Download Easy introduction to quantum informatics

Quantum Information,
Communication and Computing
Jan Kříž
Department of physics,
University of Hradec Králové
Doppler Institute
for mathematical physics and applied mathematics
Quantum Information,
Communication and Computing
Information Theory: does not care about the
physical realization of signals
Quantum:
description of the carriers of
information
Resources:
Taksu Cheon
Kochi University of Technology,
Japan
Private communication in 2004
http://www.mech.kochi-tech.ac.jp/cheon/q-inf/q-inf00_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
When will we have a quantum computer?
pessimists:
NEVER!
optimists:
within next
30 years
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.”
Preliminaries
Hilbert Space: we associate a Hilbert space  to each
quantum system
  is a vector space over 
  has a sesquilinear scalar product
,
z    z    z
for z,
   the
 positivity
0 for 
 0.
satisfying
condition
2
m 0 
 isn 
complete,
i.e.   , such that  n   0.
Outline
QI contains more sexy topics than boring mathematical
description…
1. Story on the quantum witch
2. Entangled states
3. Quantum teleportation
4. Quantum cryptography
5. Quantum computing
6. Quantum game theory
Prerequisity
Quantum mechanics, version 0.5
Starring
Alice
Bob
On the quantum witch
Two ways of bark analysis:
to dissolve
to burn
On the quantum witch
On the quantum witch
On the quantum witch
100%
0%
70%
30%
0%
100%
30%
70%
On the quantum witch
70%
17%
30%
83%
30%
83%
70%
17%
On the quantum witch
100%
0%
70%
30%
0%
100%
30%
70%
On the quantum witch
1.There is a “symmetry” in reddish and
greenish property !
On the quantum witch
100%
0%
70%
30%
0%
70%
100%
30%
30%
17%
70%
83%
On the quantum witch
0%
100%
30%
70%
30%
83%
70%
17%
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. Bob’s result depends
on the Alice’s choice of analysis!
On the quantum witch
On the quantum witch
On the quantum witch
70%
30%
0%
0%
On the quantum witch
0%
0%
30%
70%
On the quantum witch
11%
59%
5%
25%
On the quantum witch
25%
5%
59%
11%
On the quantum witch
same colour
70%
different colours
30%
same colour
36%
different colours 64%
On the quantum witch
Alice can send a signals to Bob
by encoding her message in her choice
of the way of analysis.
67%
same colour
67%
different colours
Bob’s guesses are better than chance!
We have proper transmission of information (although
in a “noisy channel”)
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
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
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
p(1)  cos
Preparing
device
q
2

2
,
p(1)  sin
2

Measuring 2
device
1 1 -1 1 -1
1,-1
Entangled states
q
p   cos 2

p    sin

2
2
2
(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
  cos

2
  sin

2

    ,  
Entangled states
Assume now the system of two particles,
we have four possible combinations of basis states:
  ,
1
2
  ,
1
2
  ,
1
2
 
1
2
Any superposition of these states is again
a quantum state, which can be prepared
in suitable preparing device, e.g.
1
1
S 
  
  ,
1
2
2 1 2
2
1
1
W 
  
 
1
2
1
2
2
2
Entangled states
Spins in entangled state can be send
to different places on the Earth,
they still remain entangled…
?
?
What does the measurement bring?
1
1
Measuring
W 
  
 
1
2
1
device: ↑or↓
2
2
2
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
projections
Two piecesdevice:
of “Magic
bark” = to
= a couple of spins in entangled state
Measuring device: projections to
Entangled states
x
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)
Entangled states
Since this "instanteneous
Albert 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
Einstein –
standard quantum state must
Podolsky – Rosen
be wrong.
Paradox (EPR paradox)
Modern experiments go against Albert!
Quantum teleportation
Alice wants to teleport a “spin” to Bob.
Teleporting one qubit requires one
Two-level
(spin,
entangled
pairsystem
of qubits
and two
photon
…) = qubit
bits ofpolariazation,
classical information.
q
?
?
A
E
B
1

Measuring S 
2
device
E
1
 

B
2
E

~
X
B
~
X
B
X
AE
X  AE
Y
AE
Y  AE
q
Preparing
device B
~
Y
B
1
~
Y
B
2
B
3
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.
Quantum cryptography
Top secret
Preparing Preparing
device ↑ device →
Measuring
device ↑
Measuring
device →
Measuring
device ↑
Measuring
device →
Preparing
device ↑
Preparing
device →
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)
If not…
secret sequence
of numbers. They can use it to
code messages classically.
BB84 protocol according to inventors Bennet, Brassard.
Quantum computing
How does the quantum computer look like?
Why? We have perfectly good classical computers.
Quantum computing
Why? We have perfectly good classical computers.
P. Shor converted a classical hard
task into a tracktable one…