due to measurement process leading to non
replication of information on a qubit is an alternative explanation of the Quantum No
Cloning Theorem.
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Chapter 4
Experimental Realisation of Quantum Computers
4.1 Materials of Low dimensionality- Quantum Dot as a promising candidate
In 1959 physicist Richard Feynman gave an after-dinner talk exploring the limits of
miniaturization. The scientist wrote about the potential for nanoscience in an
influential 1959 talk “There’s Plenty of Room at the Bottom.” thus ushering a new
era of the field of low dimensional materials. One thing common about all the
conducting nanomaterials is that, in spite of the contraction in the size of the material,
the basic characteristics of the material remains the same- the energy levels remain
the same. This fact remains true as long as the size of the material undergoes
noticeable change. The number of energy levels increases and they start shifting
towards ‘blue’ region. By this, it is meant that the wavelength is smaller than green,
yellow, orange and red. But with the decrease in wavelength, the frequency increases.
Now we know that the energy level E is given by the equation:
E = h ν where h: Planck’s constant and ν is frequency.
As the frequency increases by shifting towards blue, the energy increases. Therefore,
these materials are capable of storing more energy and are said to exhibit
nanoproperty. The cause of the above said phenomena is stated below:
There is a wavelength called Compton Wavelength, which decides at which energy
levels of the materials change.
λc = h/ (m0 * c)
h: Planck’s constant
m0: rest mass
c: velocity of light.
If the size of any material is comparable with its Compton’s wavelength, it exhibits
nanoproperty and the materials are called Nanomaterial. This is found to be the order
10-9 m. This is the reason why these materials exhibit nanoproperty. Thus, we come
to the conclusion that as we move from macro to nano dimensions, we observe the
following:
1. There is an additional constraint in the movement of electrons and protons.
2. There is reduction in the degree of freedom.
Therefore, these materials are also called as materials of low dimensionality, in which
the charge density increases by decreasing the degree of freedom. The uses of
nanomaterials in the manufacturing of computers justify Moore’s Law, which states that:
26
“In every eighteen months, the capacity of processing and carrying information of
the computers is doubled.”
This is possible by decreasing the number of electrons required to carry one bit of
information. Most of the nanomaterials are characterised by the decrease of the degrees
of freedom. Effective reduction of geometry to two or fewer dimensional systems is
possible by a strong spatial localisation to a plane, line or point( i.e. confinement of an
electron in at least one direction at the de-Broglie wavelength) occurs only in the case of
atoms and electrons localised on crystal imperfections( e.g. on impurities).
Quantum Wells are one of the three basic components of quantum devices, which are
ultra thin, quasi 3-D planes. A narrow strip sliced from one of the planes is a 1-D
quantum wire. Dicing up a 1-D wire yields a 0-D quantum dot. Reducing the number of
dimensions in this manner forces electron to behave in a more atom like manner. All the
three devices mentioned are the quantum devices i.e. they have nanostructures. Thus, we
can say that with the evolution of nanotechnology comes the era of Diminishing
dimensions from 3-dimensions, which was commonly found, to 2-dimensions in the form
of Quantum Wells. Quantum wells were followed by quantum wires at 1-dimensional
level. And now the level 0-dimension can be attained by the concept of quantum dots.
Fig 7: Quantum Dots
Quantum Dots are major contenders for being used as a building blocks of future
Quantum Computers as compared to others like Nuclear Magnetic Resonance technique,
Trapped Ion technique etc because of the fundamental fact that decoherence of these
systems is very less hence even if they are prepared to act as quantum bits they
degenerate very fast and interact with the environment thus loosing their coherence
behaviour. Quantum Dots being 0-Dimensional entities have practically no degree of
freedom hence their interaction with the environment is also ideally zero thus they can act
as qubits to a larger extent. Further analysis and stability criterion are analysed in the
subsequent sections.
4.2 Need for Modified Coulomb Potential and its Analysis
The highly successful theory of electricity and magnetism amalgamated with the laws
of classical mechanics, leading to classical electrodynamics is plagued with a serious
problem right from its inception. The problem lies in the interaction of charge particles
when the distances tend to zero. The energy of the particle becomes infinite, its field
momentum becomes infinite and its equation of motion leads to solutions like run-away
solutions and pre-acceleration phenomenon, which are non-physical. This has been
termed as “Problem of Divergences” in the classical electrodynamics.
27
It was envisaged that the advent of Quantum Theory would resolve all these difficulties.
However, this did not prove true. All these problems are still present in the quantum
version of the theory and appear in an even more complicated manner. In the last ten
decades, even since Lorentz identified the problem in 1903, a number of attempts have
been made to circumvent divergence problem of Classical Electrodynamics. Out of all
the prescriptions, the renormalization technique and Rohrlich procedure of invoking
boundary conditions have drawn special attention. During the course of present
investigation, I am bringing to the forefront an attempt which has been made to
reformulate the theory of electricity and magnetism. In the process, a simple but an
elegant prescription proposed by Prof. Y M Gupta has been used. For the development of
proposed formalism a stand has been taken that the problem as enunciated above must be
solved initially at the classical level and only then the reformulation be carried forward to
the quantum level. This procedural development automatically ensures the basic stringent
requirement that the future theory must be a covering theory of the present one: invoking
Bohr Correspondence Principle- “In the limit when the proposed modification is
withdrawn, the proposed theory reduces to the existing one.”
The proposed formalism is based on the observation that in the realm of known
formalism of Classical Electrodynamics potential function, field intensity, field
momentum and interaction force, all, are singular at the origin(R=0) i.e. all these physical
quantities become infinite in the limit of zero distance. It is this singularity, which is the
crux of the problem. In the present formalism we choose a potential function such that it
is suitably and correctly chosen such that it is regular at the origin as well as all other
space points, we automatically get physical quantities free from singularity. This is a
logical way to obtain a whole set of dependent quantities which are regular ab initio. But
at the same time it is also clear that all attempts at changing the basic equations of
Electrodynamics (Maxwell equations) have proved unsuccessful. As a matter of fact, it is
required to keep the basic equations of electrodynamics intact and yet look for a suitable
potential function (solution of basic equations) that provides divergence-free solutions to
the problems of Classical Electrodynamics (field energy, field intensity, field momentum
etc.). To achieve this objective, prescriptions proposed by Prof. Y M Gupta have been
followed. The important features of the prescriptions are1. The basic equations of Electrodynamics (Maxwell equations) remain intact.
2. The form of potential function, as obtained, is such that it is regular at all
space points including origin. Notwithstanding this, it also gives rise to field
intensities, field energies and interaction forces that are regular at all space
points ab initio.
3. No free parameter is introduced into the theory- it is basically a parameter free
theory.
4. The prescription very gently obeys Bohr Correspondence Principle. In other
words, in the limit of large distances the proposed potential function reduces
to usual Coulomb potential function.
5. The prescription is quite general in nature and not restricted to
electrodynamics alone. Indeed, it can be applied to any inverse square force
law. (E.g. Newtonian gravitational force and Yukawa Nuclear force.)
When the whole problem is analysed very closely, it is observed that the crux of the
problem in hidden in the fact that in the realm of known formalism of classical
28
electrodynamics potential function, field intensity, field momentum and interaction force
is infinite at the origin (R=0). It is this singularity in the behaviour of these physical
quantities that must be removed systematically for the fundamental development of the
theory of electrodynamics. If the potential function is suitably and correctly selected, we
shall automatically get physical quantities that are regular at the origin as well as other
space points. This is a logical way to remove singularities in a systematic way. Once we
are able to obtain the suitable potential function, we can then attempt to carry forward the
reformulation to the quantum level. This procedural development will ensure that, the
basic stringent requirement, that the future theory must be a covering theory of the
previous one, is automatically fulfilled.
The present formalism retains the basic laws of electrostatics as given below:
∇. E = ρ/ε0
∇xE=0
E = -∇ φ
∇2 φ = - ρ/ε0
Following Prof. Y M Gupta it is found that the last equation above of the potential
has the solution:
φ = 1/2π2 ε0 Si (Y|r-r’|)/ |r-r’| ρ(r’) d3 r’
Further lim|r-r’|->0 Si (Y|r-r’|)/ |r-r’| = Y
And lim|r-r’|->0 φ = qY/2π2 ε0, q = ρ(r’) d3 r’
This formalism makes the potential function (φ) regular at the point R=0. For a point
charge ρ(r’) = qδ (r’)
where q is the charge of the particle. The Equations are given as below:
φpoint = (q/2π2 ε0) Si(YR)/R
Epoint = (q/2π2 ε0) [{Si (YR) –sin (YR)}/R2 ]
Fpoint = (q0 q/2π2 ε0) [{Si(YR) – sin(YR)}/R2 ]
4.3 Analysis of Quantum Dots using Modified Coulomb Potential.
Quantum Dots is a 0-D system and this system can be analysed as a two body bound
state problem like a hydrogen atom. As the systems keep on becoming smaller and
smaller the potential function of the system changes at a particular radius and
assumes a different form. The length at which the potential function changes form is
determined by the Compton Wavelength of the system. Thus the Compton
wavelength plays a major role in determining the modification in potential function.
The modified Coulomb potential is given by:
V = ½ kh r2 –V0 for r < rc
= -1/4π0 e2/r for r> rc
Where rc determines the size of quantum dot, rc is in the range of 10nm to 100nm, this
has been experimentally determined.
29
The expression of rc is given by:
rc = h/meff c
h: Plancks constant
meff: Effective mass
c: Velocity of light
Fig 8: The graphical representation of the Modified Coulomb potential
Due to the reduction of the size at the nano scale the neighbours also try to exert force
on the particles just like a person is trying to move in a congested street with walls on
both sides feels the presence of the closeness of the walls. Similarly in the physical
systems the potential function changes gradually from Coulombian to the one shown
in above equations i.e.:
V = ½ kh r2 –V0 for r < rc
= -1/4π0 e2/r for r> rc
Hence in the nanomaterials the effective potential energy is given by the following
word equation:
Potential energy = potential energy between the constituents + potential energy
due to boundary.
The above equation can be written as:
V = ½ kh r2 –V0
Hence even experimentally also the form of potential function is found to be
harmonic in nature for distances less than the Compton wavelength (λ0) of the
system. The estimation of parameters of the potential energy function can be found
out in the following way:
V| r < rc = V| r > rc at r = rc
∂V/∂r | r < rc = ∂V/∂r | r > rc at r = rc
Solving these two sets of equations that is the boundary conditions and the
differentiability of the potential function leads to finding the solutions to potential
function. Also kh and V0 can be found out b numerical values. These variables can be
found out by:
30
R l (r) | r < rc = R l (r) | r > rc at r = rc
∂R l (r)/ ∂r | r < rc = ∂R l (r)/ ∂r | r > rc at r = rc
Here we can mention that we can also solve the Schrodinger Wave Equation on the
modified Coulomb Potential for a quantum dot system and get the analytical form of
the wave function. If the majority of the wave function lies inside the Compton
wavelength region then it is a very high probability that the quantum dot is stable in
that region and also gives support to the formulation of potential function. At very
low distances potential acts as a harmonic potential, this also suggests that at the
fundamental level Nature may be harmonic and this may in some way lead to the
origin of Heisenberg Uncertainty Principle due to the fluctuating harmonic nature of
the potential at quantum levels.
Fig 9: Plot of Wave Function Vs. distance in a Quantum Dot
4.4 Study of Quantum Wires using Modified Coulomb Potential
As already discussed above that Quantum Wires are 1-D systems with the degree of
freedom in 1 dimension only. So as the analysis of 0-D Quantum Dots has been done
I have also analysed Quantum Wires using the Modified Coulomb Potential by
applying the Schrodinger time independent wave equation given by
∇2 ψ+ 2m/h’ (E-V) ψ = 0
where V = ½ kh r2 –V0 for r < rc
= -1/4π0 e2/r for r> rc
and rc (Compton Wavelength) = h/ (m0 * c)
Thus solving the Schrodinger Wave Equation for the given potential and the form of
wave function ψ ≡ ψ(r,θ
θ, z) and using the variable separable method we could get the
wave function in ‘z’ direction as a free particle wave function since the equation
which the Schrodinger Wave Equation reduces to is given by:
∂2χ/∂z2 + λχ(z) =0 which is a famous second order differential equation
with the solution given by
31
χ(z) = A e+iλz which is the wave function of the free particle system. This is also
logically deducible since Quantum Wire is a 1-D system with complete freedom in 1
dimension.
y
r
θ
(r, θ,z)
x
Fig 10: Quantum Wire as a 1-D system
The analysis of Quantum Wires is more complicated than Quantum Dots due to the
increase in the dimension of the system hence a preliminary study is done of this
system unlike the rigorous study done for Quantum Dots. It is expected from the
wave function expression of the Quantum Wire in modified Coulomb potential that
Compton Wavelength (λ0) plays a fundamental role in this system. Thus by the study
of the two systems we have done using the modified Coulomb Potential we can
conclude that Compton Wavelength is going to play a major role in Physics may be
as important as the fundamental constants of Nature.
4.5 Visit to the Nano Technology Lab in Barkatullah University, Bhopal
We had a short three day visit to a state of the art one of its kind Nano Technology
Lab in Barkatullah University in Bhopal. It was an informative visit in which we saw
various experiments done with nanomaterials. We also saw different fabrication
techniques for nanomaterials. We saw and learnt the concept of Atomic Force
Microscope (AFM), Scanning Tunnelling Microscope (STM), X Ray Diffractometer
running on the principle of Compton scattering and then using the diffraction pattern
the inter planar distance between the planes of atoms is calculated. We operated upon
the Atomic Force Microscope and also got pictures of the surface of nanomaterials.
We learnt about three kinds of materials based on the degree of freedoms ranging
from 2-D systems like Quantum Wells to 1-D systems like Quantum Wires to 0-D
systems like Quantum Dots. How these quantum structures are developed using Wet
Chemical method which is the most common one and various other methods. Overall
it was a highly informative visit to the Nanotechnology Lab and we could appreciate
the practical ways by which work is going on in the field of nanomaterials and
nanostructures and how this field is going to revolutionise the coming technologies.
This field experience was worth doing in the summer project. The functioning of
AFM and the numerical analysis we did in the Lab are reproduced in the coming
pages and is suggestive of the work we had undertaken.
32
Fig 11: Quantum Dots as seen from an Atomic Force Microscope
Also the basic principle of use of Scanning Probe method is the principle of Quantum
Tunnelling which leads to induction of potential and hence production of current
which thus leads to scanning of the surface of the nanomaterial. We had a visit to the
Nanotechnology Lab and we saw the Scanning Tunnelling Microscope and its
principle is reproduced below:
Fig 12: The tip of the Scanning Tunnelling Microscope
Fig 13: AFM Plots of Quantum Dots prepared in laboratory
33
Chapter 5
Future Directions of Research
5.1 Explanation of Measurement Problem by Symmetry breaking process
Symmetry is the most fundamental property of physical systems. It is a very general term
with regard to physical systems. There are various symmetries in Nature like local space
time symmetries and global space time symmetries. Global space time symmetries apply
to all the laws of Physics. Here in this section we will attempt to give a direction on
which a further thought can be developed for the resolution of the famous Measurement
Problem in Quantum Mechanics. It is the Measurement Problem which lies at the heart of
various problems and observation anomalies in Quantum Mechanics. Here I am trying to
give an explanation of Measurement Problem using Symmetry breaking process. The
basic equation of Quantum Mechanics i.e. AB ≠ BA valid for conjugate variables shows
some kind of irreversibility in the system. Now by the Leibniz’s Principle of Sufficient
Reason (PSR): If there is no sufficient reason for one thing to happen instead of another,
the principle says that nothing happens or in a situation with certain symmetry evolves in
such a way that, in the absence of an asymmetric cause, the initial symmetry is preserved.
In other words, a breaking of initial symmetry cannot happen without a reason or an
asymmetry cannot originate spontaneously. Asymmetry is what creates a phenomenon.
Thus symmetry breaking follows Principle of Causality that is “For every effect there is a
cause.” By the Measurement process we purposefully throw the Quantum System as a
“choice of information” by the observer unlike the collapsing of the wave function into
an uncertain Eigen state. Thus in a Measurement process the system does not lose
information like classical systems but is thrown in a particular eigen state based on the
information choice made by the observer. Hence we need to introduce a parameter in
Quantum Mechanics which takes care of this “choice of information” rather the “loss of
information” whose parameter is entropy in classical mechanics. Curie’s Principle also is
an interesting way to represent symmetry requirements in a system. The symmetry
elements of the causes must be found in their effects, but the converse is not true, that is
the effects can be more symmetric then their causes. Conditions valid for Curie’s
Principle are:
1. The causal connection must be valid.
2. Cause and Effect and their respective symmetries must be well defined.
The there must be a sufficient reason for the Measurement Problem to occur and by the
Leibniz’s Principle of Sufficient Reason (PSR) we can conclude that there must be some
symmetry breaking phenomenon occurring as a reason for the asymmetry in the system
and thus leading to Measurement Problem. Thus it is a matter of further research to find
the precise way how symmetry breaks in the system? What type of symmetry exists? Is it
a local or a global symmetry which breaks in the Reduction(R) Process of the wave
function as it is called in the Quantum Mechanics parlance? Thus introducing symmetry
in the Reduction Process in Quantum Mechanics is a novel idea and can put some more
light into the reason and the precise process by which wave function collapse occurs.
Thus this may fundamentally give the reason of Heisenberg Uncertainty Principle. It is
34
yet to be seen that it is the Uncertainty Principle which is more fundamental or the
symmetry breaking which is occurring in the system.
5.2 EPR Correlation: Interacting Hamiltonian Vs Non Linear Wave function
Basically there are two ways to introduce the effect of correlation in an EPR situation in
the definition of wave function of the Quantum System those are:
1.
In the form of Schrodinger Time Dependent Wave Eq.
ih/2π
π (∂
∂/∂
∂t) ψ = H ψ
We can introduce the correlation effect in the modification of the Hamiltonian (H) of
the system by introducing a potential term in the expression of H of the system and
keeping wave function (ψ) as it is defined by the Expansion Theorem.
2. In the second case we can keep Hamiltonian (H) as it is and change the definition
of wave function (ψ) by introducing a δ term to account for correlation effect of
EPR states.
Hence we have attempted the second case in our formalism of accounting for correlation
in the definition of wave function (ψ) of the system. Though both the approaches are new
to the scene and have not been attempted for this type of a situation. Hence the
application of first case is an area of further research in this formalism.
Eigen states are essentially entangled or correlated in an EPR situation and since EPR
implies non-local communication and apparent breakdown of Heisenberg Uncertainty
Principle hence it may lead to the modification of Linear Superposition Principle’s
linearity gets violated and different eigen states start interfering with each other and lead
to the introduction of δ in the definition of wave function (ψ).
EPR states
modification of Linear Superposition Principle
of wave function (ψ) or introduction of a new parameter altogether.
new definition
We find from the above two possible scenarios that EPR correlation can be incorporated
in the Schrodinger time dependent wave Eq. by modifying the Hamiltonian and
introducing the δ term due to correlation i e. The word eq. can be written as:
Hamiltonian = Kinetic energy term + Potential energy + energy due to EPR
correlation
Thus this can be one of the attempted explanations for introducing correlation in the
scene. In the second case we can put the Hamiltonian as it is without “energy due to EPR
correlation” term and can rather modify the wave function (ψ) in a non-linear way by
introducing the δ term and accounting for EPR correlation as it has already been done in
previous sections.
5.3 Possibility of third paradigm in Quantum Mechanics
It is a well known fact that though Quantum Mechanics gives correct mathematical
results of quantum phenomena but the interpretation of Quantum Mechanics is very
peculiar and is still not complete till date because of paradoxes like Schrodinger Cat
paradox and Einstein Podolsky Rosen (EPR) Paradox, Wave Function Collapse and
35
Measurement Problem. In this we attempt to give a new direction of thinking for a new
formulation of the fundamental principle of Uncertainty in Quantum Mechanics since all
peculiarities of Quantum Mechanics are derived from the Uncertainty Principle. We
describe the two mechanics we know of and predict the possibility of the third one:
1.
In Classical Mechanics the fundamental concept is the concept of Determinism
which says that given a state of the system at one point of time the state of the system at
some later time can be predict with 100% accuracy. This can be reformulated in a
statement including dynamical variables position (x) and momentum (p) as
“Both position (x) & momentum (p) can be measured to any desired accuracy.”
2.
In Quantum Mechanics we have the fundamental Uncertainty Principle given by:
∆x.∆p >= h/2π
It is interpreted as “ Either position (x) or momentum (p) can be measured with
desired accuracy but not both at the same time.”
3.
Here we give a new line of thinking to the above formalisms by saying that:
“Neither position (x) nor momentum (p) to any degree of accuracy can be
measured.”
This is a very interesting statement since in case 1 above we have determinism in the
definition of the system. In case 2 we have probability but still it is limited probability
since one of the parameters can still be measured to desired degree of accuracy. But in
case 3 we have brought in a complete indeterminism in which both the variables are
completely indeterministic. This formalism requires a new definition of Schrodinger
Wave Eq., Uncertainty Principle etc. It may be possible that Quantum Mechanics is a
special case of this third mechanics as we move from complete indeterminism to limited
indeterminism (Quantum Mechanics) to complete determinism (Classical Mechanics).
This concept of complete indeterminism may be related in some way to Consciousness of
the observer which is hot area of research in Quantum Mechanics these days. This last
proposal was given by Stephen Hawking as a new direction of thinking on the
modification of Quantum Mechanics right at the fundamental level.
5.4 Conclusion and Future Scope
The foundations of the subject of quantum computation have become well established,
but everything else required for its future growth is under exploration. That covers
quantum algorithms, logic gate operations, error correction, understanding dynamics and
control of decoherence, atomic scale technology and worthwhile applications.
Reversibility of quantum computation may help in solving NP problems, which are easy
in one direction but hard in the opposite sense. Global minimization problems may
benefit from interference effects (as seen in Fermat’s principle in wave mechanics).
Simulated annealing methods may improve due to quantum tunneling through barriers.
Powerful properties of complex numbers (analytic functions, conformal mappings) may
provide new algorithms. Theoretical tools for handling many-body quantum
entanglement are not well developed. Its improved characterization may produce better
implementation of quantum logic gates and possibilities to correct correlated errors.
36
Though decoherence can be described as an effective process, its dynamics is not
understood but an attempt has been made in the present project work in the form of
Symmetry breaking argument or need for an entropy like parameter or function to
account for irreversibility in the system. To be able to control decoherence, one should be
able to figure out the eigenstates favored by the environment in a given setup. The
dynamics of measurement process in not understood fully, though the attempt is also
made in this regard in this project. Measurement is just described as a non-unitary
projection operator in an otherwise unitary quantum theory. Ultimately both the system
and the observer are made up of quantum building blocks, and a unified quantum
description of both measurement and decoherence must be developed. Apart from
theoretical gain, it would help in improving the detectors that operate close to the
quantum limit of observation. For the physicist, it is of great interest to study the
transition from classical to quantum regime. Enlargement of the system from microscopic
to mesoscopic levels, and reduction of the environment from macroscopic to mesoscopic
levels, can take us there. If there is something beyond quantum theory lurking, there it
would be noticed in the struggle for making quantum devices. We may discover new
limitations of quantum theory in trying to conquer decoherence. Theoretical
developments alone will be no good without a matching technology. Nowadays, the race
for miniaturization of electronic circuits in not too far away from the quantum reality of
nature. To devise new types of instruments, we must change our view-points from
scientific to technological-quantum effects which are not for only observation; we should
learn how to control them for practical use. The future is not foreseen yet, but it is
definitely promising.
REFERENCES
1.
2.
3.
K.Brading, and E.Castellani (2003), Symmetries in Physics: Philosophical
Reflections, Cambridge University Press, 2003.
Sanjeev Kumar (2002), Reformulation of Classical Electrodynamics, Jiwaji
University, Gwalior, INDIA.
Michael Nielsen, and Isaac Chuang, Quantum Computation, Cambridge
University Press.
37
Annexure:
Bound state problem under modified Coulomb potential
We are taking an electron hole pair as the system and we perform our calculations on this
system. From numerical analysis we find the following behavior of modified Coulomb
potential (attraction).
V(r) = ½ khr2 – V0
V(r) = - kc/r
for r
for rc
rc,
r
V0 is positive
kc= 1/(4 0 ) Z e2 -----(1)
Boundary condition
V(r)lhl = V(r)rhl and V’(r)lhl = V’(r)rhl at r = rc where rc = Compton radius
Thus we calculate kh = 1/(4 0 ) e2/rc4
And
V0 = 3/2 * 1/(4 0 ) * e2/rc
The radial equation for two body problem is given by
d2 Rl(r)/d r2 + (2/r) d Rl /d r + { 2 / 2 (E – V(r)) - l(l+1)/r2} Rl(r) = 0 (2)
for bound state E = -B ( B is positive )
With the help of equation (1), equation (2) becomes
d2 Rl(r)/d r2 + (2/r) d Rl /d r + { 2 /
and
d2 Rl(r)/d r2 + (2/r) d Rl /d r + { 2 /
2F0 (ac,1+ac–bc ,1/(2
2
(-B – ½ khr2 – V0) - l(l+1)/r2} Rl(r) = 0 -----(3)
2
(-B + kc/r) - l(l+1)/r2} Rl(r) = 0 ---------(4)
r). 1F1(ah,bh, r2) [ r- ]
= (ah/bh)(2 r) 1F1(ah+1,bh+1, r2) . 2F0 (ac,1+ac–bc ,1/(2 r) +
(- ac/(2 r2)) 2F0 (ac+1,2+ac–bc ,1/(2 r)] -----------------------------------(5)
where ah = ½ (l+3/2) – /2 , bh = l+3/2
=k2/(2 ), k2 = 2 /
2
(V0-B) ,
2
=2 /
2
* kh
We have to find the value of 1/(2 r) for which LHS is equal to RHS for equation (5).
By finding this value we can find the energy eigen values B for the particular equation.
This will be the value of the stable energy eigen value for the electron hole pair static
quantum dot.
38
Val = 1/(2 r)
l(Azimuthal
Quantum no.
.04119194
.05201122
.08165704
.04566950
.06203910
.15786181
.20713222
Not found
.11385872
.18893625
.08575267
.25269888
.11484041
.07116760
.13074870
.08839940
.06195164
.09749363
.07394641
.19056947
.05548334
.08040687
.06458225
.11987688
.05063503
.06963503
.21120285
.05791610
.09352694
.04683321
.06208580
.12605788
.05287561
.07876733
.04375221
.05643507
.09732910
.04890059
.06904407
.03902194
.04843254
.07147330
.04297285
.05669535
.11132382
0
0
0
1
1
1
0
1
0
1
0
0
1
0
0
1
0
0
1
1
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
0
0
0
1
1
1
r ( distance
between
electron& hole)
10 nm
10 nm
10 nm
10 nm
10 nm
10 nm
1nm
1nm
2nm
2nm
3nm
3nm
3nm
4nm
4nm
4nm
5nm
5nm
5nm
5nm
6nm
6nm
6nm
6nm
7nm
7nm
7nm
7nm
7nm
8nm
8nm
8nm
8nm
8nm
9nm
9nm
9nm
9nm
9nm
11nm
11nm
11nm
11nm
11nm
11nm
39
B
(Energy V0 = 3/2 * 1/(4
2
eigen value)
0 ) * e /rc
1.8006e-20
1.121197e-20
4.581994e-21
1.464837e-20
7.937994e-21
1.2225992e-21
7.121090e-20
2.304e-20
2.304e-20
2.304e-20
2.304e-20
2.304e-20
2.304e-20
2.304e-19
5.891816e-20
2.13962e-20
4.616405e-20
5.316094e-21
2.574007e-21
3.770137e-20
1.116984e-20
2.443561e-20
3.184169e-20
1.285728e-20
2.234949e-20
3.365077e-21
2.756856e-20
1.312662e-20
2.03475e-20
5.905659e-21
2.43188e-20
1.285834e-20
1.397803e-21
1.781454e-20
7.128074e-21
2.176476e-20
1.238446e-20
3.004151e-21
1.707462e-21
7.694308e-21
1.970411e-20
1,184291e-20
3.981725e-21
1.577352e-20
7.912323e-21
1.658207e-20
1.07642e-20
4.942751e-21
1.36731e-20
7.855277e-21
2.037418e-21
1.152e-19
1.152e-19
7.679999e-20
7.679999e-20
7.679999e-20
5.76e-20
5.76e-20
5.76e-20
4.608e-20
4.608e-20
4.608e-20
4.608e-20
3.84e-20
3.84e-20
3.84e-20
3.84e-20
3.291428e-20
3.291428e-20
3.291428e-20
3.291428e-20
3.291428e-20
2.88e-20
2.88e-20
2.88e-20
2.88e-20
2.88e-20
2.56e-20
2.56e-20
2.56e-20
2.56e-20
2.56e-20
2.094545e-20
2.094545e-20
2.094545e-20
2.094545e-20
2.094545e-20
2.094545e-20
.03715315
.04546430
.06413234
.04068532
.05245294
.09023859
.03871289
.04898401
.07754975
.03552240
.04295338
.05856849
.034083607
.04079518
.05415846
.11111741
.03699031
.04608237
.06884917
.03280219
.03891556
.05055576
.09046542
.03546969
.04361094
.06241206
0
0
0
1
1
1
0
0
0
1
1
1
0
0
0
0
1
1
1
0
0
0
0
1
1
1
12nm
12nm
12nm
12nm
12nm
12nm
13nm
13nm
13nm
13nm
13nm
13nm
14nm
14nm
14nm
14nm
14nm
14nm
14nm
15nm
15nm
15nm
15nm
15nm
15nm
15nm
1.53705e-20
1.026451e-20
5.158519e-21
1.281754e-20
7.711512e-21
2.605521e-21
1.432684e-20
9.798517e-21
5.270199e-21
1.206268e-20
7.534360e-21
3.006036e-21
1.341821e-20
9.366297e-21
5.314385e-21
1.262471e-21
1.139225e-20
7.340338e-21
3.288427e-21
1.264983e-20
8.966288e-21
5.312733e-21
1.659182e-21
1.079306e-20
7.139509e-21
3.485958e-21
1.92e-20
1.92e-20
1.92e-20
1.92e-20
1.92e-20
1.92e-20
1.772308e-20
1.772308e-20
1.772308e-20
1.772308e-20
1.772308e-20
1.772308e-20
1.645714e-20
1.645714e-20
1.645714e-20
1.645714e-20
1.645714e-20
1.645714e-20
1.645714e-20
1.536e-20
1.536e-20
1.536e-20
1.536e-20
1.536e-20
1.536e-20
1.536e-20
5.00E+36
4.00E+36
3.00E+36
Series1
2.00E+36
Series2
1.00E+36
Series3
0.00E+00
-1.00E+36
1
3
5
7
9
-2.00E+36
40
11
13 15 17
2.00E+41
0.00E+00
-2.00E+41
1
3
5
7
9
11 13 15 17
-4.00E+41
Series1
-6.00E+41
Series2
-8.00E+41
Series3
-1.00E+42
-1.20E+42
-1.40E+42
So we observe that there are three points of intersection. So there exist three values
of Val where a solution exists which indirectly means that there are three possible
values of B (stable energy Eigen values for the quantum dot).
Thus we can effectively conclude that
1. Qdots have highly stable energy eigen states below the Compton Wavelength
hence perturbations are almost negligible.
2. Decoherence time (time to decohere) of Quantum Dots is much more than other
techniques due to their 0-dimensional nature.
3. Qubits, Teleportation, storage of information, processing of information can be
done efficiently on Quantum Dots.
4. A large ensemble of Quantum Dots can be used for making Quantum Computers
because of their property of almost negligible decoherence amongst themselves
and from the environment.
Acknowledgements
First of all let me take this opportunity to thank God for with out Him nothing can happen
in this mortal world.
I also pay my gratitude to Professor Y M Gupta who is a person from whom I learned
so much on so many aspects of life. His self-less preaching influenced almost every
aspect of my thought. On the research front, almost all of my works originated from his
insightful notes. He always supported and motivated me to do hard work during my B.
Tech and especially for this dissertation by guiding me through the right path.
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