Transcript Classical and Quantum Information Theory

ELEC 590 Directed Study
Level: Ph.D.
Course Instructor:
Dr. T. A. Gulliver
University of Victoria
Department of Electrical and Computer Engineering
August 3, 2012
Classical and Quantum
Information Theory
Information and Computation
of micro and mega-scale subjects
via Maxwell’s Demon metric
By:
Philip B. Alipour
Introduction
• In this study, we have conducted a set of surveys on real-time events
occurring on small and macro scales:
• Brownian motion
• Random events as randomly moving particles (attributing to Brownian
motion)
• Matter state physics defining particle motion based on their energy
signatures in the system
• We visualize events using simulation methods (Mathematica v.8.0)
presenting models to study and observe the statistics attached to
them:
• Maxwell’s Demon
• Some models used in Quantum and Classical Information Theories (QIT
and CIT) by incorporating the demon entity as an observer and
information measurer i.e. entropy, probability analysis, polynomial,
unconventional Hamiltonian path analysis, distance weights, Markov
chains, and the relative kinetic physics representing information.
• We at first introduce information, then its measure in terms of
entropy, plus relevant comparisons between CIT and QIT techniques.
What is Information?
• In both CIT and QIT fields:
• From [1, 2], one could conceive the general concepts known
within the CIT and QIT fields by referring to a simplistic summary
model,
Information content = decodable data in terms of 0’s and 1’s into
meaningful characters
The processing of information = quantum or classical computation
• Methodological comparisons in measuring information:
• Classically, information is encoded in a sequence of bits, i.e.,
entities which can be in two distinguishable states, which are
conventionally labeled with 0 and 1: High voltage = 1, low voltage
=0
• In a quantum system, we use parallel operators for multiple
states of logic, information is encoded based on energy states : 0
for spin-up, 1 for spin-down as a quantum bit (qubit).
Information Processing
• One of the famous parallel operators in a quantum system is a
reversible logic of a qubit called the controlled-NOT or CNOT gate.
• The following relative to above illustrates the computational
differences between Classical and Quantum Mechanical
Information Processing
Information Processing
• The simplistic case of a single qubit, its state representation is
|𝜓 = 𝛼|𝜓0 + 𝛽 |𝜓1
The two parameters α and β are both complex numbers, and 𝜓 is a
wavefunction, and is a probability amplitude in quantum mechanics
describing the quantum state of a particle and how it behaves. (the Bloch
sphere above)
Information Processing (a QIT example)
•
A semiconductor quantum dot containing an extra electron
acquires a net spin. The quantum state of the electron's spin is
represented by a vector (bold arrow) from the origin of the Bloch
sphere to a point on its surface:
•
•
•
the spin-up and spin-down states are at the north and south
poles, respectively; and the spins that correspond to equal
superpositions of the spin-up and spin-down states are in the
equatorial plane.
The spin rotation about the z axis is achieved by applying a static
magnetic field (B) along the z axis; the spin rotation about the x
axis is produced by a circularly polarized optical pulse injected
along the x axis.
A quantum computer with just 30 qubits would have
1,073,741,824 possible states, and a quantum computer with
300 qubits would have roughly the same number of possible
states as the total number of atoms in the observable universe,
also known as 1081 lesser than Shannon number 10120 denoting
the lower bound on the game-tree complexity of chess (number
of possible moves to make in the game).
Information Representation
• Would it be possible to represent probable events as they cooccur in form of 00, 01, 10, 11’s and interpret as information?
•
•
Yes, this parallel logic in a quantum system could be given as covariance and multi-variance representation of events (next slide).
We have formulated, defined and employed in our Simulation
Scenarios A and B this information representation as follows:
Maxwell’s Demon Metric
• The logic could also be incorporated classically by
identifying and averaging a particular parameter in terms
of a key distance d travelled by a body or particle
between two, pairs of two or 2n co-occurring events.
• We call this Maxwell’s Demon metric in our scenarios which is
the metric described as a distance function which defines a
distance measured between elements of a set of events Ei to En
(previous slide’s figure)
• So, the probability of such events could be calculated to obtain
information about the probable existence of event
(existentiality). We have classified them as:
• Uniform co-variance or pairwise events
• Multi-variance or multi-pairwise uniform events
Our definition of probability
• Uniform co-variance or pairwise events: extended to more than two events as disjoint
independent events with uniform P, or mutually exclusive i.e. 𝐸𝑖 ∩ 𝐸𝑗 = ∅ for all 𝑖 ≠ 𝑗 then
𝑛
𝑃𝑏𝑜𝑑𝑦 𝐸1 ∪ 𝐸2 ∪ ⋯ ∪ 𝐸𝑛 = 𝑃(𝐸1 ) + 𝑃(𝐸2 ) + ⋯ + 𝑃(𝐸𝑛 ) =
𝑃𝑏𝑜𝑑𝑦 𝐸𝑖
(3.1)
𝑖=1
where its distribution mean appears as
𝑛
𝑃(𝐸1 ) + 𝑃(𝐸2 ) + ⋯ + 𝑃(𝐸𝑛 )
1
𝑃
𝐸
=
𝑏𝑜𝑑𝑦
𝑖
𝑛
𝑛
(3.2)
𝑖=1
• Multi-variance or multi-pairwise uniform events: extended to more than 2 events up to n
events simultaneously (quite synchronized probabilities) as independent pairwise events
(two are not and collide or intersect) where at least each pair of the events out of the
product of all events depend on each other values computed by their probability product
with respect to time. For example, is the deuce going to appear in two consecutive events
knowing what each event contains separately as a card value? If so, what is its probability
of their mutual product, and if the product of a targeted value in both events is not equal to
the mutual product, then these two events are not independent.
𝑃𝑏𝑜𝑑𝑦 (𝐸1 ∩ 𝐸2 ) ∩ (… ∩ 𝐸𝑛 )
𝑛
= 𝑃(𝐸1 ∙ 𝐸2 ) ∙ 𝑃(… ∙ 𝐸𝑛 ) =
𝑃𝑏𝑜𝑑𝑦 𝐸1 ∙ 𝐸𝑖
𝑖=1
(3.4)
Our QIT and CIT probabilities coded in the
simulation
• 𝐸𝑐ℎ𝑜𝑠𝑒𝑛 = 𝑃 𝑑 𝑝 𝐸𝑖 , 𝑝 𝐸𝑛
=𝑃
𝛼 2 𝐸𝑖 +𝛽2 𝐸𝑛
Δ𝑑
𝑠
(3.6)
which calculates a key distance of P of E’s in the random function as
applied.
• This equation implies that there is a key random distance from the
generated random walk indicating a strong event candidate to occur
compared to its other mutual concurring events in the same system.
• For our next case determining a body candidate in Maxwell’s demon
scenario, we employ (3.6) into
• 𝜓𝑐ℎ𝑜𝑠𝑒𝑛 = 𝑃 𝑑 𝑝 𝛼 , 𝑝 𝛽 , 𝑡 = 𝑃
𝑠
−𝑐 2 Δ𝑡 2 +𝛼2 𝑛 𝜓+𝛽2 𝑚 𝜓
Δ𝑑
(3.7)
which represents an existing 𝜓 or eventful wave-particle candidate, a
moving particle with high energy compared to others in its partition.
Maxwell’s Demon
• In 1867 A.D., the Scottish physicist James C. Maxwell
formulated the thought experiment that seems to violate the
2nd law of thermodynamics. Can we cleave into it deeper?!
Quantum phenomenon as
light travelling faster than
particles in Chamber A or B
Is there another demon
observing and measuring
prior to ours?! Probably…!
Why Not on a cosmic scale?
2nd Law of Thermodynamics, Entropy and
the Demon’s Paradox
•
•
•
•
•
• ~ "In all energy exchanges, if no energy enters or leaves the
system, the potential energy of the state will always be less
than that of the initial state." ~
This is also commonly referred to as entropy a measure of
disorder, where in the process of energy transfer, some energy
will dissipate as heat (the demon’s paradox problem).
Scenario B focuses on this type of information measurement
incorporating our entropy in its analysis
Our entropy is evaluated based on the energy-matter type
Boltzmann, and information-theoretic classical type Shannon:
Shannon’s entropy rate 𝐻(𝑋) = − 𝑛𝑖=1 𝑝𝑖 log 2 𝑝𝑖
Boltzmann entropy 𝑆 = −𝑘 𝑛𝑖=1 𝑝𝑖 log 𝑝𝑖 , 𝑘= 1.38×10–23J K–1
2nd Law of Thermodynamics, Entropy and
the Demon’s Paradox
• Scenario A: We observe and measure random walk + heat, and
analyze the kinematics of the particles without studying the initial
forces causing them, reaching a state of equilibrium (vs. paradox).
• Scenario B: We involve the thermodynamic properties of the
particles studying the P’s of heat dissipation, transfer + entropy.
Maxwell’s Demon in Scenario A
• Scenario A: The simulation shows a gas-filled container
that is divided into two compartments with a trapdoor in
the wall that separates them.
• A “demon” selects the molecules passing through the bidirectional trapdoor according to their velocities relative
to the projected light.
• Those with more kinetic energy than the mean energy of the
molecules in the first compartment pass to the second and
equally from right to the first.
• One chamber ends up with highly kinetic molecules, whilst the
first chamber with less kinetic ones.
• The simulation conveys to the nature of an equilibrium state
tackling the paradox.
Maxwell’s Demon in Scenario B
• Scenario B: shows a condensed matter state of gas molecules
in a container as we super-freeze its interior down to 0o K =
– 273.15o C. (Bose-Einstein Condensate.)
• Radiation is applied through lensing effect to excite the atoms.
• Some are active and the rest remain idle relative to the
container’s temperature (according to Planck’s blackbody
radiation effect, temperature gradually increases).
• A “demon” selects the atoms passing through the trapdoor
according to their velocities relative to their energy signature.
• Those with more kinetic energy than the atoms’ mean energy
in the first chamber pass to the second, whereas the latter
ends up with a population of energetic ones and the former,
with an emptied space
• We measure the atomic speed relative to the projected light
as a preemptive approach of measurement on behalf of the
demon.
Maxwell’s Demon Paradox
Tackled in Scenario A
• The demon must perform measurements on the molecules moving
between the chambers in order to determine their velocities.
• The result of this measurement must be stored in the demon’s
memory where it could run out of space since it is finite.
• The analysis shows that the system’s entropy could be decreased by
the actions of the demon, thus ensuring that the 2nd law of
thermodynamics is obeyed and memory is not running out of space.
• Scenario A maintains the equilibrium state of the environments
while quantum computation and classical information are conducted
based on probability calculus, thereby harnessing energy to address
the paradox by formulating efficient ways of computation i.e.
reversible circuit and enormously computable data stored in space
with respect to time based on single co-variant and multi-variant
events’ parameters.
Simulation Scenario A
• Hypothesis 1. Let our demon be a multi-variant observer, monitoring
events emerging from α and β, measuring them based on particle
motion, such that the total number of particles is evenly distributed
inclusive of travel distances from α-to-β bi-directionally. Therefore, at
the end, molecules with higher velocities end-up in the second container
and slower molecules in the first, so that the temperature is greater in β
than in α, contrary to what the second law of thermodynamics states.
• Conjecture 1. The second law is not violated by Hypothesis 1, because
no matter what method is used by the hypothetical demon, entropy is
increased by the work done by the demon in monitoring the particles
(pp. 54 and 55 of [4]), whilst reaching a state of equilibrium by the
sorted particles, gaining us information described as the range of energy
states, forming a group of highly kinetic particles on the β side, and a
group of the least kinetic particles on the α side within the QIT context.
Simulation Scenario A
Simulation Scenario A
• At the final steps of the simulation, we achieve a state of
equilibrium between energy states of both α 1:1 β contents
Water into Ice
CIT Analysis finding
key distance d
Water into Boiling
Steam
Results and Analysis
• We have conducted our analysis for the 25 molecules scenario
(a total of 50 in both partitions) by default.
Particle population is
easily changeable based
on user’s preference to
slide the bar “molecules
on each side”
between 2 and 2n, as
programmed within the
simulation program’s
dynamic modules.
Scenario A Formulaic Basis
• The simulation’s polynomial products allowed us to determine
key distance d values for particles moving about in α and β on
behalf of the demon.
• The polynomial expressions were symmetric and homogenous,
reflecting the characteristics and properties of our particles,
evenly distributed and transacted between α and β chambers.
• We have written the key formula based on (3.6) and (3.7) in code,
according to our CIT and QIT probabilities (previous slides), as
• 𝐄𝐱𝐩𝐚𝐧𝐝 𝐒𝐪𝐫𝐭
𝛼2
left +
𝛽2 left
𝐀𝐛𝐬 𝐌𝐞𝐚𝐧 𝐋𝐨𝐠 posl
𝑠𝑡𝑒𝑝𝑠
• An excerpt of our table from the main report showing different
steps of the simulation:
Scenario A Results and Analysis
Step
Sort
#
#
1
1
2
2
3
3
d inputs for
25 molecules on each α , β side
1.06*Sqrt[25.`4.*α^2 + 25.`4.*β^2]
,
1.11*Sqrt[25.`4.*α^2 + 25.`4.*β^2]
29.91*α^2 + 29.91*β^2 ≡ 𝐸𝛼→𝛽
High P selection;
Pw (Ed)   (d, t) ≥
0.5~1
1/25 = 0.04 < 0.5
, not eligible (ne)
1/25 = 0.04 < 0.5
𝐸𝛼 ∪ 𝐸𝛽 = {¼ =0.25 , ½
Outputs
H(αiβi)
dw
ne
0
− 𝑖 𝑝𝑖 log 2 𝑝𝑖 1
=
1.5
, 𝑛1 𝑛𝑖=1 𝐸𝑖 =31.835 = 0.5} , 𝑛1 𝑛𝑖=1 𝑃 𝐸𝑖
−0.375 log 2 0.375
2
= 0.375
33.76*α^2 + 33.76*β^2 ≡ 𝐸𝛽→𝛼
∴ since s – 1, =0.53
P
31.55*α^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2] +
½ = 0.5
2
31.55*β^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2]
, 28.8765 0.375
P 0.53
1.5
26.203*α^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2] +

26.203*β^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2]
¼ = 0.25
1
Scenario A Results and Analysis
Step
#
8
d inputs for
25 molecules on each α , β side
18.22*^6*α^8 + 4.90*^6*α^6*β^2 +
7.35*^6*α^4*β^4 + 4.90*^6*α^2*β^6 +
1.2*^6*β^8
, 3750000
649284.99*α^8 + 2.6*^6*α^6*β^2 +
3.89571*^6*α^4*β^4 + 2.6*^6*α^2*β^6 +
649284.99*β^8
High P selection;
Pw (Ed)   (d, t) ≥ 0.5~1
1 – 0.2 = 0.8
0.85
1 – 0.1 = 0.9 easier to
mutually reach and select
Entropy
H(αiβi)
0.199
dw
5
3
1
Results and Analysis
Step
#
Sort
#
8
8
d inputs for
25 molecules on each α , β side
1.22*^6*α^8 + 4.90*^6*α^6*β^2 +
7.35*^6*α^4*β^4 + 4.90*^6*α^2*β^6 +
1.2*^6*β^8
, 3750000
649284.99*α^8 + 2.6*^6*α^6*β^2 +
3.89571*^6*α^4*β^4 + 2.6*^6*α^2*β^6 +
649284.99*β^8
Interpretation
Out of a total
population of 8
energetic particles, 2
particles are the high
candidates to pass
through the trapdoor
with a mean distance of
3750000.
Summary of Scenario A results
• The polynomial degree distribution of α and β for each
distance value, obtaining a key distance coefficient, is
correspondingly symmetric, where the co-variance property of
α and β denotes this fact.
• The scalar of 3 events-point forms polynomial groups, one as
the α group, the other as β group plus their mean. Probability
weight Pw is derived based on the most and least travelled
distances by one or more particle candidates.
• Weight measurement provides a sectionalized area for each
group, αi and βi , as a set of 1 or more candidates, as partitions
of the polynomial within chamber α and β. Contents of each
chamber is selected based on the evaluation of weights.
• Hence, a specific candidate or particle is selected within the
sectionalized area, from a group, quite similar to a zooming
effect made on some region of path points (positions).
Summary of Scenario A results
• So, in a CIT sense, entropy is eventually 0; in a QIT sense where energy
states matter, selection of different states of energy within the same
chamber after the final sort (in this case step # 50), requires the demon
to measure from the start in further sorting the partition content.
• Scenario B becomes prioritized in measuring information with a total
entropy that once again grows in sub-partitioning the sorted groups
based on their energy signatures. Entropy in the QIT sense is measured
as states of energy, which is in turn > 0 in Scenario B.
• Within this context, the change of energy relative to temperature
contributes to our final measurement involving light. The current
scenario has an overall entropy change between steps 50 and 16, H =
H50 – H16 = –0.2488 minimum labor was required for the demon to sort
particles in the remaining steps, whereas, H = H16 – H2 = 0.395 as a
great labor for the demon was required to sort particles between steps
16 and 2 (step 1 is not eligible since no sorting was made).
Simulation Scenario B
• The demon prioritizes the selected event with less probability
weight (lighter weight)…
• The demon opens the door once the fast light, travelling faster
than all particles is projected and received, and decided upon
by motion measurement:
• It is the projection of light relative to particle motion in delivering
bits to the demon for real-time decisions.
• The demon knows entropy calculus based on information
content I(X), such that the information compilation and
interpretation is conducted through the ratio of logs according
to our entropy (next slides). Thus, quantifying a message
between α and β assessing whether the overall system ended
up having information.
• Thus, distance values as real values are measured, quantified
based on a derived P mean known to the demon.
Simulation Scenario B
Simulation Scenario B
Scenario B P’s and our entropy
2𝑗
2𝑖+1
∀𝑎𝑖 , 𝑎𝑗 ∈ 𝐴
;
𝐻 𝛼 → 𝛽 ≡ ∆ 𝐻𝑎 = −
∆ 𝑝𝑖,𝑗 log ∆ 𝑝𝑖,𝑗
𝑖=1
𝑗=𝑖+1
where
𝑃(𝑎7 , … , 𝑎𝑛 ) 𝑃 𝑎1 , 𝑎2 + 𝑃 𝑎3 , 𝑎4 + 𝑃 𝑎5 , 𝑎6
∆ 𝑝𝑖,𝑗 =
−
𝑛
3
𝐴 = 𝐴 𝑇↓ ∪ 𝐴 𝑇↑
,
such that the next atom is the j-th atom in the atoms A set given that the
present atom is the i-th atom, and H is our entropy rate for a multi-covariant probability pi,j as atoms/selection made by the demon.
• We have written the formula in Mathematica as follows:
(4.15)
𝐄𝐱𝐩𝐚𝐧𝐝 𝐒𝐪𝐫𝐭 (−(𝑐 ∗ 𝑖 2 ) ∗ rays ∗ lengths/(𝑠𝑡𝑒𝑝𝑠))
Results and Analysis
Simulation Step # 12 computing values at 12o K (–261.15 o C) for a particle population ratio
of
𝑚 𝑟𝑎𝑦𝑠
𝑛𝐴
= 1 by velocity magnitude
P’s and entropy
(atoms/selection)
a1, a2
relative to (a3, a4) , (a5, a6 )
and n A
Sqrt[3.6`4.*^8 - 1.141*^42*α^2]
a3, a4
relative to (a1, a2) , (a5, a6 )
and n A
Sqrt[3.6`4.*^8 - 1.79*^43*α^2]
a5, a6
relative to (a1, a2) ,
(a3, a4 ) and n A
Sqrt[3.6`4.*^8 - 6.11*^43*α^2]
∆ 𝒑𝒊,𝒋
∆ 𝑯𝒂
(198/200)–
(1+0+0)/3 = 0.65
0.91
2
Sqrt[3.6`4.*^8 - 1.53*^43*α^2]
Sqrt[3.6`4.*^8 + 1.5*^-43*α^2]
Sqrt[3.6`4.*^8 + 8.9*^44*α^2]
(198/200) – ( ½ +
½+0)/3 = 0.65
ne
3
Sqrt[3.6`4.*^8 - 3.51*^43*α^2]
Sqrt[3.6`4.*^8 - 2.28*^43*α^2]
Sqrt[3.6`4.*^8 - 1.46*^43*α^2]
(194/200)– (1/3 +
1/3+½ )/3 = 0.581
ne
4
Sqrt[3.6`4.*^8 + 1.4*^-43*α^2]
Sqrt[3.6`4.*^8 + 9.8*^-44*α^2]
Sqrt[3.6`4.*^8 + 1.4*^42*α^2]
(198/200)–
(0+0+1)/3 = 0.65
0.91
5
Sqrt[3.6`4.*^8 - 1.941*^42*α^2]
Sqrt[3.6`4.*^8 - 1.065*^42*α^2]
Sqrt[3.6`4.*^8 - 5.97*^42*α^2]
2.7
6
Sqrt[3.6`4.*^8 +
1.810811476199739*^43*α^2]
Sqrt[3.6`4.*^8 +
1.4079291227104131*^43*α^2]
Sqrt[3.6`4.*^8 - 2.1518*^43*α^2]
(194/200) – (1/3
+ ½ + 1/3)/3 =
0.581
ne
n
1
ne
Summary of results
• Atoms with higher velocities were spotted during transformations of
P’s indicating excited particles compared to the rest of the n
population (1 to 6 co-variant particles out of n) identified as eligible.
• Some particles continue to apprise to get transferred to β, whilst
others remain with the least energy state (kinetics of idle a’s <
kinetics of apprised a’s) performance in α by the demon until they
become eligible.
• In QIT, we have information as energy states, since its signatures for
each particle selection and transfer per simulation step was
processed and measured according to (4.15), ending with an H > 0,
• From a CIT viewpoint, the computed information compared to
Scenario A remains 0, since no error correction satisfying an erasure
of information was attempted, nor the conservation of energy states
from chamber α to chamber β maintained, whilst selections were in
place i.e., ne P’s against the eligible P’s.
Summary of information relation
analogues in CIT and QIT
Summary of information relation
analogues in CIT and QIT
Conclusions
• Some technical aspects of CIT and QIT computation and information were given.
• We have made comparisons relevant to both contexts and distinguished the
computational energy aspect of processing information from its content.
• We have examined key concepts such as Maxwell’s demon, and employed CIT
and QIT methods in terms of two simulation scenarios A and B.
• The objective was to interpret the gained information through even-distribution
of probability states of events, as well as directional probability distribution
relative to issues raised from within the fields of thermodynamics.
• We finally demonstrated that Maxwell’s Demon metric as a key distance
measured between the elements of a set of events, travelled by subjects:
• no matter how stationary but relativistically travelling distances by other
neighboring co-occurring events useful to determine the whereabouts and
selection of near future events.
• Ergo, the choice of subject candidates (like kinetic particles) in manifesting a
pending or immediate event, is in our grasp.
~ One could say, prediction is an art when one crafts it beforehand with reliable P’s
as far as the predication comes true, otherwise, the P system still lacks in
embedding a useful metric to provide us a reliable prediction. ~
P. B. Alipour
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