1. 1
Thermal Entanglement and state transfer in
Heisenberg XXX Spin ½ Chain
Department of Electronics Science
Acharya Prafulla Chandra College
under the affiliation of
West Bengal State University.
By
Siddhartha Ray Choudhuri Sayantan Mukherjee
Roll No. APC/PG(S4)/14/ELTSC/09 and Roll No. APC/PG(S4)/14/ELTSC/18
Registration no. 10112222114000031 Registration no. 10112222114000044
Under the Supervision & Guidance of
Dr. Sankha Subhra Nag
Department Of Physics, Sarojini Naidu College for Women

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Abstract
The focus of this project is to study the entanglement generation in Heisenberg XXX Spin
Chain under the influence of external magnetic field for a bipartite system. The entangled
pair may act as a physical resource to implement quantum computation and communication.
We have measured the Fidelity (tool to check similarity between the input state and output
state) and Concurrence (measure of mutual entanglement between the pair of Qubits) of the
system under some applied external magnetic field with increasing time and temperature.
Maximally entangled state is very hard to reach and highly desirable for quantum
computation.

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Acknowledgement
I like to offer my profound gratitude and sincere regard to our supervisor Dr. Sankha Subhra
Nag, Department Of Physics, Sarojini Naidu College for Women, West Bengal State
University for his cheerful personality and constant encouragement that have turned my
enthusiasm in undertaking the present work. We are grateful for his invaluable guidance and
spirit of challenge at every stage of work for which this project has come to a success.

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Contents Page No.
Abstract 2
Acknowledgements 3
Contents 4
1. Introduction 5
1.1. Quantum Bit 5
1.2. Superposition 5
1.3. Bloch Sphere 6
1.4. Quantum Gate 6
1.4.1. Single Qubit quantum gate 6
1.4.2. Two Qubit quantum gate 7
1.5. Quantum Algorithm 8
1.6. Quantum Parallelism and Entanglement 8
1.7. Quantum State transition 9
2. Spin Chain as Quantum Channel 11
3. Entanglement Transfer in Spin Chain under external magnetic field 12
3.1. Heisenberg 2 Qubit spin ½ Hamiltonian equation 12
3.2. Density Matrix 12
3.3. Concurrence 14
3.4. Fidelity 14
4. Result and Discussion 15
4.1. Fidelity vs. Time 15
4.2. Fidelity vs. Magnetic Field 17
4.3. Concurrence vs. Temp & Field 19
5. Further planning 20
6. Appendix 21
6.1. Application and Current trends of Quantum Computing 21
6.2. Decoherence: A major barrier in the path of Quantum Computing 23
7. Bibliography 24

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1. Introduction
Quantum computation and quantum information are of great current interest in Computer Science,
Mathematics, Physical Sciences and Engineering. They will likely lead to a new wave of technological
innovations in communication, computation and cryptography. Information can be identified as the
most general thing which must propagate from a cause to an effect. It therefore has a fundamentally
important role in the science of physics. The theory of quantum information and computing puts this
significance on a firm footing, and has led to some profound and exciting new insights into the natural
world. Among these are the use of quantum states to permit the secure transmission of classical
information (quantum cryptography), the use of quantum entanglement to permit reliable transmission
of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of
irreversible noise processes (quantum error correction), and the use of controlled quantum evolution
for efficient computation (quantum computation). The common theme of all these insights is the use of
quantum entanglement as a computational resource.
Unlike classical computers using transistors to crunch the ones and zeroes individually, quantum
computers can handle both one and zero simultaneously via what are known as superposition quantum
states. A superposition state is a state of matter which we may think of as both one and zero at the
same time. Quantum computers use the strange superposition states and quantum entanglements to
do the trick of performing simultaneous calculations and extracting the calculated results. The spooky
phenomena of quantum entanglement and superposition are the key that enables quantum computers
to be superfast and vastly outperform classical computers.
Analogous to the fundamental concept of bit in classical computation and classical information, we have
its counterpart, quantum bit, in quantum computation and quantum information. Quantum bit is called
qubit for short. Just like a classical bit with state either 0 or 1, a qubit has states |0> and |1>. However,
the real difference between a bit and a qubit is that besides states |0> and |1>, a qubit may take the
superposition states,
Where are complex numbers and called amplitudes satisfying =1. That is, the
states of a qubit are unit vectors in a two-dimensional complex vector space, and states |0> and |1>
consists of an orthonormal basis for the space and are often referred to as computational basis states.
As a classical computer is built from an electrical circuit consisting of wires for carrying information
around the circuit and logic gates for performing simple computational tasks, a quantum computer can
be created from a quantum circuit with quantum gates to perform quantum computation and
manipulate quantum information. A number of physical systems are being investigated for building
quantum computers. These include optical photon, optical cavity quantum electrodynamics, ion traps
and nuclear magnetic resonance with molecules, quantum dots, and superconductors.

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1.3 Bloch Sphere
In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of two-
level quantum mechanical system (Qubit).
The Bloch sphere is a unit 3D-sphere, with each pair of antipodal points corresponding to mutually
orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to
correspond to the standard basis vectors |0> and |1>, respectively which in turn might correspond e.g.
to the spin-up and spin-down state of an electron. Any point | > on the Bloch sphere is represented by
| > = |0> + |1>
1.4 Quantum Gates
“i ple u ita ope atio s o u its a e alled ua tu logi gates . For example, if a qubit evolves
as| >→| >, | >→exp (i t) | >, the afte ti e t e a sa that the ope atio , o gate has been
applied to the qubit, where θ = t. This a also e itte P (θ) =|0><0|+exp (iθ) |1><1|. Here are
some elementary quantum gates:
1.4.1 Single Qubit Quantum Gate:
Hada a d Gate: Ca e ie ed as a efle tio a ou d /8 i the eal pla e. I the o ple
plane it is actually a - otatio a out the /8 a is. The Hadamard Gate is one of the most
important gates.

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Rotation Gate: This rotates the plane by θ.
NOT Gate: This flips a bit from 0 to 1 and vice versa.
Phase Flip:
The phase flip is a NOT gate acting in the|+> = | >+| > , |−> = (|0>−|1>) basis. Indeed,
)|+> = |−> and )|−> = |+>.
1.4.2. Two Qubit Quantum Gate: A very basic two qubit gate is the controlled-not gate or the CNOT.
The fi st it of a CNOT gate is alled the o t ol it, a d the se o d the ta get it. This is e ause i
the standard basis) the control bit does not change, while the target bit flips if and only if the control bit
is 1.
The CNOT gate is usually drawn as follows, with the control bit on top and the target bit on the bottom:

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1.5 Quantum Algorithms
They are described by quantum circuits that take input qubits and yield output measurements for the
solutions of the given problems. As a classical algorithm is a step-by-step problem solving procedure,
with each step performed on a classical computer, a quantum algorithm is a step by step procedure to
solve a problem, with each step executed by a quantum computer. Although all classical algorithms can
also be carried out on a quantum computer, we refer to quantum algorithms as the algorithms that
utilize essential quantum features such as quantum superposition and quantum entanglement.
1.6 Quantum Parallelism and Entanglement
In general, while a string of n classical bits can be in any of 2n
Boolean configurations (x = 000 . . . 0
through x = 111 . . . 1), a string of n qubits can exist in any state of the form:
Where are complex numbers such that . The quantum superposition principle clearly
emerges from the above equation: while n classical bits can store only a single integer x, a n-qubit state
can be prepared in any superposition of the 2n
quantum states |x>. The superposition principle opens
up new possibilities for computation: when performing a classical computation, different inputs require
separate runs. In contrast, exploiting this huge parallelism, a quantum computer can perform a
computation for exponentially many inputs on a single run.
The other quantum feature at the basis of the quantum computational speedup is entanglement.
Entanglement quantifies genuine quantum correlations that can be established between two or more
subsystems of a quantum system, and it is definitely one of the most distinctive properties of quantum
mechanics, lying at the basis of quantum information and quantum computation theory. It is the key
ingredient of quantum dense coding, quantum teleportation and of many quantum key distribution
protocols; moreover, the efficiency of quantum algorithms which admit computational time and
memory speedup with respect to their classical analogs is related to it. Entanglement is the most
spectacular and counterintuitive manifestation of quantum mechanics that is observed in composite
quantum systems: it signifies the existence of non-local correlations between measurements performed
on well-separated particles. After two classical systems have interacted, they are in well-defined
individual states; in contrast, after two quantum particles have interacted, they can no longer be
described independently of each other. The prototypes of two qubit entangled states are the so called
Bell states:

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They are characterized by the fact that the reduced density matrix of each of the two particles is a
multiple of the identity:
This means that any local measurement performed by two persons A or B cannot give any information
on the global state of the system: it completely resides in non-local correlations between the two
parties.
1.8 Quantum state transition
Quantum communication protocols generally require the ability to reliably transfer a quantum state
between distant parties. F or example, quantum teleportation and quantum dense coding require a pair
of particles in a maximally entangled state, such as one of the four Bell states, to be shared between the
two communicating parties A and B, which will be hereafter referenced to as Alice and Bob. In a second
stage, this shared entanglement is used to achieve, respectively, transmission of a qubit with two
classical bits or transmission of two classical bits with a qubit. Transferring a quantum state from one
qubit to another is also required in scalable quantum computing based on the quantum network, in
order to link several small quantum processors for large-scale computation.
Quantum Teleportation is a process by which we can transfer the state of a qubit from one location to
another, without transmitting it through the intervening space. We illustrate the phenomenon as
follows. Alice and Bob together generated a Bell state long ago. Each took one qubit of the Bell state
when they split. Now they are far away from each other. The mission for Alice is to deliver a qubit | >
to Bob, while he is hiding, and she can only send classical information to Bob but does not know the
state of the qubit | >. Quantum teleportation is a way that Alice utilizes the entangled Bell state to
send a qubit of unknown state to Bob, with only a small overhead of classical communication. A few
important remarks about quantum teleportation are in the line.
 First, quantum telepo tatio does ot i ol e a t a sfe of atte o e e g . Ali e s pa ti le
has not been physically moved to Bob; only its state has been transferred.
 “e o d, afte the telepo tatio Bo s u it ill e o the telepo ted state, hile Ali e s u it
will become some undefined part of an entangled state. In other words, what the teleportation
does is that a qubit was destroyed in one place but instantaneously resurrected in another.

10. 10
Teleportation does not copy any qubits, and hence is consistent with the no-cloning theorem
(which forbids the creation of identical copies of an arbitrary unknown quantum state).
 Third, in order to teleportate a qubit, Alice has to inform Bob of her measurement by sending
him two classical bits of information. These two classical bits do not carry complete information
about the qubit being teleported. If the two bits are intercepted by an eavesdropper, he or she
may know exactly what Bob needs to do in order to recover the desired state. However, this
information is useless if the ea esd oppe a ot i te a t ith the e ta gled pa ti le i Bo s
possession. Also the requirement of sending two bits of information via classical channel
prevents quantum teleportation from transmitting information faster than the speed of light.
In view of the great potentialities of solid-state quantum computing, attention is recently focusing on
the possibility to realize quantum channels by using condensed-matter systems. In particular, spin
chains are being considered as coherent data buses: the quantum transmission of state is achieved
between two spins at the two ends of the chain.

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2. Spin Chain as Quantum Channel
Recently, the interest in quantum communication channels has experienced a rapid increase, also in
view of their use as a bus between registers and processor within a quantum computer. Despite the fact
that the majority of protocols for quantum communication rely on photons, because of their weak
interaction with the environment, and of the well developed optical fiber technology, it is not always
possible to use photons when one needs a frequent exchange of information between distant qubits.
For example, in the case we have mentioned, the communication between different parts of a quantum
computer would require a continuous conversion of stationary qubits (i.e. the information stored in the
components of the quantum computer) into flying qubits (i.e. photons), in order to transmit
information. This procedure leads to several interfacing problems between the two different kinds of
physical systems that could be avoided by using, as a quantum channel, the same kind of physical
system that is used for realizing the quantum computer.
Indeed, a seminal paper by Bose [S. Bose, Phys. Rev. Lett. 91, 207901 (2003)], suggested to use spin-
chains as a quantum channel for short or mid-range communication, showing that, by means of the
magnetic interaction between the spins composing the chain, the information is transferred by only
letting the system evolve dynamically, without the requirement of any external control. After the first
proposal, there has been a spread of works investigating the dynamical behavior of spin chains and
possible optimization techniques for enhancing its performances as a quantum channel.
Assume that the sende Ali e is lose to the fi st spi the se de spi , n= 1) at one end of the chain,
while the receiver Bob is located close to the last spin (the e ei e spi , n = N) at the opposite end of
the chain; all the other spins in between act as channel spins. In order to transfer an unknown state to
Bob, Alice replaces the existing sender spin with a spin encoding the state to be transferred. After
waiting for a specific amount of time, the unknown state placed by Alice travels to the receiver spin with
some fidelity; then Bob picks up the receiver spin to obtain a state close to the one Alice wanted to
send.
Figure above shows the quantum communication through a one-dimensional spin chain:
Alice and Bob are at the two opposite ends. Alice places her quantum state she wants to
communicate on the leftmost spin; then, after time t, Bob receives this state with some
fidelity on the rightmost spin.

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3. Entanglement transfer in Spin Chain under
external magnetic field
Considering an anti-ferromagnetic isotropic Heisenberg XXX chain composed of N spins 1/2, described
by the Hamiltonian:
Where, are the Pauli matrices of the nth spin. For two spin ½ particles:
Where B ≥ is est i ted, a d the ag eti field o the two spins have been so parameterized that b
controls the degree of deviation from homogeneity (non uniformity). J is the coupling coefficient. J>0
corresponds to the anti-ferromagnetic case and J<0 corresponds to the ferromagnetic case.
Henceforth this project work will be based on two particle system (bipartite system).
In the standard basis {|0,0>,|0,1>,|1,0>,|1,1>}, the Hamiltonian can be expressed as,
A density matrix or density operator describes a quantum system in a mixed state, a statistical
ensemble of several quantum states. This should be contrasted with a single state vector that describes
a quantum system in a pure state. The density matrix is the quantum-mechanical analogue to a phase-
space probability measure (probability distribution of position and momentum) in classical statistical
mechanics. More precisely, suppose a quantum system is in one of a number of states | , where i is
an index, with respective probabilities . We shall call { | } an ensemble of pure states. The
density operator for the system is defined by the equation

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A quantum system whose state | is known exactly is said to be in a pure state. For pure state the
density operator is simply
Otherwise, ρ is in a mixed state; it is said to be a mixture of the different pure states in the ensemble for
ρ. A pure state satisfies trace ( ) =1 while a mixed state satisfies trace ( ) < 1.
The thermodynamic entropy is a key concept in Statistical Mechanics, and measures disorder and
randomness in a macroscopic system. The thermodynamic entropy ST for a quantum system in thermal
equilibrium at temperature T is defined as
Where,
is (thermal density) matrix, of the Gibbs ensemble at temperature T, and is the
Gibbs partition function. At zero temperature, the system is in its ground state (with an at most finite
degeneracy) and in this limit the thermodynamic entropy vanishes, We have considered thermal
density matrix for bipartite system for our purpose.
However, one problem with density matrix formulation is that they quickly become very large. Say, for
instance, each single-particle state vector is expressed using a 1xk dimensional vector. If we then have N
particles, each state must be expressed as a kN
vector: they scale exponentially with the number of
particles. Therefore it is needed to make simplifying approximations or to find a way to deal with the
system in terms of its components. One way to break the system into smaller chunks is to use Reduced
Density Matrices. Reduced density matrix is a descriptive tool for sub-systems of a composite quantum
system. Suppose we have physical systems A and B, whose state is described by a density operator AB
.
The reduced density matrix for system A is defined by:
A = trB ( AB )
where trB is a map of operators known as the partial trace over system B. The partial trace is defined by
trB(|a1><a2|⊗|b1><b2|) = |a1><a2|tr(|b1><b2|)
where|a1> and |a2> are any two vectors in the state space of A and |b1> and |b2> are any two vectors
in the state space of B. Similarly for system B reduced density matrix is :
B = trA ( AB )

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Any two-qubit entanglement measure depends on the reduced density matrix, AB of the marked pair of
qubits A and B. The pair wise concurrence is a measure that quantifies the mutual entanglement of the
marked pair of qubits. It carries the information of entanglement sharing between qubit A and any other
qubit. If the concurrence between A and B is one, would imply neither of these two qubits is entangled
with the rest of the system but if the concurrence is less than one that would imply qubit A is sharing
entanglement with qubit B and the rest of the system too.
The concurrence measure is given as
,
where are the eigenvalues in decreasing order of the matrix , where is the time-
reversed matrix,
The above concurrence measure captures the essential information of mutual entanglement of the
marked pair of qubits. The o u e e is a aila le, o atte hethe is pu e o i ed.
The measure that can be used to determine how close one state is to another is based on the notion of
the amount of statistical overlap between two distributions, called the Fidelity. To determine the quality
of transfer state, fidelity is defined as
The values of fidelity range from zero to one. When F=0, information is completely distorted in the
transmission process, and when F=1 the final state is identical to the initial state, thus denoting the ideal
communication transmission process. In common situation, 0<F <1, information is distorted in some
extent after being transmitted. For quantum communication, F can be larger than 2/3, which is the
maximum of classical communication, so that quantum communication can outperform classical
communication. In this study, we only investigate the fidelity concept. Having input and output density
matrix ( A , B ) and using the above equation we determine the fidelity.
We have studied both the fidelity and concurrence for bipartite system.

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4. Result and Discussion
Fidelity vs. Time Graph:
FIG.1
Fig.1 shows the fidelity vs. time curve for M=0.5.

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Fidelity vs. Time Graph:
FIG.2
Fig .2 shows the fidelity vs. time curve for M=1.5.
It is found that F for different b behaves similarly but has different amplitudes and periods with
respect to T.As M increases the minimum fidelity goes below 2/3. Which is inferior to classical
communication.
Thus we can conclude from the above graphs that smaller the non uniform magnetic field, the
more efficient is the information transmission.

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Fidelity vs. non uniform magnetic field Graph:
FIG.3
Fig.3 shows the fidelity vs. non uniform magnetic field curve for t=5

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Fidelity vs. non uniform magnetic field Graph:
FIG.4
Fig.4 shows the fidelity vs. non uniform magnetic field curve for t=15.
From the graph we can observe that when magnetic field is less than 0.5 then F >2/3. Which is
desired to exploit the power of quantum communication.

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Concurrence vs. Temp vs. Non uniform Magnetic Field Graph:
FIG.5
Fig.5 shows the Concurrence vs. Temperature vs. magnetic field curve where t is varied from
0.2k to 0.5k and b is varied from 0.5tesla to 1.1tesla
From the graph we can observe that when temperature is increasing the concurrence or mutual
entanglement is decreasing. Concurrence also decreases for increasing external applied magnetic
field.

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5. Further planning
We have studied the entanglement based transport for bipartite system. In future we will like
to investigate thermal entanglement on multipartite system (with GHZ state and W state)
under influence of external magnetic field.

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6. Appendix
6.1 Application and Current trends of Quantum Computing
With the global forces of computer competition, encryption technology for national security, new
applications, and the thermodynamics of computer systems changing as they are, there is a rush
toward the new quantum technology to produce the first viable quantum computer. The world is
moving toward a place that no classical computer has gone before, nor can go. Quantum Computation
has a wide range of applications like:
 Quantum Communication: Quantum communication is the art of transferring a quantum state
from one location to another. It allows a sender and receiver to agree on a code without ever
having to meet in person. From an application point of view the major interest is Quantum Key
Distribution (QKD), as this offers for the first time a provably secure way to establish a
confidential key between distant partners. This key is then first tested and, if the test succeeds,
used in standard cryptographic applications. The uncertainty principle, an inescapable
property of the quantum world, ensures that if an eavesdropper tries to monitor the signal
in transit it will be disturbed in such a way that the sender and receiver are alerted.
 Quantum Cryptography: Quantum cryptography was discovered independently in US and
Europe. The American approach, was based on coding in non-commuting observables,
whereas the European approach was based on correlations due to quantum entanglement.
This has the potential to solve a long-standing and central security issue in our information
based society. Ironically the same technology also poses current cryptography techniques a
world of problems. The implications of Shor's factoring algorithm on the world of cryptography
are staggering. The ability to break the RSA coding system will render almost all current
channels of communication insecure.
 Artificial Intelligence: The theories of quantum computation have some interesting implications
in the world of artificial intelligence. The debate about whether a computer will ever be able to
be truly artificially intelligent has been going on for years and has been largely based on
philosophical arguments. Those against the notion suggest that the human mind does things
that aren't, even in principle, possible to perform on a Turing machine. The theory of quantum
computation allows us to look at the question of consciousness from a slightly different
perspective. The first thing to note is that every physical object, from a rock to the universe
as a whole, can be regarded as a quantum computer and that any detectable physical
process can be considered a computation. Under these criteria, the brain can be regarded as
a computer and consciousness as a computation. The next stage of the argument is based in
the Church-Turing principle and states that since every computer is functionally equivalent and

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that any given computer can simulate any other, therefore, it must be possible to simulate
conscious rational thought using a quantum computer. Ultimately this suggests that
computers will be capable of simulating conscious rational thought. These theories provoke
a minefield of philosophical debate, but maybe the quantum computer will be the key to
achieving true artificial intelligence.
 Encryption Technology: The speed of quantum computers also jeopardizes the encryption
schemes that rely on impracticably-long times to decrypt by brute force methods. Encryption
schemes that may take millions of years to guess and check are now vulnerable to quantum
computers that may reach a solution within one year.
 Ultra-secure and Super-dense Communications: It is possible to transmit information without a
signal path by using a newly-discovered quantum principle, quantum teleportation. There is no
way to intercept the path and extract information. Ultra-secure communication is also possible
by super-dense information coding, which is a new technology in its own right. Quantum bits
can be used to allow more information to be communicated per bit than the same number of
classical bits.
 Improved Error Correction and Error Detection: Through similar processes that support
ultra-secure and super-dense communications, the existing bit streams can be made more
robust and secure by improvements in error correction and detection. Recovering
informational from a noisy transmission path will also be a lucrative and useful practice.
 Molecular Simulations: Richard Feynman showed in 1982 that classical computers cannot
simulate quantum effects without slowing down exponentially; a quantum computer can
simulate physical processes of quantum effects in real time. Molecular simulations of
chemical interactions will allow chemists and pharmacists to learn more about how their
products interaction with each other, and with biological processes such as how a drug may
i te a t ith a pe so s eta olis o disease. Pharmaceutical research offers a big market to
such applications.
 True Randomness: Classical computers do not have the ability to generate true random
u e s. The a do u e ge e ato s o toda s o pute s a e pseudo-random
generators—there is always a cycle or a trend, however subtle. Pseudo-random generators
cannot simulate natural random processes accurately for some applications, and cannot
reproduce certain random effects. Quantum computers can generate true randomness, thus
give more veracity to programs that need true randomness in their processing. Randomness
plays a significant part of applications with a heavy reliance on statistical approaches, for
simulations, for code making, randomized algorithms for problems solving, and for stock market
predictions, to name a few.

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6.2 Decoherence: A major barrier in the path of Quantum Computing
In quantum mechanics, quantum Decoherence is the loss of coherence or ordering of the phase angles
between the components of a system in a quantum superposition. One consequence of this de-phasing
is classical or probabilistically additive behavior. Quantum Decoherence gives the appearance of wave
function collapse (the reduction of the physical possibilities into a single possibility as seen by an
observer) and justifies the framework and intuition of classical physics as an acceptable approximation:
Decoherence is the mechanism by which the classical limit emerges from a quantum starting point and it
determines the location of the quantum-classical boundary. Decoherence occurs when a system
interacts with its environment in a thermodynamically irreversible way. This prevents different elements
in the quantum superposition of the total system's wave function from interfering with each other.
Decoherence can be viewed as the loss of information from a system into the environment (often
modeled as a heat bath), since every system is loosely coupled with the energetic state of its
surroundings. Viewed in isolation, the system's dynamics are non-unitary (although the combined
system plus environment evolves in a unitary fashion). Thus the dynamics of the system alone are
irreversible. As with any coupling, entanglements are generated between the system and environment.
These have the effect of sharing quantum information with—or transferring it to—the surroundings.
Decoherence represents a challenge for the practical realization of quantum computers, since such
machines are expected to rely heavily on the undisturbed evolution of quantum coherences. Simply put,
they require that coherent states be preserved and that Decoherence is managed, in order to actually
perform quantum computation.

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7. Bibliography
 https://www.wikipedia.org/
 www.cs.berkeley.edu/~vazirani/f04quantum/quantum.html - Lectures of Umesh
Vazirani
 http://theory.caltech.edu/people/preskill/ph229/ - Lectures of John Preskill
 Quantum Computation and Quantum Information by Michael Nielsen and Isaac Chuang
 Consistent Quantum Mechanics by Robert B. Griffith
 Quantum Computer Science: An Introduction by N. David Mermin
 Quantum Computing Explained by David McMahon
 REDUCED-DENSITY MATRIX MECHANICS: WITH APPLICATION TO MANY-ELECTRON
ATOMS AND MOLECULES. ADVANCES IN CHEMICAL PHYSICS. VOLUME 134. Edited by
DAVID A. MAZZIOTTI
 An introduction to the spectrum, symmetries, and dynamics of spin-1/2 Heisenberg
chains by Kira Joel, Davida Kollmar, and Lea F. Santos
 Quantum Communication Through Spin Chain Dynamics: An Introductory Overview by
Sougato Bose
 Entanglement of the Heisenberg chain with the next-nearest-neighbor interaction Shi-
Jian Gu,Haibin Li,You-Quan Li and Hai-Qing Lin
 Enhanced-teleportation through anisotropic Heisenberg quantum spin chains by Xiang
Hao and Shiqun Zhu
 Transport of entanglement through a Heisenberg–XY spin chain by V. Subrahmanyam ,
Arul Lakshminarayan
 Quantum state transfer via a two-qubit Heisenberg XXZ spin model by Jia Liu, Guo-Feng
Zhang, Zi-Yu Chen
 Long-range interactions and information transfer in spin chains by Rebecca Ronke, Tim
Spiller , Ire e D’A ico

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 Optimized quantum state transfer through an XY spin chain by Yang Liu and D. L. Zhou
 Fidelity spectrum and phase transitions of quantum systems by P. D. Sacramento,
N.Pau ković, V. R. Vieira
 Entanglement and Dynamics of Spin Chains in Periodically Pulsed Magnetic
Fields:Accelerator Modes by T. Boness, S. Bose, and T. S. Monteiro
 Thermal Entanglement and Quantum State Teleportation: Mohsen Balvasi, Enayatollah
Taghavi moghadam, Farshad Baharvand