Quantum Information, Communication and Computing

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Quantum Information, Communication and Computing

• Jan Krí

Department of physics, University of Hradec
Králové Doppler Institute for mathematical
physics and applied mathematics
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Quantum Information, Communication and Computing

• Information Theory does not care about the
  physical realization of signals
• Quantum description of the carriers of
  information

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Resources
Taksu Cheon Kochi University of Technology,
Japan Private communication in 2004
http//www.mech.kochi-tech.ac.jp/cheon/q-inf/q-inf
00_e.html
Reinhard F. Werner Technical University of
Braunschweig, Germany Course Conceptual and
mathematical foundations of quantum information
given at Bressanone (Italy) in 2007
http//www.imaph.tu-bs.de/qi/qi.html
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When will we have a quantum computer?
optimists within next 30 years
pessimists NEVER!
IBM (in 1998) Probably in the next millenium
R.F.Werner Even if the Quantum Computer proper
were never to be built, the effort of building
one, or at least deciding the feasibility of this
project, will turn up many new results, likely to
have applications of their own.
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Preliminaries
Hilbert Space we associate a Hilbert space ? to
each quantum system
Sorry!!!

• ? is a vector space over ?
• ? has a sesquilinear scalar product ?????,
• for z??,
• satisfying the positivity condition
• ? is complete, i.e.

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Outline
QI contains more sexy topics than boring
mathematical description

• Story on the quantum witch
• Entangled states
• Quantum teleportation
• Quantum cryptography
• Quantum computing
• Quantum game theory

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Prerequisity
Quantum mechanics, version 0.5
Starring
Alice
Bob
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On the quantum witch
Two ways of bark analysis
to dissolve to burn
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On the quantum witch
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On the quantum witch
Oughhh!
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On the quantum witch
0
100
70
30
100
0
30
70
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On the quantum witch
30
70
17
83
70
30
83
17
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On the quantum witch
0
100
70
30
100
0
30
70
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On the quantum witch
??????
1.There is a symmetry in reddish and greenish
property !
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On the quantum witch
0
100
70
30
100
0
30
70
30
70
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83
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On the quantum witch
100
0
30
70
70
30
83
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On the quantum witch
??????
1.There is a symmetry in reddish and greenish
property !
2.There is no symmetry in ways of analysis,
i.e. Bobs result depends on the Alices choice
of analysis!
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On the quantum witch
IMPOSSIBLE MACHINES, Corp.
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On the quantum witch
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On the quantum witch
70
30
0
0
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On the quantum witch
0
0
30
70
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On the quantum witch
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59
5
25
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On the quantum witch
25
5
59
11
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On the quantum witch
70
same colour
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different colours
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same colour
64
different colours
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On the quantum witch
Alice can send a signals to Bob by encoding her
message in her choice of the way of analysis.
same colour
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different colours
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Bobs guesses are better than chance! We have
proper transmission of information (although in a
noisy channel)
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On the quantum witch
However, Alice (in Amsterdam) and Bob (in
Boston) can carry out their experiments at the
same time (or even Bob can do his
measurements sooner than Alice).
Transmission of information in infinite velocity!
CONTRADICTION with Einstein causality
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On the quantum witch
Transmission of information in infinite velocity!
CONTRADICTION with Einstein causality
This may happen in the story, where the
crucial roles are played by
By the way, nobody can be forced to accept
Einstien causality as a fundamental principle
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Entangled states
Experiment in quantum mechanics
Preparing device
Measuring device
(produces particles)
(perfectly classical output, changes the state of
particle)
Object of QM predict the probabilities of the
outcomes
Example spin projection
Preparing device
Measuring device
1
1
-1
1
-1
q
1,-1
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Entangled states
q
(Arbitrary) state q can be thus interpreted as
some mixture of states ? and ?
Such mixture in QM - SUPERPOSITION
On the other hand any (normalised)
superposition of quantum states is again a
legitimate quantum state
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Entangled states
Assume now the system of two particles, we have
four possible combinations of basis states
Any superposition of these states is again a
quantum state, which can be prepared in suitable
preparing device, e.g.
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Entangled states
Spins in entangled state can be send to different
places on the Earth, they still remain entangled
?
?
What does the measurement bring?
Measuring device ?or?
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Entangled states
Thus, we can translate the story on the quantum
witch to QM
Quantum witch a person (traditionally called
Eve) who possesses a preparing device for the
entangled state W?
Measuring device projections to
Two pieces of Magic bark a couple of spins
in entangled state
Measuring device projections to
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Entangled states
x
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Entangled states
really impossible machine
However, the impossibility to construct it is
not a consequence of Einstein causality
breakdown.
It follows from QM itself! (known as No Cloning
Theorem)
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Entangled states
Since this "instanteneous comunication" between
faraway Alice and Bob is a direct result of the
fundamental principle of quantum mechanics, and
also this is against the local causality, it
could only be that either quantum physics or the
interpretation of the standard quantum state must
be wrong.
Albert
Einstein Podolsky Rosen Paradox (EPR
paradox)
Modern experiments go against Albert!
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Quantum teleportation
Alice wants to teleport a spin to Bob.
Teleporting one qubit requires one entangled pair
of qubits and two bits of classical information.
Two-level system (spin, photon polariazation, )
qubit
q
?
?
A
E
B
Measuring device
1
2
3
q
Preparing device
B
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Quantum cryptography
Alice wants to send a secret message to Bob
Eve is now a rival of Alice Observes the signals
of Alice and tries to send the identical signals
to Bob. Has all quantum devices as Alice and Bob.
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Quantum cryptography
Top secret
Measuring device ?
Measuring device ?
Preparing device ?
Preparing device ?
Measuring device ?
Measuring device ?
Preparing device ?
Preparing device ?
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Quantum cryptography
Top secret
? ? ? ? ??? ? ??
1 0 1 1 0 1 0 0 0 1 ? ? ? ? ?? ? ? ? ?
1 1 1 1 0 1 0 0 0 0
If these bits match 100, OK.
In such a way Alice and Bob can obtain shared
(random) secret sequence of numbers. They can use
it to code messages classically.
If not
BB84 protocol according to inventors Bennet,
Brassard.
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Quantum computing
How does the quantum computer look like?
Why? We have perfectly good classical computers.
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Quantum computing
Why? We have perfectly good classical computers.
P. Shor converted a classical hard task into a
tracktable one