Quantum Information, Communication and Computing Jan Kříž Department of physics, University of...

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…

Prerequisity

Quantum mechanics, version 0.5

Starring

AliceBob

On the quantum witch

Two ways of bark analysis:

to dissolve to burn


100% 0%

70% 30%

0% 100%

30% 70%


70% 30%

17% 83%

30% 70%

83% 17%


100% 0%

70% 30%

0% 100%

30% 70%


1.There is a “symmetry” in reddish andgreenish property !


100% 0%

70% 30%

0% 100%

30% 70%

70% 30%

17% 83%


30% 70%

83% 17%

0% 100%

30% 70%


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!


70%

30%

0%

0%


0%

0%

30%

70%


11%

59%

5%

25%


25%

5%

59%

11%


70%

30%

36%

64%

same colour

different colours

same colour

different colours


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”)


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!


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

x

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.

Quantum computingWhy? Why? We have perfectly good classical computers.

P. Shor converted a classical hardtask into a tracktable one…

Quantum Information, Communication and Computing Jan Kříž Department of physics, University of...

Documents

Transcript of Quantum Information, Communication and Computing Jan Kříž Department of physics, University of...