“year 2020” -topics in information theory for further studies · 2006-10-12 · three concepts:...
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“Year 2020” -Topics inInformation Theory for Further
Studies
Comprestimation
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 104
Comprestimation
lossy compression of ”non-natural”images (regular lossy compression usesMSE)
compression of images so that thestatistical inferences on thecompressed images remain valid
E.g. compression of microarray images
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 105
Microarray images
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 106
Comprestimation cont.
also known as ”Multi-terminal datacompression”
T. Han & S.Amari; R. Jörnsten & B. Yu;
Algorithmic Information Theory
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 108
Algorithmic Information Theory
…as used by Chaitin for ”meta-mathematics”
incompleteness theoremsGödel (logic)Turing (algorithm)The Halting Probability Omega (information,
randomness)
http://www.umcs.maine.edu/~chaitin/
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 109
Physics, information and games
2020: Quantum Odyssey“On Quantum Computing and information transmission”
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 111
Motivation Computers as physical systems Technological issues
miniaturization and speedup - Moore’s lawneed for energy efficiency
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 112
Why would we bother?
Cryptography: QC can break RSA codes Communication of messages that betray
the presence of eavesdropping Teleportation: moving qubits around
without having them ever beingtransmitted over an insecure channel
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 113
Central concepts Superposition: a “blend” of 0 and 1
simultaneously, i.e., quantum parallelmode
http://www.quantiki.org Reversible computing: logical
irreversibility implies thermodynamicirreversibility (i.e., heat dissipation)
Charles Bennett0
10 0 1
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 114
Wave/particle duality
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 115
Mach-Zehnder interferometer
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 116
The Capabilities of Computers
(Deterministic)Turing Machine
ProbabilisticTuring Machine
QuantumTuring Machine
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 117
Proving vs. providing proof
QTM can simulate a TM - QTMuniversal
TM provides a proof as the sequence ofsteps performed
QTM can provide an answer without aproof trace (worse: if you try to “peek”QTM that would disrupt the proof!)
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 118
Bits and Qubits
Each bit is represented by the state ofa simple 2-state quantum system e.g.,spin state)
We need finite dimensional Hilbertspace
ω2
ω1
ω0
“Complex linear vector space”
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 119
Bra-ket
For a simple two-state system you canwrite the state as a “ket (vector)”
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Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 120
Unitary operators
2-state system has 2 eigenstates called|ψ0> and |ψ1> (basis)
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Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 121
Unitary operators continued To change the quantum world one needs
an operator, e.g. NOT
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One can also have non-classical gatessuch as NOT
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 122
Universality In classical computation AND and NOT
are enough to build any circuit In quantum computing it is enough to
use a 2-qubit gate (Barenco et al)
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Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 123
Fundamentals One can have quantum interference
whenever there is more than one way ofobtaining a particular result
measuring a quantum system:if the system is in eigenstate the outcome
is one of the eigenvaluesif the system is in superposition state the
result is given by
0 20 40 60 80 100
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Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 124
“A good quantum calculation”
Create a superposition of registerelements
Calculate in “one shot” all functionvalues F(j)
Do something clever with all the F(j)values(Use interference to increase the amplitudes andthus probabilities of the solution states)
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 125
Quantum entanglement (EPR)
If two systems (particles) are“Quantum correlated” one talks aboutentanglement
For entangled particles their joint stateis not factorizable as the directproduct of two simpler states
Produced by conservation of someattribute
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 126
Teleportation
1010010...
•dissociation•information transmission•reconstitution
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 127
Well, at least a qubit ...
Three Concepts: Information ‘06 © Petri Myllymäki & Henry Tirri 2002-2006 128
Further topics
quantum search quantum cryptography dense coding random number generation breaking unbreakable codes quantum complexity theory ...