Classical and Quantum Information TheoryInformation and Computation of micro and mega-scale subjects via Maxwell’s Demon metric By:

Philip B. Alipour

ELEC 590 Directed StudyLevel: Ph.D.Course Instructor: Dr. T. A. Gulliver

August 3, 2012

University of Victoria 

Department of Electrical and Computer Engineering

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

The two parameters α and β are both complex numbers, and 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 co-

occur in form of 00, 01, 10, 11’s and interpret as information? • Yes, this parallel logic in a quantum system could be given as co-

variance 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. for all then

where its distribution mean appears as

• 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.

Our QIT and CIT probabilities coded in the simulation

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

which represents an existing 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:

• • , J 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 bi-

directional 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

Water into Boiling Steam

CIT Analysis finding key distance d

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

• An excerpt of our table from the main report showing different steps of the simulation:

Scenario A Results and AnalysisStep #

Sort#

d inputs for 25 molecules on each α , β side

High P selection; Pw (Ed) (d, t) ≥

0.5~1

OutputsH(αiβi) dw

1 1 1.06*Sqrt[25.`4.*α^2 + 25.`4.*β^2] ,1.11*Sqrt[25.`4.*α^2 + 25.`4.*β^2]

1/25 = 0.04 < 0.5 , not eligible (ne)1/25 = 0.04 < 0.5

ne 0

2 2 29.91*α^2 + 29.91*β^2 , 31.835

33.76*α^2 + 33.76*β^2

= {¼ =0.25 , ½ = 0.5} , = 0.375 since s – 1, P

=

0.53

11.52

3 3 31.55*α^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2] + 31.55*β^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2]

, 28.876526.203*α^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2] + 26.203*β^2*Sqrt[25.`4.*α^2 + 25.`4.*β^2]

½ = 0.5 0.375 ¼ = 0.25

P

0.53

2 1.5 1

Scenario A Results and Analysis

Step #

d inputs for 25 molecules on each α , β side

High P selection; Pw (Ed) (d, t) ≥ 0.5~1

Entropy H(αiβi) dw

8 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

, 3750000649284.99*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

1 – 0.2 = 0.80.851 – 0.1 = 0.9 easier to mutually reach and select

0.199

53 1

Results and Analysis

Step#

Sort#

d inputs for 25 molecules on each α , β side

Interpretation

8 8 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

, 3750000649284.99*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

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

where

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-co-variant probability pi,j as atoms/selection made by the demon.• We have written the formula in Mathematica as follows:

(4.15)

Results and Analysis Simulation Step # 12 computing values at 12o K (–261.15 o C) for a particle population ratio

of by velocity magnitudeP’s and entropy (atoms/selection)

n a1, a2 relative to (a3, a4) , (a5, a6 ) and n A

a3, a4 relative to (a1, a2) , (a5, a6 ) and n A

a5, a6 relative to (a1, a2) , (a3, a4 ) and n A

1 Sqrt[3.6`4.*^8 - 1.141*^-42*α^2]

Sqrt[3.6`4.*^8 - 1.79*^-43*α^2]

Sqrt[3.6`4.*^8 - 6.11*^-43*α^2]

(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]

(194/200) – (1/3 + ½ + 1/3)/3 = 0.581

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]

ne 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

References1. M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information, pp. 136 to 164,

Cambridge University Press, 2000. 2. G. F. Lawler, Random Walk and the Heat Equation, SAMS American Mathematical Society, 2010. 3. J. Stolze and D. Suter. Quantum Computing, A Short Course from Theory to Experiment, Wiley-VCH,

2004.4. C. Seife, Decoding the Universe, Penguin Group Publishers, Inc. 2006.5. D. Joyner (Ed.) Coding Theory and Cryptography, from Enigma and Geheimschreiber to Quantum

Theory, Springer, 1999. 6. S. Roman. Introduction to Coding Theory and Information Theory, Springer, 1997.7. Wolfram Mathematica i. simulation toolkit projects: Boltzmann Gas (Accessed, May 2012); ii. T.

Rowland. "Velocity Vector." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/VelocityVector.html

8. L. Kirkup, Experimental Methods: An Introduction to the Analysis and Presentation of Data, John Wiley & Sons, 1994

9. K. Edamatsu, Quantum physics: Swift control of a single spin, Nature 456, 182-183 (13 November 2008)

10. D. A. McQuarrie, (2000). Statistical Mechanics (revised 2nd ed.). University Science Books. pp. 121–128. ISBN 978-1-891389-15-3.

11. P. J. Mohr, N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants," or visit http://arxiv.org/abs/1203.5425 as the 2012 version.

12. A. Gray, "Metrics on Surfaces." Ch. 15 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 341-358, 1997.