QUANTUM ENTANGLEMENT

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QUANTUM ENTANGLEMENT AND IMPLICATIONS IN
INFORMATION PROCESSING Quantum TELEPORTATION
K. Mangala Sunder Department of Chemistry IIT
Madras
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Contents
1. Introduction
2. Bits / Qubits/ Quantum Gates
3. Entanglement
4. Teleportation / Teleportation through gates /
Experimental Realization
5. Application of teleportation
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Introduction

• What are quantum computation and quantum
• information processing?

• They are concerned with computations and the
  study of the information processing tasks that
  can be accomplished using quantum mechanical
  systems

M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information, Cambridge
Univ. Press
V. Vedral et al, Prog. Quantum Electron 22
(1998), 1-39
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Introduction

• One must understand the difference
• between classical computation and
• quantum computation.

M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information, Cambridge
Univ. Press
V. Vedral et al, Prog. Quantum Electron 22
(1998), 1-39
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Introduction

• conventional computer can do anything a
• quantum computer is capable of.

• However, quantum computation offers an
• enormous advantage over classical
• computation in terms of the available data
• that a computer can handle.

M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information, Cambridge
Univ. Press
V. Vedral et al, Prog. Quantum Electron 22
(1998), 1-39
6
Introduction

• Also by Moores Law quantum effects will
• show up in the functioning of electronic
• devices as they are made smaller and
• smaller

M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information, Cambridge
Univ. Press
V. Vedral et al, Prog. Quantum Electron 22
(1998), 1-39
7
Bits / Qubits

• bit is a fundamental unit of classical
• computation and classical information

• only possible values for a classical bit
• are and

• quantum analogue of classical bit
• quantum bit or qubit

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Bits / Qubits

• qubits are correctly described by two quantum
• states can be a linear combination of
  
• and . The two states here are the only
  
• possible outcomes when you measure the
• state of the qubit.

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a matrix representation for the states.
Bits / Qubits
Remember that there are other quantum
states where the number of outcomes can be one
of many rather than one of two. Those are
not considered here.
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where and are complex numbers such
that

• state of a qubit is a vector of unit length in
• a two dimensional complex vector space

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• state of a qubit cannot be determined i.e.
• a and b cannot be determined from a single
• measurement.

• there is nothing one can experimentally do
• to them to reveal their states

• using the normalization condition

The result above can be derived from simple
quantum mechanics of spins
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where and represent a point on the
unit three dimension sphere known as Bloch sphere

• single qubit operation can be described within
• the Bloch sphere picture

M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information, Cambridge
Univ. Press
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Multiple Qubits

• two classical bits can take four different
  possible values

• two qubit system has four computational basis
  states

• thus the state can be represented as

such that
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Bits qubits
0,1
00,01,10,11
000,001,010,100, 011,101,110,111
This means that quantum computer can, in only one
computational step, perform the same
mathematical operation on different input
numbers encoded in coherent superposition of n
qubits
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• what distinguishes classical and quantum
  computing is how the information is
• encoded and manipulated

• quantum parallelism

F

• conventional gates are irreversible in
  operation. Quantum gates
• are reversible. (will elaborate later)

• existing real world computers dissipate energy
  as they run

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Quantum Gates

• classical computer electrical circuit
  containing wires and logic gates

Irreversible (except NOT Gate)

• quantum computer quantum circuit containing
  wires and elementary
• quantum gates

reversible

• quantum analogue of NOT Gate is X-Gate

• it acts on the state of the qubit to interchange
  the role of computational
• basis state

X
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• in matrix form

Z Gate
Z

• it leaves unchanged but flips the sign of

Z

• in matrix form

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• Hadamard Gate

H

• turns and halfway between
  and

and

• in matrix form

• its operation is just a rotation of about
  the y-axis followed by a reflection
• through x-y plane

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• can all 2 dimensional matrices be appropriate
  for quantum gates for single
• qubits? Think about it.

• matrix representing the single qubit gate
  must be unitary

Controlled Not gate

• two qubit gate consists of two input qubits
  known as controlled qubit and
• target qubit

A B A B
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
target wire
control wire
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• in matrix form

• operation will be reversed by merely repeating
  the gate

B A XOR B
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Entanglement

• entanglement is a quantum mechanical phenomena
  in which the quantum
• states of two or more particles have to be
  described collectively without
• being able to identify individual states

• it introduces the correlation between the
  particles such that measurement
• on one particle will affect the state of other

• algebraically if a composite state is not
  separable it is called as an
• entangled state

Non-separable for specific values of coeffs.

• entanglement is at the heart of the quantum
  computation and information
• processing

A. Peres, Phys. Rev. Lett. 77 (1996), 1413-15
M. Horodecki et al, Phys. Lett. A 223 (1996), 1-8
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• responsible for the exponential nature of
  quantum parallelism

singlet state

• neither of the subsystem in singlet state can be
  attributed by a pure state

• any measurement on subsystem one leads to two
  possibilities for the first qubit

1. with probability ½ and post
measurement state
2. with probability ½ and post
measurement state

• any subsequent measurement on second sub system
  will yield
• in former case and in latter case
  respectively

E. Rieffel et al, ACM Computing Surveys, 32
(2000) ,300-335
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Quantum network to prepare two and three particle
entangled states
H
Bell states
GHZ states
H
G. Brassard et al, Physics D, 120 (1998), 43-47
Quantum Teleportation and multiphoton
Entanglement, Thesis by J. W. Pan, Univ. of Sc.
And Tech., China
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• using single qubit operations and controlled not
  gates a suitable quantum
• network can be constructed to produce maximally
  entangled states

e.g. Bell states for two particle system
C-NOT 12
H gate
1
Play a flash movie here for one of them)
input
output

• in a similar way all the four Bell states can be
  visualized

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Entangled state analysis

• photon 1 is in input mode e and photon 2 is in
  input mode f

state of photon 1
state of photon 2
g
e
Photon 1
Photon 2
f
h

• each photon has the same probability to get
  either reflected or transmitted
• by beam splitter

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• four different possibilities are

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Teleportation

• branch of quantum information processing

• transmission of information and reconstruction
  of the quantum state
• of the system over arbitrary distances

• process in which an object disintegrates at one
  place and its perfect replica
• occurs at some other location no cloning

• techniques for moving things around, even in the
  absence of communication
• channels linking the sender of quantum state to
  the recipient

• process in which an object can be transported
  from one location to another
• remote location without transferring the medium
  containing the unknown
• information and without measuring the
  information content on either side of
• transport

C. H. Bennett et al, Phys. Rev. Lett. 70 (1993),
1895-99
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Bob (receiver)
Alice (sender)

• Alice wants to communicate enough information
  about to Bob

• how to do this?

• quantum systems cannot be fully determined by
  measurements

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2
3
Quantum carrier to be used
particle 2
particle 3

• to couple her particle 1 with EPR pair Alice
  performs Bell state measurement
• on her particles

• complete state of three particles before Alices
  measurement is

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• measurement basis

• expressing each direct product in
  Bell operator basis

• all the four outcomes are equally likely,
  occurring with equal probability 1/4

• having Alice tell Bob her measurement outcome,
  he can recover the
• unknown state

• If the outcome is Bob has to do nothing

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• in all other cases Bob has to use appropriate
  unitary transformation

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Teleportation through gates

• Quantum circuit for teleporting a qubit

here is the unknown state to be
teleported and M is probabilistic classical bit
G. Brassard et al, Physics D, 120 (1998), 43-47
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• the state input into the circuit is

where first two qubits belongs to Alice and third
qubit belongs to Bob

• Alice sends her qubits through a C-NOT gate,
  obtaining

• she then sends the first qubit through the
  Hadamard gate, obtaining

i.e.
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• depending on Alices measurements Bobs qubit
  will end up in one of
• these four equally likely possible states with
  probabilities 1/4

• if Alice performs a measurement and obtain a
  result then Bobs qubit
• will be in state which is identical
  to input state with Alice

thus

• knowing the measurement outcome Bob can fix up
  his state by the
• application of appropriate gate operation as

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C-NOT on qubit 12 with 3 remaining unchanged
The unitary transformation is
Use this in the next two transparencies!!
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• the state input into the circuit is

C-NOT
1-2
H gate on 1
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Experimental Quantum Teleportation
Pictorial representation of original scheme
Click
D. Bouwmeester et al, Nature 390 (1997), 575
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Pictorial representation of actual set-up
D. Bouwmeester et al, Nature 390 (1997), 575
P. G. Kwait et al, Phys. Rev. Lett. 75 (1995),
4337-4341
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Application

• quantum communication is centered on the ability
  to send data over large
• distances quickly

• teleportation has practical application in the
  field of quantum information
• processing that hold promises for making
  computing both much faster and
• secure

• it exploits the concept of quantum entanglement
  which is at the heart of
• quantum computing

• during the process the quantum state of an
  object will be destroyed and not
• the original object

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The most obvious practical application of
teleportation is in cryptography. It can provide
a completely secure communication between two
distant components. Sending photons entangled in
a quantum state makes it impossible for an
eavesdropper to intercept the message because
even if intercepted the message would be
unintelligible unless it was intended for a
specific recipient.
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I wish to thank Mr. Atul Kumar, my Ph. D. Scholar
for his enthusiasm to learn this and do further
research in this area.Thank you all.