Transcript QUANTUM ENTANGLEMENT AND IMPLICATIONS IN INFORMATION PROCESSING: Quantum TELEPORTATION K. Mangala Sunder

QUANTUM ENTANGLEMENT
AND IMPLICATIONS IN INFORMATION PROCESSING:
Quantum TELEPORTATION
K. Mangala Sunder
Department of Chemistry
IIT Madras
Contents
1. Introduction
2. Bits / Qubits/ Quantum Gates
3. Entanglement
4. Teleportation / Teleportation through gates
/ Experimental Realization
5. Application of teleportation
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
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
Toc H Institute of Science
and Technology, Kerala
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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
Toc H Institute of Science
and Technology, Kerala
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Introduction
• Also by Moore’s 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
Toc H Institute of Science
and Technology, Kerala
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Bits / Qubits
• bit is a fundamental unit of classical
computation and classical information
• only possible values for a classical bit
are 0 and 1
• quantum analogue of classical bit :
quantum bit or qubit
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and Technology, Kerala
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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.
  a  b 
Bits / Qubits
a matrix representation for
the states.
1
where  
0
 
0
  
1
and
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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and Technology, Kerala
9
a
  
b
| a |  | b | 1
2
2
where a and b are complex numbers such that
• state of a qubit is a vector of unit length in
a two dimensional complex vector space
• 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
  cos

2
i
  e sin

2

The result above can be derived from simple
quantum mechanics of spins
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and Technology, Kerala
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where  and  represent a point on the unit
three dimension sphere known as Bloch sphere

z


x

y

• 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
Multiple Qubits
• two classical bits can take four different possible values
00, 01, 10, 11
• two qubit system has four computational basis states
 ,  ,  , 
• thus the state can be represented as

12
such that
 a 
12
+b 
12
+c 
12
a  b  c  d 1
2
2
2
2
 d 
12
• Bell States : play an important role in information processing schemes


12


12
 
12
and
2
Bits
12


12
 
12
2
qubits
a  b 
0,1
a 
00,01,10,11
000,001,010,100,
011,101,110,111



123
12
+b 
12
+c 
12
 d 
12
 x0   x1   x2   x3 

  x4   x5   x6   x7 


123
This means that quantum computer can, in only one computational step,
perform the same mathematical operation on 2 n different input numbers
encoded in coherent superposition of n qubits
• what distinguishes classical and quantum computing is how the information is
encoded and manipulated
• quantum parallelism
a1 
 a2 
F
a1F 
b1 
 a2 F 
b2 
 a3 
 a3 F 
 a4 
 a4 F 
=
b3 
b4 
• conventional gates are irreversible in operation. Quantum gates
are reversible. (will elaborate later)
• existing real world computers dissipate energy as they run
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
X
• it acts on the state of the qubit to interchange the role of computational
basis state
  a  b 
X
a  b 
• in matrix form
Z Gate :
0 1
X 

1
0


Z
• it leaves  unchanged but flips the sign of 
  a  b 
Z
a  b 
• in matrix form
1 0 
Z 

0

1


• Hadamard Gate
• turns  and  halfway between 
 
 
• in matrix form
and
2
H
and

 
 
2
1 1 1 
H
1 1
2

• its operation is just a rotation of 900 about the y-axis followed by a reflection
through x-y plane
• can all 2 dimensional matrices be appropriate for quantum gates for single
qubits?
Think about it.
• matrix U representing the single qubit gate must be unitary
U U  I
Controlled Not gate
• two qubit gate : consists of two input qubits known as controlled qubit and
target qubit
A B A’ B’
A
B
control wire
A'
B'
0
0
0
0
0
1
0
1
1
0
1
1
1
1 1 0
target wire
• in matrix form
U CN
1
0

0

0
0
1
0
0
0
0
0
1
0
0

1

0
• operation will be reversed by merely repeating the gate
A
A
B
B'  B
B’ = A XOR B
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


A

B
AB


A
separable state
B
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
• responsible for the exponential nature of quantum parallelism
singlet state


12


12
 
12
2
• 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.

2. 
1
1
with probability ½ and post measurement state 
with probability ½ and post measurement state
12

12
• any subsequent measurement on second sub system will yield 
in former case and in  2 latter case respectively
E. Rieffel et al, ACM Computing Surveys, 32 (2000) ,300-335
2
Quantum network to prepare two and three particle entangled states
 ,
H
Bell states
 ,
 ,
H
 ,
GHZ states
 ,
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
• 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
1
  12   12 
 12 H gate
2
1
1
 
2
C-NOT 12
Play a flash movie here for one of them)
input
12
 
output
• in a similar way all the four Bell states can be visualized


12
12
1
 
2
1
 
2
12
12
 
 
12
12



12
1
 
2
12
 
12

12


12
1

1 0
2 0

0
1 1
    
 0 1  0  2
0
1
0
0
0
0
0
1
0  1 
0  0 
 
1  1 
 
0  0 12
1
 
1 0
2 0
 
 1 12
H Gate
on 1st
1  1 1   1   1  
1 1  0    0  
2
  1   2 
C-NOT
on 1-2
 1   0  
    
1  0   0  

2  0   0  
    
 0 1  1  2 
1  1   1  
     
2  11  0  2 
1
 
2
12
 
12

Entangled state analysis
• photon 1 is in input mode e and photon 2 is in input mode f
state of photon 1
a  1 b 
1
state of photon 2
c
2
Photon 1
e
Photon 2
f
2
d 
g
h
• each photon has the same probability to get either reflected or transmitted
by beam splitter
• four different possibilities are:
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
 1  a  1 b 
1
unknown state
Alice (sender)
Bob (receiver)
• Alice wants to communicate enough information about 
1
to Bob
• how to do this?
• quantum systems cannot be fully determined by measurements
3

23
2
1
 

2
23
 
23

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 Alice’s measurement is

123

a
 
2
1

2
 3
1

2
 3  
b
 
2
1

2
 3 
1

2
 3 
• measurement basis


12

1
 
2
 
12


12 
• expressing each direct product

123
 
1
 
2   

1
2


12
1
 
2
12
 
12

in Bell operator basis
 a   b      a   b  
 b   a      b   a   

12
3
3
3
12
3

12
3
3
12
3
3
• 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 

12
Bob has to do nothing
• in all other cases Bob has to use appropriate unitary transformation



12
 -a 
Z




b
 3
 -b 
Y

 


12 a
 


12
12

12
b
X
 
 a  
 
 1 0   -a 




 0 -1  b 

  3
 -a 
12  b 
 3
 0 -i   -b 
 
 i 0   a  

  
 0 1  b 

 



12 1
0  a 


 -ia 
 i 
12  ib 


a
12  b 
 

a
12  b 
 
Teleportation through gates
• Quantum circuit for teleporting a qubit

1

M1
H

M2


23
X M2
Z M1

here  1 is the unknown state to be teleported and M is probabilistic
classical bit


23
1
 

2
G. Brassard et al, Physics D, 120 (1998), 43-47
23
 
23

1
• the state input into the circuit is
0 
1 
a

2
1
 
23
 
23
  b   
1
23
 
23

where first two qubits belongs to Alice and third qubit belongs to Bob
• Alice sends her qubits through a C-NOT gate, obtaining
1 
1 
a

2
1
 
23
 
23
  b   
1
23
 
23

• she then sends the first qubit through the Hadamard gate, obtaining
 2   a   1  
2
1
i.e.
2
1
  
23
 
1   12  a   b  3  
 
2     a   b  
12
3

23
12
  b 
a 
1
 
b 

3
1
  
 
12
23
 
a 
23

b 

3



• depending on Alice’s measurements Bob’s 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 Bob’s qubit
will be in state 
which is identical to input state with Alice
3
thus

 a  b 
,

 a  b 

 a  b 
,


a  b 
• knowing the measurement outcome Bob can fix up his state by the
application of appropriate gate operation as
  a   b 
Z

 a  b 
  a   b 
  a   b 
X

 a  b 
Z
X

a   b 
C-NOT on qubit 12 with 3 remaining unchanged:
The unitary transformation is
1
0

0

0
0
1
0
0
1
0

0
0 0

0 0   1 0   0



0 1 0 1 0


1 0
0
0

0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
Use this in the next two transparencies!!
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0 
0

0
0

1
0

0
• the state input into the circuit is
0 
0
1 
a

2
 
12
1
1
 
 
a 1 0
b 0 0

0  0 
1  0
2  1
2  1
 
 
 1  23
 1  23
H gate on 1
1
 0
0
 0
 
 
0
 0
 
 
a 1
b  0



0
2
2  0
 
 
0
 
1
0
1
 
 
0
 123
 0 123
 
12
C-NOT
1-2
  b   
1
0

0

0
0

0
0

0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
12
0
0
0
0
0
0
1
0
 
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
12


0
1
 0 




 0 
0
0

 
 

0
 0 
0


 
 
0
1
a
b
 
 0 

 2 0
0
2  1 


 
 
1
0

 
 0 

0
 0 
0


 
  
0 123 
0
 
 1  123
1
0

0

1 0
2 1

0
0

0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0 1 0 0 0
1
0
0
0
0 0 1 0 0 
 
 
0
0
0 0 0 1 0

 
 
1 0 0 0 1
a 1
b 0


0 1 0 0 0 
2 0
2 0

 
 
0 0 1 0 0 
0
1
0
1
0 0 0 1 0 

 
 
1 0 0 0 1123
 0 123
 0 123
1
0
0
1
 
 
0
1
 
 
a 1 b  0 

2 1 2  0 
 
 
0
 1 
0
 1 
 
 
1
 
0
1
0
 0
 0
0
 1
 0
 1
 0
0  1
1  0
0  0
1 
   a    b       a    b       a    b       a    b   
 
 
 
 
 
 
 
 
0
 2  0 3 2  1 3   0 
 2  1 3 2  0 3   1 
 2  0 3 2  1 3   0 
 2  1 3 2  0 3 
 
 
 
 
 0 12
 0 12
 0 12
 1 12
2
1    a   b      a   b 
 
2     a   b  

    a 
 b  


Experimental Quantum Teleportation
Pictorial representation of original scheme
Click
D. Bouwmeester et al, Nature 390 (1997), 575
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
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
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.
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.
Toc H Institute of Science
and Technology, Kerala
44