Arun Kumar Pati

1
QUANTUM CRYPTOGRAPHY

• Arun Kumar Pati
• Institute of Physics
• Bhubaneswar-751005, Orissa
• and
• Bhabha Atomic Research Center, Mumbai, India
  

email akpati_at_iopb.res.in
2
QUANTUM THEORY
? Quantum theory is the most successful physical
theory of our time. This theory could explain
nearly all the physical phenomena of the everyday
world. ? The applications include transistors,
lasers, nanodevices, and many more. Quantum
theory explains behavior of electrons, nuclei,
atoms, molecules, solids, and even macroscopic
systems such as superconductors. ? Quantum
theory now enters the domain of information
theory. Unification of these two theories has
lead to a new area of research --- what we call
quantum information theory.
3
Quantum Mechanical Postulates

• State of an isolated system is described by a
  state vector ?? in a Hilbert space H, with
  dim(H) N. The state vector contains all the
  information about a quantum system.
• Evolution of a closed system is described by a
  unitary evolution ?? ? ?? U ??
  (Schrödinger equation).
• To every observable there is associated a linear
  Hermitian operator A. This satisfies an
  eigenvalue equation A n? a n?, n1,2N,
  where n? are the eignstates and a are the
  non-degenerate eigenvalues of the observable A.

n
n
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Postulates...

• If the system is in an arbitrary state ?? ? C
  n? and a measurement of A is being performed,
  then it will yield an eigenvalue a with a
  probability C .
  (i)
  If we know the measurement outcome, then the
  state makes a transition to the nth eigenstate,
  i.e, ?? ? n? . (ii) If we do not
  know the measurement outcome, then the state
  after measurement is described by a mixture,
  i.e., ?? ?? ? ?
  C n?? n .

n
n
2
n
n
2
n
n
This is von Neumanns collapse postulate, also
known as reduction of the state vector. We
need this, yet we do not know how and when does
this occur?
5
Postulates...

The Hilbert space of a composite quantum
system is the tensor product of the individual
Hilbert spaces of the physical systems. If we
have a composite system consisting of two
subsystems, then the total Hilbert space H H1 ?
H2 .
1
. When we have a composite system consisting of
two or more subsystems, then the state of the
composite object could be in an entangled state.
This is basically superposition of product basis
states that itself cannot be written as a direct
product of two or more pure states for individual
subsystems.
6
Quantum Bit (QUBIT)

• A classical bit can remain either in 0 or in 1,
  but a qubit can remain simultaneously in ?0 ? and
  ?1?.
• Take ?H ? ?0 ? and ?V ? ?1? as two distinct
  horizontal and vertical polarization states of a
  photon. Then by linear superposition principle
  any arbitrary state such as ???
  ? ?H ? ??V ? ? ?0? ??1 ? is also a logical
  state.
• Any arbitrary state of a two state system is a
  qubit. An electron spin state such as ??? ? ?? ?
  ??? ? is also a qubit.
• This inherent parallelism is responsible for
  quantum computation and host of other tasks.

7
Qubit on a Bloch sphere
?0?
z
???
?
x
?
y
?1?
??? ? ?0 ? ??1? cos ?/2 ?0 ? sin ?/2 exp(
i ? ) ?1 ?

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A qubit keeps its privacy

• If we know the complex amplitudes (for a qubit
  two real numbers), then the qubit is known.
  That is we know its exact location on the Bloch
  sphere. If we do not know them, then a qubit is
  in an unknown state.
• How many bits do we need to specify a qubit? We
  need infinite amount of bits! (As we need to
  specify two real numbers in bits.) A qubit
  contains both quantum and classical information.
  Even though it has large amount of information,
  we can extract only one bit by a measurement!
  Most of the information is hidden from us.

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Determining a qubit

• To determine the state of a qubit we need an
  ensemble of identically prepared quantum states
  (infinite number of them).
• If we have a finite N number of identical
  prepared qubits the fidelity of state
  determination is only (N1)/(N2).
• Given a single qubit we cannot determine its
  state completely.
• Can one make copies of a single qubit in an
  arbitrary state and then determine its state?

S. Masar and S. Popescu, PRL 74 (1995) 1259.
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DETERMINISM VS INDETERMINISM
? Quantum system suffers two kinds of changes
(i) Deterministic and (ii)
probabilistic. ? Unitary evolution (the
Schrödinger equation) is completely deterministic
(as deterministic as Newtons equation) and
reversible. ? Probabilstic element enters when
we perform a measurement on the quantum system.
Quantum theory does not tell you with certainty
what will be the outcome of an measurement in a
single run of the experiemnt. ? Measurement
process is not a unitary and hence irreversible.
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QUANTUM ENTANGLEMENT
(ANOTHER WIERDEST FEATURE)
? If a composite system consisting of two or
more subsystems is in a superposition of product
basis states that itself cannot be written as a
direct product of two or more pure states for
individual subsystems then it is an entangled
state, i.e., it cannot be written as ??? ???
????. ? This is an attribute of a composite
system where we cannot associate an individual
pure state to its subsystems. ? A pure
two-qubit entangled state can always be written
as ??? a ?0??0? b ?0??1?c ?1??0?d ?1?
?1?
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SCHRÖDINGER

• Schrödinger (1935) I would not call
  entanglement one but rather the characteristic
  trait of quantum mechanics, the one that enforces
  its entire departure from classical lines of
  thought.

13
ENTANGLMENT IS A RESOURCE

• Quantum mechanical entangled states are at the
  center stage of quantum information theory
  because they have many fundamental and practical
  applications.
• These include quantum computing, quantum
  cryptography, quantum dense coding, quantum
  teleportation, remote state preparation, remote
  state measurement, telecloning, quantum secret
  sharing and so on.
• Without quantum entanglement these tasks will be
  impossible or may require infinite amount of
  other resources.

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ENTANGLEMENT IS PHYSICAL
Information is physical R. Landauer.

• Storage and processing of entanglement requires
  physical systems and physical laws.
• Entanglement can be used to do informational
  work.
• Entanglement properties are independent of
  physical representations. For example one unit of
  entanglement (ebit) can be manifested in a
  variety of ways

1. Two-spin half particles 2. Photon
polarizations 3. Atomic states
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Quantum vs Classical

• Quantum information differs from classical
  information in many ways.
• Classical information can be copied and deleted
  but quantum information cannot be.
• Classical information can be read without
  disturbance but quantum information cannot be.

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Cloning of a Quantum state

• We can copy classical information perfectly. For
  example, a classical bit can be cloned via 0 0 ?
  0 0 and 1 0 ? 1 1.
• Can we clone an unknown qubit?
• Quantum cloning operation for a qubit is a linear
  operator that acts jointly on the input, blank
  and ancilla states as ?????0 ? ??A ? ?
  ????????A??
• Here, ?0 ? is the blank state, ?A ? and A?? are
  the initial and final states of the cloning
  machine.
• If one can satisfy the above transformation for
  an arbitrary input, then we could design a
  quantum cloning (Xerox) machine.

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Quantum Xerox Machine
???
???

CLONING
?0 ?
???

• By the linearity of quantum evolution we can show
  that an arbitrary state of a single quantum
  cannot be cloned perfectly. This is called the
  no-cloning Theorem.
• However, if we know a state we can clone it
  perfectly. For example, if a qubit is either in
  ?0 ? or?1 ?, then it can be cloned perfectly.

W. K. Wootters and W. H. Zurek, NATURE, 299
(1982) 802.
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No-Cloning from Linearity

• Let there be a cloning machine that copies a
  single qubit in an orthogonal state
• 0?? 0? ?A? ? 0??
  0??A0?.
• 1? ?0? ?A?
  ? 1?? 1??A1?.
• If we send an unknown qubit through the cloning
  machine, we will have ??? 0?? A? ?
  ?0??0??A? ??1 ??0??A?
  ? ? 0??0??A0? ? 1??1??A1?.
• Ideally the cloning machine should produce a
  state ????????A?? ?0??0??1??1? ?
  ?(0??1?1??0?) ?A??
• These states cannot be equal. Thus, it is
  impossible to copy an unknown state perfectly
  the no-cloning Theorem.

2
2
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Importance of No-cloning

• No-cloning is one of the hall mark feature of
  quantum information.
• Since it is fundamental to understand the
  limitations on quantum information, i.e. what we
  can do and what we cannot do. There stands the
  no-cloning principle.
• It is the no-cloning principle which provides
  security to information stored in quantum states
  and has great application in quantum cryptography.

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No cloning from unitarity

• Can we clone two non-orthogonal states ? If so,
  we would have ?1? 0? A? ? ?1? ?1? A1?
  ?2? 0? A? ?
  ?2? ?2? A2?
• Unitarity must preserve the inner product. This
  implies we must have ??1 ?2? ? ??1 ?2?
  which is a contradiction.
• Thus two non-orthogonal states cannot be cloned
  by a unitary machine. However, a qubit in any
  one of the two orthogonal states can be cloned
  unitarily.

2
H. P. Yuen, PLA 113 (1986) 405.
21
Deleting quantum information
???
???

DELETING
???
?0 ?

• Suppose Alice prepares two copies of a qubit and
  asks Bob (who do not have the complete knowledge)
  to delete a copy, keeping the other intact. Can
  Bob do that?
• It is impossible to delete a copy from two
  identical copies The no-deleting theorem.

A. K. Pati and S. L. Braunstein, NATURE, 404
(2000) 164.
22
Deletion and erasure

• In classical world if we have two identical bits
  such as 00 or 11, then we can delete one copy
  against the other in a reversible manner via 0 0
  ? 0 0 and 1 1 ? 1 0.
• Primitive deletion or erasure operation
  resets the last bit to a standard bit
  irrespective of all others, e.g., 01101 ? 01100 .
  This can take a collection of unordered bits to a
  collection of ordered bits, hence
  thermodynamically irreversible.
• Erasure of a single bit at temperature T needs k
  T log 2 amount of energy a result known as
  Landauers principle.
• Classical deletion against a copy takes an
  ordered set to an ordered set, hence logically
  reversible.

R. Landauer, IBM J. Res. Dev. 5 (1961) 183.
23
Deleting an unknown qubit

• Can we delete an unknown qubit from two identical
  copies?
• ??????? ??A ? ? ?????0? ?A?? ?
  
• Here ?0? is the blank state and ?A ? and A??
  are the initial and final states of the cloning
  machine.
• We do not want that quantum information should be
  hidden anywhere in the deleting machine, or any
  other part of the universe .
• By the linearity of quantum evolution one can
  show that a single copy of an arbitrary state of
  a qubit cannot be deleted perfectly.

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No deleting Theorem

• Action of the deleting machine on a pair of
  qubits in the orthogonal states
  
  ?0??0??A ? ? ?0??0?A0?
  and ?1??1??A ? ? ?1??0?A1?
• If two qubits are not identical, or entangled
  then the final state can be an entangled state of
  the two qubits and the ancilla
  1/?2 (?0??1? ?1??0?)?A ? ? ???
• Because of the linear nature of the deleting
  transformation it is not the time reverse of
  cloning operation.
• If we send an unknown state then by linearity we
  will have

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No deletion...

• The output state is a quadratic polynomial in ?
  and ?. The desired state is only (? ?0? ? ?1?
  )A??. Since the actual and desired states are in
  general different, we cannot design an all
  purpose deletion machine.
• However, there is a choice that makes the actual
  and desired state equal
• Since the final state is normalized for all ? and
  ?, this implies that the ancilla states A0? and
  A1? are orthogonal. But this choice suggests
  that the final state of the ancilla is not
  independent of ? and ?.

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No deletion...

• This choice is not deletion but swapping of an
  unknown qubit onto two-dimensional subspace of
  the ancilla (hiding of quantum information in the
  deleting machine).
• Thus linearity allows one to move around quantum
  information instead of perfect deletion. This is
  called the no-deleting Theorem.
• Like cloning, if we know a qubit we can delete
  it perfectly. For example, if we have two
  identical qubits either in ?0 ? or 1 ?, then
  we can transform ?0??0? ? ?0??0? and ?1??1? ?
  ?1??0?. Thus it can be deleted perfectly.

A. K. Pati and S. L. Braunstein, NATURE, 404
(2000) 164.
27
Information gain and Disturbance

• Given two non-orthogonal state, any operation
  revealing the identity must disturb the state.
• One can prepare an ancilla, couple to system and
  evolve ?1? A? ?
  ?1?A1? and ?2? A? ? ?2? A2?.
• After evolution, Eve may take the ancilla and
  leave the original system. Then she may try to do
  measurement on the ancilla and learn about the
  quantum states.
• Unitary evolution preserves the inner product. We
  must have ??1 ?2?
  ??1 ?2? ?A1 A2?
• If ??1 ?2? ? 0, then ?A1 A2? 1, i.e., A1?
  A2?. So final state of the ancilla is
  independent of the input. Thus, measurement of
  ancilla will not tell anything about the identity
  of the state.

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QUANTUM INFORMATION SCIENCE
QUANTUM CRYPTOGRAPHY
QUANTUM COMPUTATION

QUANTUM COMMUNICATION
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QUANTUM INFORMATION THEORY
Quantum Communication
Quantum Computation
Quantum Cryptography
Teleportation
Algorithms
BB84 scheme
Dense Coding
Computational Complexity
Entangled state based scheme
Remote state preparation
Decoherence Error Correction
Secret sharing
Remote control
Experimental Implementations

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QUANTUM INFORMATION
? One of the main goal in quantum information
theory is how well we can store, process and
transfer the vast amount of information contained
in the quantum state using principles of quantum
theory. ? Quantum mechanical features like
superposition, entanglement, and non-locality are
exploited for practical applications and
sometimes doing amazing tasks which are
impossible otherwise.
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QUANTUM COMMUNICATION
Sending quantum information according to quantum
rules using given resources. Alice and Bob are
not allowed to send quantum particles directly
but allowed to do local operation and classical
communication (LOCC).
Classical Channel
Unitary/ Measurement operations
Unitary / Measurement operations

Quantum Channel
ALICE
A

BOB
32

• Quantum information theory has revolutionized the
  way in which information is processed using
  quantum resources such as entangled states, local
  operations and classical communications (LOCC).
• ? Examples Quantum Teleportation, Remote state
  preparation, Secret sharing, Quantum Cryptography
  and many more

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Cryptography (Classical)

• Cryptography may be defined as the art of
  encrypting and decrypting messages in codes in
  order to ensure their confidentiality and
  authenticity.
• The fundamental task of cryptography is to allow
  two users to render their communication
  unintelligible to any third party, while for the
  two legitimate users the message remain
  intelligible.

34
Symmetric-key Cryptography

• Data encrypted and decrypted with same key
• Classical examples Caesar cipher and one-time
  pad

35
Caesar Cipher (n3) (Review)

• If we use the algorithm of simply moving each
  letter n places down the alphabet (here n3) then
  the original alphabet we were using, or the Plain
  Text becomes the following Cipher Text, as
  follows

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Encrypting a Message

• Bob wants to send a secret message to his friend
  Alice. He encrypts his message with the key of
  n3
• "This is a secret message" becomes
• "Wklv lv d vhfuhw phvvdjh"

37
Decrypting a Message

• Alice receives Bob's encrypted message. If she is
  knows the key (n3) she decrypts the message by
  reversing the encryption process. She takes the
  ciphertext
• "Wklv lv d vhfuhw phvvdjh"
• and applies the Caesar Cipher using her key to
  render it
• "This is a secret message"

38
Vernam Cipher (One time Pad)

• Vernam (1917) proposed a cryptosystem where each
  letter is advanced by a random of positions in
  the alphabet.
• These random numbers form the cryptographic key
  that must be shared between Alice and Bob.
• Even though the Vernam cipher offers
  unconditional security against Eve possessing
  unlimited computational power, it faces the
  problem of how to securely distribute the key.

39
One Time Pad

• The principle of the cipher is that if a random
  key is added to a message, the bits of the
  resulting string are also random and carry no
  information about the message.
• The encryption algorithm E can be written as
  EK(M) (M1K1, M2K2, MnKn) mod 2, where M
  (M1, M2, Mn) is the message to be encrypted and
  K (K1, K2, Kn) is the key consisting of random
  bits. The message and key are added bitwise
  modulo 2.
• The decryption of cipher text C EK(M) is given
  by M DK(C) (C1K1, C2K2, CnKn) mod 2.

40
Security of One time Pad

• Three requirements on key (1) The key must be as
  long as the message, (2) it must be purely
  random, (3) it may be used only once.
• The drawback of One time pad is the necessity to
  distribute a secret key as long as the message.
• Here, quantum information theory comes in handy
  and readily offers a solution. Quantum
  cryptography helps to generate a secret key.
• Also, it gives the power to detect eavesdropping.

41
BB84 CRPTOGRAPHY PROTOCOL
Random measurements
CLASSICAL CHANNEL
Random bits and bases
QUANTUM CHANNEL
In the end they share a secret key
ALICE
BOB
C. H. Bennett and G. Brasard, IEEE Conference on
Computers, Systems and Siganls Processing,
Bangalore, India (1984), p 175-179.
42

• Bennett and Brasard (1984) originally used
  photon polarization states to transmit the
  information.
• The sender (Alice) and the receiver (Bob) are
  connected by a quantum communication channel
  which allows quantum states to be transmitted.
• In the case of photons this channel can be an
  optical fibre or simply free space. In addition,
  they need a public classical channel (broadcast
  radio, the internet or phone).
• Neither of these channels need to be secure.
  BB84 protocol is designed with the assumption
  that an eavesdropper (Eve) can interfere with
  both channels.

43

• The security of the protocol comes from encoding
  the information in non-orthogonal states.
• These states cannot in general be measured
  without disturbing and cannot be cloned.
• BB84 uses two pairs of states, with each pair
  conjugate to the other pair, and the two states
  within a pair orthogonal to each other. Pairs of
  orthogonal states are referred to as a basis.
• Qubit pairs used are either Z basis (0? and 1?
  ) or X basis (? and -?), where Z and X are
  Pauli matrices with z and x components.

44
BB84 PROTOCOL

• Alice and Bob have agreed that 0? and ? stand
  for bit value 0, and 1? and -? stand for a
  bit value 1.
• Alice generates a sequence of random bits, and
  randomly and independently for each bit she
  chooses her encoding basis, Z or X.
• Physically, it means that she transmits qubits
  in four states 0?, 1?, ?, and -? with
  equally distributed frequency.

45
BB84

• Bob randomly and independently of Alice, chooses
  his measurement bases, either Z or X.
• Statistically, their bases coincide in 50 of
  cases, when Bobs measurements provide
  deterministic outcomes and perfectly agree with
  Alices bit.
• Alice and Bob use a public channel to tell each
  other what basis they had used for each
  transmitted and detected qubit. This classical
  channel may be tapped, because it transmits only
  information about the used bases, not about the
  particular outcomes of the measurement.

46
BB84

• Whenever their bases coincide, Alice and Bob keep
  the bit. The bit is discarded when they chose
  different bases or Bob did not receive the qubit
  (if it was lost somewhere on the way). This way
  they generate a secret key.
• Any eavesdropper (Eve) who listens this
  conversation can only learn whether Alice and Bob
  choose Z or X basis, but not whether Alice had
  sent a 0 or 1.
• If Eve has gained any information about the
  qubit, this will have introduced errors in Bobs'
  measurements. If more bits differ they abort the
  key and try again, possibly with a different
  quantum channel, as the security of the key
  cannot be guaranteed.

47
BB84 Table
48
Eve

• The security of BB84 quantum cryptography
  protocol comes from laws of quantum theory.
• Eve cannot make copy of quantum states and learn
  (the no-cloning theorem). Nor can she learn the
  identity of quantum states without disturbance.
• If Alice is sending identical secret messages to
  two people, then Eve cannot delete one message
  (the no-deleting theorem).

49
World Premiere Bank Transfer via Quantum
Cryptography

• Press conference and demonstration of the
  ground-breaking experiment
• 21 April 2004, 1130, Vienna City Hall
  Steinsaal
• A collaboration of group of Professor Anton
  Zeilinger, Vienna University ARC Seibersdorf
  research
• GmbH City of Vienna Wien Kanal
  Abwassertechnologien GmbH and Bank Austria
  Creditanstalt
• Today, the Bank Austria Creditanstalt has, on
  behalf of the City of Vienna, performed the
  Worlds first bank transfer encoded via quantum
  cryptography. This novel technology was
  demonstrated by the group of Professor Anton
  Zeilinger, Vienna University in collaboration
  with the group Quantum Technologies (Information
  Technologies Division) of Seibersdorf research.
  The bank transfer was initiated by Viennas Mayor
  Dr. Michael Häupl, and executed by the Director
  of the Bank Austria Creditanstalt, Dr. Erich
  Hampel. The information was sent via a glass
  fiber cable, laid by the company Wien Kanal
  Abwassertechnologien from the Vienna City Hall to
  the Bank Austria Creditanstalt branch office
  Schottengasse.

50
SUPER DENSE CODING
? If we send a single qubit we can convey one
bit of classical information. ? The presence of
entanglement between sender and receiver can
double the capacity of classical information. If
a qubit is entangled with another qubit
previously shared with receiver then sender can
communicate two classical bits----called super
dense coding. ? So entanglement not only plays an
important role in quantum communication but also
in classical communication.
C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett.
69, 2881 (1992).
51
2 CBITS
2 CBITS
QUBIT
M
Ui
DECODING
EPR
ENCODING
52
PROTOCOL FOR SUPER DENSE CODING
? Alice and Bob have previously shared an EPR
pair or any one of the four Bell states (say)
B0? 1/?2(?? ? ?? ? ?? ? ?? ? ) ? Two
classical bits such as 00, 01, 10, and 11 are
encoded in four unitary operators I, ?x, i?y
and ?z. ? Alice applies one of these unitary
operators to her particle locally. When she
applies I, we have I ?I B0? B0? When she
applies ?x, we have ?x ?I B0? B2? When she
applies i?y, we have i?y ?I B0? B3? When she
applies ?z, we have ?z ?I B0? B1?
53
? After applying unitary operator Alice sends her
qubit to Bob. ? Bob has two qubits in the state
which is either B0?, B1?, B2? or B3? .
Because these are mutually orthogonal, he can do
a projective measurement on the two qubits and
extract two classical bits with certainty. ?
Thus, by sending a single qubit Alice can
communicate two classical bits of information to
Bob. This enhancement of classical capacity is
called super dense coding. ? Note that without
previously shared entanglement this is
impossible. That is, if the qubit that is being
sent by Alice is not a part of an entangled pair
this cannot happen.
54
QUANTUM COMMUNICATION
? Quantum Cryptography ? Quantum Secret Sharing ?
Quantum Data Hiding ? Quantum Remote Control ?
Quantum Tele Cloning ? ? ?
55
CONCLUSION
? Quantum state contains a vast amount of
inaccessible information. How well we can store
and process this information? ? Quantum
information cannot be copied and deleted. Any one
trying to tamper can be detected. ? Quantum
Computing, Quantum Communication and Quantum
Cryptography are some of the emerging areas of
future research. ? Quantum theory provides us a
handy way to share secret key. ? Its security is
guaranteed by fundamental laws of quantum
theory ? And there are many more amazing things
one can do with quantum information and
communication.
56
THANKS