Summer Research Fellowship Programme of India's Science Academies 2017
SUMMER PROJECT REPORT
SUMMER RESEARCH FELLOWSHIP PROGRAMME-2017,
INDIAN ACADEMY OF SCIENCES,
BANGALORE, INDIA.
UNCERTAINTY RELATIONS AND BASICS OF
QUANTUM INFORMATION
Submitted by
Heramb Bhusane
Fergusson College, Pune-411004
Under the guidance of
Dr. Pankaj Agrawal
Institute of Physics,Bhubaneswar.
An Autonomous Research Institute of Department of Atomic Energy,
Government of India.
Summer Project
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Summer Project
Acknowledgement
I would like to express my thanks to Indian Academy of Science, Bangalore to give me the
golden opportunity to do this wonderful summer project in Institute of Physics, Bhubaneswar.
I would like to thank Dr. Pankaj Agrawal, Institute of Physics,Bhubaneswar for his time to
time valuable guidance.Dr. Agrawal’s guidance motivate me in the field of quantum mecah-
nics.
This project would have been impossible without Chandan Datta, Arpan Das and Dr. Sujit
Chaudhari who helped me clearing the douts and difficulties in the project.
Finally, I will thank to the all staffs of Institute of Physics, Bhubaneswar for providing good
environment for study.
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Abstract
In this report, I will discuss Uncertainty relations and basic concepts in quantum information.
Before this, I willl give historical background and formalism of quantum mechanics.
Uncertainty relations were first proposed by Heisenberg . His proposal made significant
impact on the understanding of quantum mechanics. Later, mathematical elaboration was
given by, Kennard, Robertson and Schr
¨
odinger.
In quantum information I will discuss about quantum entanglement. To do this I will give
brief account of qubits, composite systems, entangled states and Bell states.
Then, I will discuss No-cloning theorem and a basic quantum communication protocol:
teleportation.
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1.Introduction
Quantum mechanics is specially designed branch of physics to describe behavior of world
at much smaller scale (compared to size of atom or molecule). Quantum mechanics is the
generalization of classical physics over smaller scale. It covers the region, where classical
mechanics fails. Physical quantities can be discrete and one can not known all the quantities
about system at the same time in quantum mechanics . This makes quantum mechanics so
abstract and non-intuitive.
Quantum mechanics not only helps us to understand the world better, but also it has
profound applications in fields like condensed matter, quantum computation and informa-
tion, chemistry, solid state, etc. It enables us to make the use of small world for our benefits.
Scientist are hoping we can understand the life using quantum biology.
In the future, quantum information will ensure that we can make the of smallest quantum
system like atoms to store and process the information. Which in turn will reduce the size of
the components along with increase in speed.
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1.1.Origin of quantum mechanics
a)Blackbody radiation and Planks law
Underdstanding the radiation emmited by a blackbody was one of the unsolved question of
nineteenth century. Physicists tried to solve this question using classical physics. But theory
was unable to explain Blackbody radiation. Theory was predicting that the body should
emmit infinte energy at higher frequencies (ultraviolet catastrophe). In the year 1900, Max
Planck resolved this question. He adopted an odd assumption. His assumption was energy
exchange between matter and radiation should be discrete.
Planck imagined energized electric resonators were producing heat radiation.
Ordinary resonators of classical physics can take continuous range of frequencies. But Planck
assumed these energized electric resonators should take discrete energies. He then postulated
that the energy of the radiation (of frequency
ν
) emitted by these resonators must come only
in integer multiples of hν:
E =nhν where n=0,1,2,3,...
Here
h
is called as Planks constant, which is a fundamental constant in quantum physics.(h=6
.
626
×
10
34
J.s
)The number is very small suggesting quantum effects are to be expected in the small
scale.
Using Plancks concept of quantization many scientist like Einstein, Compton, de-Broglie
and Bohr explained many other intriguing phenomena.
b)Photoelectric effect
In 1887, Hertz discovered photoelectric effect:The electrons are liberated from surface of some
metal upon the incidence of light of suitable frequency. Einstein in 1905 used the concept of
quantization of light to explain this phenomenon. He saw the similarity between the ideal
gases and radiation of high energy. Then he argued that like ideal gases, light must consist
of discrete units. Using this concept of quantization he successfully described aspects of
this phenomenon like why it starts after certain frequency, why photocurrent increases with
intensity of light and saturates after certain level and why it is not an time lagging process (as
described by classical physics).
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c)Niels Bohrs atomic model
In 1913, Bohr used concept of quantization of angular momentum of an electron to obtain
atomic model. The orbits in which angular momentum is integral multiple of
nħ
are only
allowed.
Next, he supposed electrons can jump up and down between these allowed orbits. Light of
suitable energy can shift a electron in higher energy orbit. But electron can not stay in this
orbit, it will emmit the light of specific frequency that is equal to energy difference between
these two orbits. Bohr assumed the amount of energy of this light agrees with Plancks
formula: E =hν
This model enables to explain discrete spectra of an atom.
d)de-Broglie Hypothesis
In blackbody radiation problem, photoelectric effect, Compton effect, one sees the particle
aspect of radiation. Can other way round case betrue? Can matter show wave like properties ?
In 1923, de-Broglie provided hypothesis that every massive particle moving with momentum
p is associated with wavelength given by,
λ =
h
p
.
This hypothesis was confirmed experimentally by Davission-Germer and later by Thomson.
Matter wave soon became an good idea to describe the behaviour of quantum world.
e)Schr
¨
odinger Equation
Based on the idea of matter wave Schr
¨
o
dinger in 1926 developed an wave equation which
describes this wave behaviour.
Consider, simple case of standing waves on a string. Only the frequencies which are integral
multiple of fundamental frequency will survive. The same condition applies to the electron
in hydrogen atom. The electron is trapped in the positive potential of nucleus. This gives
answer to the question of stable orbits in Bohrs theory.
Solution of this differential equation gave the energy spectrum and wave function of system
under consideration.
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f)Matrix mechanics
Before the wave mechanics formulation of quantum physics, Heisenberg, Max Born, and
Pascual Jordan in 1925 developed the matrix mechanics. Expressing dynamical quantities
such as energy, position, momentum and angular momentum in terms of matrices, he
obtained an eigenvalue problem that describes the dynamics of microscopic systems; the
diagonalization of the Hamiltonian matrix yields the energy spectrum and the state vectors
of the system. Matrix mechanics successfully described observed atomic spectra.
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1.2.Formalism of Quantum Mechanics
a)Postulates of Quantum mechanics
1. State of the system:-
A quantum mechanical system ia described by a ket vector
|ψ >
which belongs to a Hilbert space. All the information about the system is contain within
|ψ >.
2. Observables
Position operator
ˆ
X and momentum operator
ˆ
P obey commutation rule;
[
ˆ
X ,
ˆ
P] =i ħ.
An observable A is represented by Hermitian operator
ˆ
A,
ˆ
A =ω(
ˆ
X ,
ˆ
P).
3. Measurement of Observables
A measurement of an observable A on the system gives one of the eigenvalues of hermi-
tian operator
ˆ
A.
If
|ψ
n
>
is an eigenvector of an operator
ˆ
A
with eigenvalue
a
n
then, probability of
obtaining the value a
n
is given by,
P
n
=|<ψ
n
|ψ >|
2
.
After the measurement |ψ > collapses to |ψ
n
>.
4. Time evolution:-Time evolution of the system is given by Schr
¨
odinger equation:
iħ
|ψ>
t
=
ˆ
H|ψ >.
And time evolution is unitary.
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b)Description of Postulates
Mathematical description of quantum world is based on these few assumptions. First three
postulates say about state of the system and the forth one gives information how the state of
the system will be after some time t. Systems state in quantum mechanics is represented as
vector |ψ > in Hilbert space.
|ψ|
2
is the probability density for finding the position of particle.Total probability of
should be one. Hence,
Z
+∞
−∞
|ψ|
2
d x =1.
In other words norm of |ψ > should be one.<ψ|ψ >=1.
Hermitian operators are linear operators that have real eigenvalues and the eigenvectors
belonging to distinct eigenvalues are orthonormal.
The eigenvectors of an observable are complete. It means, any function which belong to
Hilbert space can be represented by superposition of these eigenvectors.
We can write state |ψ > as,
|ψ >=
X
n
|ψ
n
><ψ
n
|ψ >=
X
n
a
n
|ψ
n
>.
The expectation value of an observable is given as,
<
ˆ
A >= <ψ|
ˆ
A|ψ >.
If eigenvalues of the operator are discrete we can write;
<
ˆ
A >=
X
n
a
n
|<ψ
n
|ψ >|
2
=
X
n
a
n
P
n
.
And if the eigenvalues are continuous then,
<
ˆ
A >=
Z
+∞
−∞
ψ
ˆ
Aψd x.
c)Measurement and commuting operators
Act of measurement perturbs the quantum mechanical system. Measurement changes the
state of the system. After carrying out a measurement, the state will change to one of the
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eigenstates of the corresponding operator.
Two observables are said to be compatible if they commute:i.e,[A,B]=0.
The two operators can be measured simultaneously for the system.
Operators which do not commute are called as incompatible operators and they can not be
measured simultaneously.
Suppose, if
ˆ
A and
ˆ
B do not commute and system is in eigenstate |ψ
(a)
n
> of operator
ˆ
A.
ˆ
A|ψ
(a)
n
>=a
n
|ψ
(a)
n
>
then if we measure the operator
ˆ
B the system will go into the one of the eigenstate of
ˆ
B .
If we measure
ˆ
A again then we will get the value differ from former value a
n
.
But, suppose
ˆ
A
and
ˆ
B
are commutating then they have same set of eigenvectors. There-
fore, measurement of A will not affect measurement of B and vice versa.
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2. Uncertainty relations
2.1.Introduction to Uncertainty principle
Microscopic world is well described by quantum mechanics. Indeterminacy is inherent in
quantum mechanics.In classical mechanics we can assign exact simultaneous values to the
position and momentum, but in quantum we can not do that. The more precisely we know
the position of a particle, less precisely we known about its momentum.
In general, the quantities which are canonical conjugates in classical mechanics can not
be simultaneously measured with an arbitary accuracy and there is lower bound on their
product of uncertainties of the order of Plancks constant. Simultaneously means at the same
instant of time and with arbitary accuracy means with infinite precision. So, we can not
measure these quantities exactly at the same moment. Uncertainty principle is more than
measurement of these quantities. Uncertainty is fundamental is nature. It tell us that these
quantities are not actually simultaneously defined at a given instant of time.
These uncertainty relations were introduced by Werner Heisenberg in 1927 in the paper
"On Physical content of quantum kinematics and mechanics". He tried to argue in this
paper that Indeterminacy is inherent in quantum mechanics, but the way in which he was
thinking is confused with observed effect in physics. Observed effect is when we measure
the system measurement disturbs the system which we are measuring. Later in the end
of year Kennard related this relation with the standard deviation of ensemble of particle
system. Uncertainty proved to be intrinsic property of the system. Schr
¨
o
dinger gave stronger
uncertainty principle.
2.2.Why Heisenberg proposed uncertainty pr inciple?
The story starts with the making of formalism for quantum mechanics. Heisenberg in 1925
gave first formalism of quantum mechanics. He used observed quantities for his formalism.
His inspiration was Plancks law of quantization and Bohr’s atomic model. His idea was that
we must use the thing which we can observe and measure. There is no point to look at the
structure of atom which we cannot see. He made use of spectral data to make his theory.
Born realized that these transition quantities obey the rules of matrices. Using this idea
Heisenberg, Born and Jordan developed "Matrix Mechanics". This theory was consistent
with observations. In this theory it was postulated that the quantities which are canonical
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conjugate in classical physics, should obey following commutation relation in quantum
mechanics:
QP PQ =i h/2π (1)
Next year schr
¨
o
odinger proposed another alternative formalism for quantum mechanics
called "wave mechanics". Wave meachanics is more intuitive than the marix mechanics.
Schr
¨
o
dinger claim his theory was intelligible. Many scientist appreciated this theory. But this
wave theory had problems of its own. Most physicists were slow to accept "matrix mechanics"
because of its abstract nature and its unfamiliar mathematics. Heisenberg was upset about
this development. He admitted that "Matrix mechanics" is not as intuitive as wave mechanics.
He concluded:
To obtain a contradiction-free intelligible interpretation, we still lack some essential feature in
our image of the structure of matter.
The purpose of his 1927 paper was to provide exactly this lacking feature.
2.3.Heisenberg argument
In 1927, he publish his paper "On physical content of quantum kinematics and mechanics".
In this paper he proposed uncertainty relations in micro physical systems. His goal was that
matrix mechanics could lay the same claim to intelligibility as wave mechanics. In his view, if
theory is experimentally consistent then we could say that theory is correct.
He adopted an assumption: terms like position of particle have meaning only if one
specifies a suitable experiment by which the "position of a particle" can be measured. He
considered the thought experiment to find the "position of the electron" using light(photon).
The shorter the wavelength of light employed, the greater the accuracy will be. But, when we
use the shorter wavelength of light we also have to consider the Compton effect, which will
disturb the momentum of the electron. Heisenberg argues using this thought experiment
that when we known the position more accurately, the momentum becomes less accurate.
He stated:
The instant of time when the p ositi on is determined, that is, at the instant when the photon
is scattered by the elec tron, the electron undergoes a discontinuous change in momentum.
This change is the greater the smaller the wavelength of the light employed, i.e., the more
exact the determination of the position. At the instant at which the position of the electron is
known, its momentum therefore can be known only up to magnitudes which correspond to
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that discontinuous change; thus, the more precisely the position is determined, the less precisely
the momentum is known, and conversely. (Heisenberg 1927: 174–5)
Let
δq
be error in determining the position, so here it is related to wavelength of light and
δp
be discontinuous change in momentum due to quantum effect. Then, there product of
uncertainties becomes:
δpδq h (2)
Heisenberg argued that this is the consequence of the postulate:
QP PQ =i h/2π
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2.3.1.Heisenberg Gamma Ray Microscope
To improve his argument, as discussed in teh lst section, Heisenberg came up with more
elaborated thought experiment. Consider, the electron is traveling from left to the right along
x direction under
γ
-ray microscope. The electron is illuminated from below with
γ
-rays as
shown in the diagram:
γ
-rays are entering the microscope making a cone whose vertex is at position of electron.
Angle of this cone is
²
. Then applying criterion for resolving power of microscope, the limit
up to which we can find the position of e
is given by,
x
λ
si n(²)
.
But to get the position of electron,
γ
-ray should collide with electron. This will cause Compton
Effect. Due to this effect the momentum of e
will change by:
p
h sin(²)
λ
.
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Multiplying the above two uncertainties we get
x p h.
In this experiment
γ
-ray can enter the microscope within any angle up to
²
. So, we are
not able to find change in the momentum of electron by using formula for Compton effect.
Therefore, Heisenberg argued that in the process of measuring the position of electron, it is
not just that the momentum of electron changes, but it changes by an unpredictable amount.
This introduces uncertainty between position and momentum.
2.3.2.Problem with Heisenberg Uncertainty
There are a number of issues with the above reasoning.
He talks about exact position and momentum before act of measurement, which is not
true in quantum mechanics.
The position and momentum of the electron is given by probability distribution. The
position and momentum of the electron are not well defined simultaneously.
We see in this argument the relation between error in position of the electron and
disturbance in momentum and this kind of error-disturbance relation follows trade-off
relation.
2.4.Kennards Inequality
In the above discussion uncertainties in the position and momentum were not precisely
defined. The inequality relating the standard deviation of position
σ
q
and the standard
deviation of momentum σ
p
was derived by Earle Hesse Kennard later that year in 1927:
σ
q
σ
p
ħ
2
The fluctuation in the quantum system exists regardless whether it is measured or not, and
this inequality does not say anything about what happens when a measurement is performed.
Spread in standard deviation represents the fluctuations in the observable in a given state.
It is not connected to concept of the “inaccuracy” of a measurement, such as the resolving
power of a microscope. This inequality says that you cannot suppress quantum fluctuations
of both position
σ
q
and momentum
σ
p
lower than a certain limit simultaneously. Kennard’s
formulation is therefore totally different from Heisenbergs formulation. Heisenberg himself,
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have been under the misapprehension that both formulations describe virtually the same
phenomenon.
Proof of Kennard inequality
Here is the proof of Kennards inequality:
Square of standard deviations are defined as;
σ
2
x
=<ψ|(
ˆ
x< x >)
2
|ψ >
σ
2
p
=<ψ|(
ˆ
p<p >)
2
ψ >
which can be written as,
σ
2
x
=
Z
−∞
x
2
.|ψ(x)|
2
(
Z
−∞
x.|ψ(x)|
2
d x)
2
σ
2
p
=
Z
−∞
p
2
.|φ(p)|
2
(
Z
−∞
p.|φ(p)|
2
d p)
2
We shift our coordinate system in a such a way that the means will vanish
σ
2
x
=
Z
−∞
x
2
.|ψ(x)|
2
d x
σ
2
p
=
Z
−∞
p
2
.|φ(p)|
2
d p
The function f (x) = x ·ψ(x) can be interpreted as a vector in position space.
Variance in x can be written as,
σ
2
x
=< f |f >
We can take
˜
g
(
p
)
=p.φ
(
p
) as a vector in momentum space. But, f(x) is in position space and
˜
g
(
p
) is in momentum space.
φ
(
p
) and
ψ
(
x
) are Fourier transforms of each other. (
iħ
d
d x
)
is momentum operator in position space. Hence,
g
(
x
)
=i ħ
d
d x
ψ
(
x
)
.
Fourier transform is
unitary. So, it conserves the inner product. We can write:
σ
2
p
=
Z
−∞
|
˜
g (p)|
2
d p
=<g | g >
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Hence,
σ
2
x
σ
2
p
=< f |f >. < g|g >
Applying Schwartz inequality;
< f |f >. < g|g > < f |g >
The modulus squared of any complex number z can be expressed as,
|z|
2
=(Re(z))
2
+(Img (z))
2
(Img (z))
2
=(
z z
2i
)
2
we let z =< f |g > and z
=<g |f > and substitute these into the equation above to get
|< f |g >|
2
(
<f |g>−<g|f >
2i
)
2
< f |g >< g |f >=
Z
−∞
ψ
(x)x.(i ħ
d
d x
)ψ(x)d x
Z
−∞
ψ
(x)(i ħ
d
d x
)xψ(x)d x
=i ħ.
Z
−∞
ψ
(x)[(x.
dψ(x)
d x
) +
d(xψ(x))
d x
]d x
=i ħ.
Z
−∞
ψ
(x)[(x.
dψ(x)
d x
) +ψ(x) +(x.
dψ(x)
d x
)]d x
=i ħ.
Z
−∞
ψ
(x)ψ(x)d x
=i ħ.
Z
−∞
|ψ(x)|
2
d x =i ħ
Therefore,
|< f |g >|
2
(
ħ
2
)
2
It follows
< f |f >. < g|g > (
ħ
2
)
2
σ
2
x
σ
2
p
(
ħ
2
)
2
σ
x
σ
p
ħ
2
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2.5. Schrodinger Inequality
Schr
¨
o
inger inequality generalizes the Robertson inequality. Roberson inequality is a general-
ization of Kennard inequality to any pair of noncommuting observables. In the derivation of
inequalities of Robertson and Kennard, one ignores the real part of complex number z, which
we use after Schwartz inequality. In Schr
¨
o
inger inequality we consider this real part. We can
derive Robertson and Kennard inequalities from this inequality.
Proof of Schrodinger Inequality
Before starting let us define some terms:
1. Symmetrized correlation function
C
s
(
ˆ
A,
ˆ
B) =
1
2
<{
ˆ
A,
ˆ
B} ><
ˆ
A ><
ˆ
B >
2. Non-symmetrized correlation function
C(
ˆ
A,
ˆ
B) =<
ˆ
A
ˆ
B ><
ˆ
A ><
ˆ
B >
3. Matrix of variation K
K =
Ã
σ(
ˆ
A)
2
C(
ˆ
A,
ˆ
B)
C(
ˆ
B,
ˆ
A) σ(
ˆ
B)
2
!
To obtained the inequality we will prove a condition for the matrix
K
. We will show that
determinant of this matrix is non-zero so that we can obtain inequality condition.
We proceed as follows. Let
z
=(
z
1
, z
2
) be the complex vector belonging to the two dimen-
sional complex space.
(z)
(K z) =(
z
1
z
2
)(
σ(
ˆ
A)
2
C(
ˆ
A,
ˆ
B)
C(
ˆ
B,
ˆ
A) σ(
ˆ
B)
2
)(
z
1
z
2
)
=(
z
1
z
2
)(
σ(
ˆ
A)
2
z
1
+C (
ˆ
A,
ˆ
B)z
2
C(
ˆ
B,
ˆ
A)z
1
+σ(
ˆ
B)
2
z
2
)
=|z
1
|
2
σ(
ˆ
A)
2
+z
1
z
2
C(
ˆ
A,
ˆ
B) +z
2
z
1
C(
ˆ
B,
ˆ
A) +|z
2
|
2
σ(
ˆ
B)
2
=z
1
z
1
(< A
2
>< A >
2
)+z
2
z
2
(<B
2
><B >
2
)+z
1
z
2
(<
ˆ
A
ˆ
B ><
ˆ
A ><
ˆ
B >)+
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z
2
z
1
(<
ˆ
B
ˆ
A ><
ˆ
B ><
ˆ
A >)
=|z
1
|
2
< A
2
>+z
1
z
2
<
ˆ
A
ˆ
B >+z
2
z
1
<
ˆ
B
ˆ
A >+|z
2
|
2
<B
2
>(z
1
z
1
< A >
2
+
z
2
z
2
<B >
2
+z
1
z
2
< A ><B >+z
2
z
1
<B >< A >)
= <(z
1
ˆ
A+z
2
ˆ
B)
(z
1
ˆ
A+z
2
ˆ
B) >(|z
1
|
2
< A >
2
+2Re[(z
1
z
2
) < A ><B >]+|z
2
|
2
<B >
2
)
=(z)
(K z) =<(z
1
ˆ
A +z
2
ˆ
B)
(z
1
ˆ
A +z
2
ˆ
B) >−|< z
1
ˆ
A +z
2
ˆ
B >|
2
A
n ×n
Hermitian matrix
M
is said to be positive semi-definite, if (
z
)
(
K z
) is real and greater
than or equal to zero for all complex numbers. Therefore
K
is positive semi-definite matrix. It
has all non-negative eigenvalues and its determinant will be greater than or equal to zero.
det[K] =σ(
ˆ
A)
2
σ(
ˆ
B)
2
C (
ˆ
A,
ˆ
B)C(
ˆ
B,
ˆ
A)
C(
ˆ
A,
ˆ
B) =<
ˆ
A
ˆ
B ><
ˆ
A ><
ˆ
B >
=
1
2
<
ˆ
A
ˆ
B +
ˆ
B
ˆ
A ><
ˆ
A ><
ˆ
B >+
1
2
<
ˆ
A
ˆ
B
ˆ
B
ˆ
A >
=
1
2
<{
ˆ
A,
ˆ
B} ><
ˆ
A ><
ˆ
B >+
1
2
<[
ˆ
A,
ˆ
B] >
=C
s
(
ˆ
A,
ˆ
B) +
1
2
<[
ˆ
A,
ˆ
B] >
Likewise,
C(
ˆ
B,
ˆ
A) =<
ˆ
B
ˆ
A ><
ˆ
B ><
ˆ
A >
=
1
2
<
ˆ
A
ˆ
B +
ˆ
B
ˆ
A ><
ˆ
A ><
ˆ
B >
1
2
<
ˆ
A
ˆ
B
ˆ
B
ˆ
A >
=
1
2
<{
ˆ
A,
ˆ
B} ><
ˆ
A ><
ˆ
B >
1
2
<[
ˆ
A,
ˆ
B] >
=C
s
(
ˆ
A,
ˆ
B)
1
2
<[
ˆ
A,
ˆ
B] >
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det [K] =σ(
ˆ
A)
2
σ(
ˆ
B)
2
©
C
s
(
ˆ
A,
ˆ
B) +
1
2
<[
ˆ
A,
ˆ
B] >
ª©
C
s
(
ˆ
A,
ˆ
B)
1
2
<[
ˆ
A,
ˆ
B] >
ª
=σ(
ˆ
A)
2
σ(
ˆ
B)
2
C
s
(
ˆ
A,
ˆ
B)
2
1
4
°
°
< A,B >
°
°
2
0
Therefore, we have
σ(
ˆ
A)
2
σ(
ˆ
B)
2
C
s
(
ˆ
A,
ˆ
B)
2
1
4
°
°
< A,B >
°
°
2
So Schrodinger Inequality is
σ(
ˆ
A)
2
σ(
ˆ
B)
2
C
s
(
ˆ
A,
ˆ
B)
2
+
1
4
°
°
< A,B >
°
°
2
Since first term on the RHS is positive, so we can drop it and obtain Robertson or
Heisenberg-Robertson uncertainty relation
σ(
ˆ
A)σ(
ˆ
B)
1
2
°
°
< A,B >
°
°
If we choose A to be a position operator and B to be a momentum operator, then we will
recover Kennards uncertainty relation.
2.6.Conclusions
There are two kinds of uncertainties in Quantum mechanics: Preparation Uncertainty
and Measurement Uncertainty.
Preparation Uncertainties are inherent in Quantum Mechanics. It cant be avoided in
any way and it has no connection to the measurement of system. Kennard, Robertson
and Schr
¨
odinger uncertainties come under this type of uncertainty.
The act of measurement disturbs the system and this causes measurement uncertain-
ties. This causes error-disturbance trade-off relation.This we have seen in Heisenbergs
argument of gamma ray microscope.
Heisenberg wanted to conceive preparation uncertainty. But the way by which he
described this uncertainty it is associated with measurement uncertainty. These two
things are totally different.
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3.Quantum Entanglement
Entanglement is one of the most mysterious phenomenon in physics. Consider, two particles
A and B. These particles somehow share some relationship with each other. Measurement
of a quantity on A affects the measurement of a quantity on B. One can make measurement
on A to find its state. But, by postulate of quantum mechanics, measurement disturbs the
system and system collapses into one of the eigenstate. This collapse of system in the act
of measurement of A will affect particle B, because of their relationship. Entanglement can
exist between two systems which are far away from each other. Entanglement is the key for
teleportation.
3.1.Quantum Bits
Quantum bits or simply qubits are the fundamental unit in quantum information. Classical
bits can have one of the values 1 or 0, while qubit is superposition of two quantum state
|
0
>
and |1 >:
|ψ > = α|0 > + β|1 >,
|
0
>
and
|
1
>
are called computational basis states.
α
and
β
are complex numbers obeying
relation,
|α|
2
+β|
2
=
1 to conserve the probability in quantum mechanics. So, a qubit can be
imagined as a vector of unit length in two dimensional vector space. It can also be written as,
|ψ >=cos
θ
2
|0 >+ e
iφ
sin
θ
2
|1 >,
The qubit |ψ > can be represented in a Bloch sphere.
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3.2. Composite system
Suppose, we have two particles A and B, each representing a qubit. Then, the state of the
system which made from the combination of A and B is given by the tensor product of the
state of A and the state of B.
|AB >=|A > |B >
If we have
|A >=α
1
|0 >+β|1 >,
|B >=α
2
|0 >+β|2 >.
Then, the composite system state |AB> is,
|AB>=α
1
α
2
|0|0 >+α
1
β
2
|0|1 >+β
1
α
2
|1|0 >+β
1
β
2
|1|1 >
This system have four computational basis states
|
00
>,|
01
>,|
10
> and |
11
>
. If system
|ψ >
is made from n components |ψ
1
>,|ψ
2
,...,|ψ
n
> then composite system state is given by,
|ψ >=|ψ
1
>⊗|ψ
2
>... |ψ
n
>
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3.3.Entangled States
The state which can not be represented by the tensor product of the states of the subsystems
is called a Entangled State. If
|ψ > 6= |A >⊗|B >,
then, |ψ > is called Entangled state.
3.4. Bell States
These are special kind states of two qubits,
|φ
+
>=
1
p
2
(|00 >+|11 >)
|ψ
+
>=
1
p
2
(|01 >+|10 >)
|φ
>=
1
p
2
(|00 >−|11 >)
|ψ
>=
1
p
2
(|01 >−|10 >)
Bell states are entangled states. To show this let us consider the Bell state
|φ
+
>=
1
p
2
(
|
00
>
+|
11
>
)
.
Let us try to express
|φ
+
>
as linear combination of two qubits
|α|
0
> +β|
1
>
and
γ|0 >+δ|1 >. We will expand and find the values of α,β, γ and δ.
(|00 >+|11 >) =(|α|0 >+β|1 >)(γ|0 >+δ|1 >)
=αγ|00 >+αδ|01 >+βγ|10 >+βδ|11 >
This implies
αγ = β δ =
1 and
αδ = β γ =
0. This is acontradiction.
Bell states can not
represented as product of two qubits. And hence Bell states are entangled states.
4. Quantum Communication Protocols
4.1. No-Cloning Theorem
Quantum Systems obey the principle of superposition. Also, the transformations are linear in
quantum mechanics.The No-Cloning theorem is consequence of the linearlity of the quan-
tum mechanics. The theorem says that we can not copy an arbitary unknown quantum state.
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It was stated by Wooters, Zurek and Dieks in 1982 and has profound implications in quantum
computing and related fields.
Proof of No-cloning theorem by Contradiction
We have an unknown arbitary state |ψ >.
|ψ >=α|0 >+β|1 >.
Suppose there exist an unitary transformation
U
which can clone the basis states of the the
system.
U|0 >=|00 >,
U|1 >=|11 >.
Question is can we also clone the state |ψ > ? We can, if
U|ψ >=|ψ >|ψ >.
Let us see if it is possible.
L.H.S. =U(α|0 >+β|1 >)
=α|00 >+β|11 >.
R.H.S. =(α|0 >+β|1 >)(α|0 >+β|1 >)
=α
2
|00 >+αβ|01 >+βα|10 >+β
2
|11 >
6=(α|0 >+β|1 >) =|ψ >
Hence,
U|ψ >6=|ψ >|ψ >.
This completes the proof of No-cloning theorem. So, there does not exist any linear transfor-
mation that can give us identical copy of the given system. It does not imply similar things
can not exist, but it says we can not copy the system which we dont known about.
No-cloning protects the uncertainty principle in quantum mechanics. If one could clone
an unknown state, then we can measure the position from one copy and momentum from
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another each with arbitary precision. Thus, in this way one can get knowledge about position
and momentum of unknown quantum state violating uncertainty principle. But, no-cloning
denies this possibility.
4.2.Teleportation
In this protocol, one party, say, Alice has a particle in an unknown state. She wishes to send
the state of the particle without sending the particle to another party, say Bob. It is possible
if they share an entangled state. So Alice and Bob are two parties sharing entangled state
|φ
+
>
. Alice wants to send unknown qubit
α|
0
>+β|
1
>
to Bob using teleportation protocol.
Combined state of this qubit and entangled state |φ
+
> is,
|Ψ >=(α|0 >+β|1 >)(
1
p
2
(|00 >+|11 >)).
In the Bell basis, we can rewrite this state as,
|Ψ >=
1
2
[|φ
+
>(α|0 >+β|1 >)+|φ
>(α|0 >β|1 >)+|ψ
+
>(α|1 >+β|0 >)+|ψ
>(α|1 >β|0 >)]
Alice will do measurement in the Bell basis. According to what she measures state of the
particle belonging to Bob now changes correspondingly:
State measured by Alice State of Bob
|φ
+
> (α|0 >+β|1 >)
|φ
> (α|0 >β|1 >)
|ψ
+
> (α|1 >+β|0 >)
|ψ
> (α|1 >β|0 >)
Alice will send the information about the state she gets in Bell basis through classical channel
using two classical bits to Bob.Then, correspondingly, Bob will apply the transformation to
get original unknown state (α|0 >+β|1 >).
State measured by Alice The classical bits sent to the bob Transformation which bob will apply
|φ
+
> 00 I(indentity transformation)
|φ
> 01 σ
z
|ψ
+
> 10 σ
x
|ψ
> 11 σ
z
σ
x
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By applying suitable transformation, Bob will change the state of his qubit to the unknown
qubit state (
α|
0
>+β|
1
>
). In this process, Alice will not retain the qubit. It will be transfered
to Bob. In this case we can see consequence of no-cloning theorem.
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References
1. The Uncertainty Principle, Stanford Encyclopedia of Philosophy.
2.
Quantum Theory and Measurement,John Archibald Wheeler, Wojciech Hubert Zurek
Princeton University Press, 1983.pages,62-66.
3.
Y. Watanabe, Formulation of Uncertainty Relation Between Error and Disturbance in
Quantum Measurement by Using Quantum Estimation Theory, Springer Theses
4.
Quantum computation and Quantum Information by Michael A. Nielsen and Issac
L.Chuang. Text book Cambridge University Press. Edition 2002
5.
Introduction to Quantum Information Science, Vlatko Vedral.Print ISBN-13: 9780199215706
6. //www.quantiki.org/wiki/no-cloning-theorem
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