Post on 27-Dec-2015
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
ComputingInformation Theory:Information Theory: does not care about the
physical realization of signals
Quantum:Quantum: description of the carriers of information
Resources:
Taksu CheonTaksu CheonKochi University of Technology, JapanPrivate communication in 2004
http://www.mech.kochi-tech.ac.jp/cheon/q-inf/q-inf00_e.html
Reinhard F. WernerReinhard F. WernerTechnical 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.”
PreliminariesHilbert Space: Hilbert Space: we associate a Hilbert space to each
quantum system
is a vector space over
has a sesquilinear scalar product ,
for z,
satisfying the positivity condition
is complete, i.e.
zzz
.002 for
0. that such nmn ,0
Outline
1.1. Story on the quantum witchStory on the quantum witch
2.2. Entangled statesEntangled states
3.3. Quantum teleportationQuantum teleportation
4.4. Quantum cryptographyQuantum cryptography
5.5. Quantum computingQuantum computing
6.6. Quantum game theoryQuantum game theory
QI contains more sexy topics than boring mathematicaldescription…
On the quantum witch
1.There is a “symmetry” in reddish andgreenish property !
2.There is no “symmetry” in ways of analysis, i.e. Bob’s result depends depends on the Alice’s choice of analysis!on the Alice’s choice of analysis!
On the quantum witch
Alice can send a signals to Bobby encoding her message in her choice
of the way of analysis.
same colour 67%
different colours 67%
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).
CONTRADICTION with Einstein causalityCONTRADICTION with Einstein causality
Transmission of information in infinite velocity!
On the quantum witch
CONTRADICTION with Einstein causalityCONTRADICTION with Einstein causality
Transmission of information in infinite velocity!
This may happen in the story, where the crucialroles are played by …
By the way, nobodycan be forced
to accept Einstiencausality as
a fundamental principle
Entangled statesExperiment in quantum mechanics:Experiment in quantum mechanics:
Preparingdevice
(produces particles)
Measuringdevice
(perfectly classical output,changes the state of particle)
Object of QM: Object of QM: predict the probabilities of the outcomes
Example: spin projection
Preparingdevice
Measuringdevice
11,-1
1 -1 1 -1
2sin)1(,
2cos)1( 22
pp
Entangled states
2
sin 2
p
2
cos2
p
(Arbitrary) state can be thus interpreted as somemixture of states ↑ and ↓
Such mixture in QM - SUPERPOSITIONSUPERPOSITION
On the other hand: any (normalised) superpositionof quantum states is again a legitimate quantum state
,2
sin2
cos
Entangled states
Assume now the system of two particles,we have four possible combinations of basis states:
21212121,,,
Any superposition of these states is again a quantum state, which can be prepared
in suitable preparing device, e.g.
2121
2121
2
1
2
1
,2
1
2
1
W
S
Entangled states
Spins in entangled state can be sendto different places on the Earth,
they still remain entangled…
2121 2
1
2
1 W
What does the measurement bring?
Measuringdevice: ↑or↓
Entangled states
Thus, we can “translate” the storyon the quantum witch to QM…
Quantum witch = a person (traditionally called Eve) who possesses a preparing devicefor the entangled state |W
Two pieces of “Magic bark” == a couple of spins in entangled state
Measuring device: projections to
Measuring device: projections to
Entangled states
…really impossible machine
However, the impossibility to construct it is nota consequence of Einstein causality breakdown.
It follows from QM itself! (known as No Cloning Theorem)
Entangled states
AlbertSince 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.
Einstein – Einstein – Podolsky – Rosen Podolsky – Rosen Paradox (EPR paradox)Paradox (EPR paradox)
Modern experiments go against Albert!
Quantum teleportationAlice wants to teleport a “spin” to Bob.
Two-level system (spin, photon polariazation, …) = qubitqubit
A
E
B
BEBE S
2
1
2
1
Preparingdevice
Measuringdevice
AEAEAEAE Y Y X X
BBBB Y Y X X
~~~~
B
1 2 3
Teleporting one qubit requires oneentangled pair of qubits and two
bits of classical information.
Quantum cryptographyAlice 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
Preparingdevice ↑
Preparingdevice →
Measuringdevice →
Measuringdevice ↑
Top secret
Preparingdevice ↑
Preparingdevice →
Measuringdevice ↑
Measuringdevice →
Quantum cryptography
1 0 1 1 0 1 0 0 0 1↑ ↑ → ↑ →→ → ↑ → ↑
↑ → ↑ → ↑→→ ↑ ↑→
Top secret
1 1 1 1 0 1 0 0 0 0
If these bits match 100%, OK.
If not…In such a way Alice and Bob can obtain shared (random)
secret sequence of numbers. They can use it tocode messages classically.
BB84 protocol BB84 protocol according to inventors Bennet, Brassard.
Quantum computing
How does the quantum computer look like?How does the quantum computer look like?
Why? Why? We have perfectly good classical computers.