QUANTUM ENTANGLEMENT IN ONE DIMENTIONAL SPIN-1/2HEISENBERG ANTIFERROMAGNETIC
BY
OLEABHIELE PROMISE IROBEKHAN
MATRIC NO: FNS/PHY/15/19160
DEPARTMENT OF PHYSICS,
FACULTY OF PHYSICAL SCIENCES,
AMBROSE ALLI UNIVERSITY,
EKPOMA, EDO STATE.
JANUARY, 2020.
A PROJECT WORK SUBMITTED TO THE DEPARTMENT OF
PHYSICS, FACULTY OF PHYSICAL SCIENCES, AMBROSE ALLI UNIVERSITY, EKPOMA, EDO
STATE IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR AWARD OF BACHELOR OF
SCIENCES (B.Sc) DEGREE IN PHYSICS.
CERTIFICATION
This
is to certify that this project work was carried out by OLEABHIELE PROMISE IROBEKHAN with Matriculation number FNS/PHY/15/19160. In the Department of
Physics, Faculty of Physical Sciences, Ambrose Alli University, Ekpoma, Edo
State.
_______________________ ______________________
DR. R. O.
OKANIGBUAN Date
(Project
Supervisor)
_______________________ ______________________
DR. R. O.
OKANIGBUAN Date
(Head of
Department)
_______________________ ______________________
(External
Supervisor) Date
DEDICATION
This Project work is dedicated to
Almighty God for His sufficiency throughout its course and to my parent for
their unwavering believe in me.
ACKNOWLEDGMENT
In the course of writing this book,
I have received tremendous support from family, friends, relations, and
colleagues. I own a heavy debt of gratitude to all of them.
I take immense pleasure in
appreciating my beloved parents, Dn. and Dns. Oleabhiele Alfred, my siblings,
Miss Oghoye, Mrs. Akanbi and Mr. Eromonsele and my Uncles, Matthew Idumonza,
Charles Idumonza, Barr. Sylvester Oleabhiele for their love, care and support.
I am particularly grateful to my
dynamic supervisor and the Head of the Department of physics, Dr. R. O.
Okanigbuan who has sacrifice his time reading through the project and making
valuable criticisms, suggestions and putting his academic experience at my
disposal.
To my beloved Lecturers, Prof.
J.E.A. Osemeikhian, Prof. S.E. Iyayi, Prof. O. Ujuanbi, Prof. I. Aigbedion, Dr.
S.I. Jegede, Dr. O.J. Ataman, Dr. I. Iyoha, Dr. S.I. Ehika, Dr. C.V.O.
Amadasun, Dr. O.E Akhirabulu, Mr. K.O. Ozegin, Mr. S. Salufu, Mr Omole Henry,
who imbibed in me the
drive for relevant knowledge
acquisition, which ensured maximum utilization of this period. I will also like
to say a big thank you to the non-academic staffs of physics department.
Equally remembered my wonderful and
amazing pastor, Emmanuel Obot, Mr and Mrs Justice Denedo for their advices and
support and my friends Obehiaghe Helen and Isunoya Ibrahim, you both are
friends indeed. God bless you all.
TABLE OF
CONTENTS
Title
Page
Certification
Dedication
Acknowledgement
Table
of Contents
Abstract
CHAPTER ONE
1.1
General
Introduction/Literature Review
1.2
The
EPR Paradox
1.3
Bell’s
Theorem
1.4
Other
Tests on Entanglement
1.5
Application
of Quantum Entanglement
1.6
Aim
of Research
CHAPTER TWO
2.1
Basic Concepts of Entangled State
2.2
Entangled Quantum State
2.3
Pure State
2.4 Mixed State
2.5 Schmidt Decomposition
2.6 Density Matrix
2.7 Entanglement Entropy
CHAPTER THREE
3.1 Entanglement Entropy for the Two Site
Heisenberg Antiferromagnet
CHAPTER FOUR
4.1 Conclusion
Reference
ABSTRACT
Quantum
Entanglement in the ground state of two-site spin 1/2 Heisenberg antiferromagnetic is studied using
the concept of density matrix. The study reveal that, the ground state is a
maximally Entangled state.
CHAPTER ONE
1.1.
INTRODUCTION AND LITERATURE REVIEW
In quantum physics, the
entanglement of particles describes a relationship between their fundamental
properties that can't have happened by chance (Gregg Jeager 2009). Knowing
something about one of these characteristics for one particle tells you
something about the same characteristic for the other, this is to say that;
Quantum entanglement is a physical phenomenon that occurs when pairs or groups
of particles are generated, interact, or share spatial proximity in ways such
that the quantum state of each particle cannot be described independently of
the state of the others, even when the particles are separated by a large
distance Grangier
et al., (1982), this could refer to states such as their momentum,
position, or polarisation.
The counterintuitive predictions of quantum mechanics
about strongly correlated systems were first discussed by Albert Einstein in
1935, in a joint paper with Boris Podolsky and Nathan Rosen (Wikipedia). In
this study, the three formulated the Einstein–Podolsky–Rosen paradox (EPR
paradox), a thought experiment that attempted to show that quantum mechanical
theory was incomplete. They wrote: "We are thus forced to conclude that
the quantum-mechanical description of physical reality given by wave functions
is not complete (Einstein
A, Podolsky B, Rosen N; Podolsky; Rosen 1935).
However, the three
scientists Einstein, Podolsky and Rosen did not coin the word entanglement, nor
did they generalize the special properties of the state they considered.
Following the EPR paper, Erwin Schrödinger (1935) wrote a letter to Einstein in
German in which he used the word Verschränkung(translated by himself as
entanglement) "to describe the correlations between two particles that
interact and then separate, as in the EPR experiment. Like Einstein, Schrödinger was dissatisfied
with the concept of entanglement, because it seemed to violate the speed limit
on the transmission of information implicit in the theory of relativity.
Einstein later famously derided entanglement as
"spukhafte Fernwirkung" or "spooky action at a distance."
Many years passed until
J. Bell put all this discussion in more solid grounds. Accepting the notion of local realism adopted
by EPR, Bell developed his famous inequality involving statistics of
measurements on composite quantum systems. From that point on, the local
realism debate could go to the labs. Sometime later the first experimental
tests of Bell inequalities started to appear and confirm the non-local aspect of
quantum mechanics. As untangled states (also called separable states) can never
violate a Bell inequality, the experimental violation of Bell inequalities can
be seen as the first observation of entanglement (Horodecki et al., 2019).
This leads to correlations
between observable physical properties of the systems. For example, it is
possible to prepare two particles in a single quantum state such that when one
is observed to be spin-up, the other one will always be observed to be
spin-down and vice versa, this despite the fact that it is impossible to
predict, according to quantum mechanics, which set of measurements will be
observed. As a result, measurements performed on one system seem to be
instantaneously influencing other systems entangled with it, but quantum
entanglement does not enable the transmission of classical information faster
than the speed of light (Horodecki et al., 2019).
The correlations
predicted by quantum mechanics, and observed in experiment; reject the
principle of local realism, which is that information about the state of a
system should only be mediated by interactions in its immediate surroundings.
Different views of what is actually occurring in the process of quantum
entanglement can be related to different interpretations of quantum mechanics.
Entanglement has been demonstrated experimentally with photons, neutrinos,
electrons molecules as large as bulky balls, and even small diamonds. On 13
July 2019, scientists from the University of Glasgow reported taking the first
ever photo of a strong form of quantum entanglement known as Bell entanglement.
The utilization of entanglement in communication and computation is a very
active area of research (Wikipedia 2019).
As an example of
entanglement: a subatomic particle decays into an entangled pair of other
particles. The decay events obey the various conservation laws, and as a
result, the measurement outcomes of one daughter particle must be highly
correlated with the measurement outcomes of the other daughter particle (so
that the total momenta, angular momenta, energy, and so forth remains roughly
the same before and after this process). For instance, a spin-zero particle
could decay into a pair of spin-½ particles. Since the total spin before and
after this decay must be zero (conservation of angular momentum), whenever the
first particle is measured to be spin up on some axis, the other, when measured
on the same axis, is always found to be spin down. (This is called the spin
anti-correlated case; and if the prior probabilities for measuring each spin
are equal, the pair is said to be in the singlet state.)
1.2.
THE EPR PARADOX
The
paradox is that a measurement made on either of the particles apparently
collapses the state of the entire entangled system—and does so instantaneously,
before any information about the measurement result could have been
communicated to the other particle (assuming that information cannot travel
faster than light) and hence assured the "proper" outcome of the
measurement of the other part of the entangled pair. In the Copenhagen
interpretation, the result of a spin measurement on one of the particles is a
collapse into a state in which each particle has a definite spin (either up or
down) along the axis of measurement. The outcome is taken to be random, with each
possibility having a probability of 50%. However, if both spins are measured
along the same axis, they are found to be anti-correlated. This means that the
random outcome of the measurement made on one particle seems to have been
transmitted to the other, so that it can make the "right choice" when
it too is measured.
The
distance and timing of the measurements can be chosen so as to make the
interval between the two measurements spacelike, hence, any causal effect
connecting the events would have to travel faster than light. According to the
principles of special relativity, it is not possible for any information to
travel between two such measuring events. It is not even possible to say which
of the measurements came first. For two spacelike separated events 
and X1 and X2 there are inertial frames in which X1 is first and others in which X2 is first. Therefore, the correlation between
the two measurements cannot be explained as one measurement determining the
other: different observers would disagree about the role of cause and effect.
and X1 and X2 there are inertial frames in which X1 is first and others in which X2 is first. Therefore, the correlation between
the two measurements cannot be explained as one measurement determining the
other: different observers would disagree about the role of cause and effect.
1.3.
BELL’S THEOREM
Physicist John Stewart
Bell In his ground breaking 1964 paper, "On the Einstein Podolsky Rosen
paradox, presented an analogy (based on spin measurements on pairs of entangled
electrons) to EPR's hypothetical paradox. Using their reasoning, he said, a
choice of measurement setting here should not affect the outcome of a
measurement there (and vice versa). After providing a mathematical formulation
of locality and realism based on this, he showed specific cases where this
would be inconsistent with the predictions of quantum mechanics theory.
In experimental tests
following Bell's example, now using quantum entanglement of photons instead of
electrons, John Clauser and Stuart Freedman (1972) and Alain Aspect et al.
(1981) demonstrated that the predictions of quantum mechanics are correct in
this regard, although relying on additional unverifiable assumptions that open
loopholes for local realism.
“Bell's theorem states that any physical
theory that incorporates local realism cannot reproduce all the predictions of
quantum mechanical theory. Because numerous experiments agree with the
predictions of quantum mechanical theory, and show differences between
correlations that could not be explained by local hidden variables, the
experimental results have been taken by many as refuting the concept of local
realism as an explanation of the physical phenomena under test. For a hidden
variable theory, if Bell's conditions are correct, the results that agree with
quantum mechanical theory appear to indicate superluminal (faster-than-light)
effects, in contradiction to the principle of locality.”
In the course of years, many scientists have written about entanglement
and we shall review some of the work;
In
1964 Bell accepted the EPR conclusion—that the quantum description of physical
reality is not complete—as a working hypothesis and formalized the EPR idea of
deterministic world in terms of the local hidden variable model (LHVM) (Bell,
1964). The latter assumes that:
Ø Measurement
results are determined by properties the particles carry prior to, and
independent of, the measurement “realism”.
Ø Results obtained
at one location are independent of any actions performed at space like
separation “locality”
Ø The setting of
local apparatus is independent of the hidden variables which determine the
local results “free will’.
Bell proved that the above assumptions
impose constraints in the form of the Bell inequalities on statistical
correlations in experiments involving bipartite systems. He then showed that
the probabilities for the outcomes obtained when some entangled quantum state
is suitably measured violate the Bell inequality. In this way entanglement is
that feature of quantum formalism which makes it impossible to simulate quantum
correlations within any classical formalism.
The present-day entanglement theory has
its roots in some key discoveries: quantum cryptography with the Bell theorem
(Ekert, 1991), quantum dense coding (Bennett and Wiesner, 1992), and quantum
teleportation (Bennett et al., 1993), including teleportation of
entanglement of EPR pairs (so-called entanglement swapping) (Yurke and Stoler,
1992a, 1992b; Z ˙ ukowskiet al., 1993; Bose et al., 1998). All
such effects are based on entanglement and all have been demonstrated in
experiments (see Mattleet al., 1996; Bouwmeesteret al., 1997;
Boschiet al., 1998; Furusawaet al., 1998; Pan et al.,
1998; Jenneweinet al., 2000; Naiket al., 2000; Tittelet al.,
2000.
Remarkably, entanglement is a
resource which, though it does not carry information itself, can help in such
tasks as the reduction of classical communication complexity (Cleve and
Buhrman, 1997; Buhrmanet al., 2001; Brukneret al., 2004),
entanglement-assisted orientation in space (Brukneret al., 2005; Bovino,
Giardina, etal., 2006b),
Entanglement plays a fundamental role in
quantum communication between parties separated by macroscopic distances
(Bennett, DiVincenzo, Smolin, et al., 1996). Entanglement has also given
new insights for understanding many physical phenomena including super-radiance
(Lambert et al., 2004), superconductivity (Vedral, 2004), disordered
systems (Düret al., 2005), and the emergence of classicality (Zurek,
2003).
Entanglement
was also used on a deeper conceptual level to derive Born’s rule with the help
of the symmetry entanglement under local unitary operations, the property
called “entanglement assisted invariance” or “envariance” (Zurek, 2005; see
also Zurek, 2009).
1.1. OTHER TEST ON
ENTANGLEMENT
In August 2014, Brazilian researcher Gabriela
BarretoLemos and team were able to "take pictures" of objects using
photons that had not interacted with the subjects, but were entangled with
photons that did interact with such objects. Lemos, from the University of
Vienna, is confident that this new quantum imaging technique could find
application where low light imaging is imperative, in fields like biological or
medical imaging.
In 2015, Markus Greiner's group at Harvard
performed a direct measurement of Renyi entanglement in a system of ultracold
bosonic atoms.
From 2016
various companies like IBM, Microsoft etc. have successfully created quantum
computers and allowed developers and tech enthusiasts to openly experiment with
concepts of quantum mechanics including quantum entanglement.
1.2. APPLICATIONS OF QUANTUM ENTANGLEMENT
Quantum entanglement has applications in the emerging
technologies such as; Quantum Cryptography usually involves a key or keys to be
used in encryption and decryption algorithms. Quantum cryptography is primarily
concerned with the secure distribution of keys using quantum communication
channels. Quantum teleportation; Teleportation which has to do with the
transmission of quantum information using a classical channel and entanglement.
It demonstrates the use of entanglement as a communication resource. The
simplest case is to consider the teleportation of a single qubit using two bits
of classical communication and one entangled pair (EPR pair).
Another is the Quantum Dense Coding. In quantum
information theory, dense coding is a quantum communication protocol to
transmit two classical bits of information (i.e., either 00, 01, 10 or 11) from
a sender to a receiver by sending only one qubit from sender to receiver, under
the assumption of sender and receiver pre-sharing an entangled state. Dense
coding is the underlying principle of secure quantum secret coding. The
necessity of having both qubits to decode the information being sent eliminates
the risk of eavesdroppers intercepting messages.
1.3. AIM OF RESEARCH
The overall aim of this study is to determine the entanglement entropy
of the ground state of two site Heisenberg antiferromagnet.
CHAPTER TWO
2.1. BASIC CONCEPTS
OF ENTANGLED STATE
Quantum systems
display properties that are unknown for classical ones, such as the
superposition of quantum states, interference, or tunneling. These are all
one-particle effects that can be observed in quantum systems, which are
composed of a single particle. But these are not the only distinctions between
classical and quantum objects there are further differences that manifest
themselves in composite quantum systems, that is, systems that are comprised of
at least two subsystems.
States that display such non-classical correlations are referred to as entangled states,
and it is the aim of this chapter to introduce the basic tools that allow to
understand the nature of Such states, to distinguish them from those that are
classically correlated and to quantify non-classical correlations.
2.2. ENTANGLED
QUANTUM STATE
CHAPTER FOUR
4.1. CONCLUSION
The Entanglement Entropy of two site antiferromagnetic
Heisenberg model have been studied using the concept of density matrix theory.
It is shown that the ground state of the Heisenberg chain
is Maximally Entangled with an Entanglement entropy of 1.
REFERENCE
A. Aspect, J.
Dalibard, and G. Roger (1982). “Experimental Test of Bell’s Inequalities Using
Time- Varying Analyzers”. Phys. Rev. Lett. 49, 1804.
A. Aspect, P.
Grangier, and G. Roger (1981). “Experimental Tests of Realistic Local Theories
via Bell’s Theorem”. Phys. Rev. Lett. 47, 460.
Alisa Bokulich,
Gregg Jaeger 2010. “Philosophy of Quantum
Information and Entanglement”. Cambridge University Press,.
C. H. Bennett
and G. Brassard 1984. "Quantum cryptography: Public key distribution and
coin tossing". In Proceedings
of IEEE International Conference on Computers, Systems and Signal Processing,
volume 175, p. 8.
Einstein A,
Podolsky B, Rosen N; Podolsky; Rosen (1935). "Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?" Phys. Rev. 47. 777.
Freedman, Stuart
J.; Clauser, John F (1972). "Experimental Test of Local Hidden-Variable
Theories". Physical Review
Letters. 28 (14):
938–941.
Horodecki M,
Horodecki P, Horodecki R; Horodecki; Horodecki (1996). "Separability
of mixed states: necessary and sufficient conditions". Physics Letters A. 223.
Horodecki R,
Horodecki P, Horodecki M, Horodecki K; Horodecki; Horodecki; Horodecki (2009).
"Quantum entanglement". Rev.
Mod. Phys. 81 (2):
865–942.
J.B. Altepeter,
E.R. Jeffrey, P.G. Kwiat, S. Tanzilli, N. Gisin and A. Ac´ın(2005).
“Experimental methods for detecting entanglement”. Phys. Rev. Lett. 95, 033601.
M. P. Hobson; et al.,(1998). “Quantum Entanglement and Communication
Complexity”pp. 1/13.
Nicolas Brunner; Daniel Cavalcanti; Stefano Pironio; Valerio Scarani;
Stephanie Wehner (2014)."Bell nonlocality". Rev. Mod.
Phys. 86 (2): 419–478.
Nielsen, Michael
A.; Chuang, Isaac L (2000). “Quantum
Computation and Quantum Information”. Cambridge University Press.
pp. 112–113.
R. F. Werner
(1989). “Quantum states with Einstein-Podolsky-Rosen correlation admitting a
hidden-variable model”. Phys. Rev. A 40, 4277.
Schrödinger
E(1936)."Probability relations between separated systems". Mathematical Proceedings of the Cambridge
Philosophical Society. 32 (3):
446-452.
Schrödinger
E(1935). "Discussion of probability relations between separated
systems". Mathematical
Proceedings of the Cambridge Philosophical Society. 31 (4): 555–563.
R. Augusiak, M.
Demianowicz, J. Tura and A. Acín (2015). "Entanglement and Nonlocality are
Inequivalent for Any Number of Parties". Phys. Rev. Lett. 115 (3): 030404.
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