Information, Entropy and Reversible Computation. Michael C. Parker 1 , and Stuart D. Walker 2. 1:Fujitsu Laboratories of Europe Columba House, Adastral Park, Ipswich IP5 3RE, U.K. [email protected] .com , [email protected] 2: University of Essex - PowerPoint PPT Presentation
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Michael C. ParkerMichael C. Parker11, and Stuart D. Walker, and Stuart D. Walker22
1:Fujitsu Laboratories of Europe 1:Fujitsu Laboratories of Europe Columba House, Adastral Park, Ipswich IP5 3RE, U.K.Columba House, Adastral Park, Ipswich IP5 3RE, U.K. [email protected] .com , [email protected]@fle.fujitsu .com , [email protected]: University of Essex2: University of Essex Dept. of Electronics Systems Engineering,Dept. of Electronics Systems Engineering, Wivenhoe Park, Colchester, Essex, CO4 3SQ, U.K. Wivenhoe Park, Colchester, Essex, CO4 3SQ, U.K.
• How ‘Physical’ is Information?• Does Information obey Physical Laws?• Can Information travel faster than speed of light?• Does Information require energy to process?• Is Reversible Computation possible?• What about Information and Entropy?• What about Quantum Information? (Ask me questions at end, if there’s time!)
• Mathematics of Information• Fourier Transforms• Complex Function Theory (Cauchy-Riemann)• Maxwell’s Equations
• Erasure of information requires energy, e.g. setting a computer register to zeroLandauer’s PrincipleLandauer’s Principle
1 1 1 10 0 0 01 10 1
0 0 0 00 0 0 00 00 0
E Input Energy Required
•Creation of information doesn’t require extra energy input, e.g. equivalent to diffusion, Brownian motion, is not dissipative, i.e. doesn’t ‘yield’ energy.•Implies ‘processing of information’ or calculation requires no intrinsic energy.
Reversible Computing - Toffoli GateReversible Computing - Toffoli Gate• Reversible Logic requires equal number of i/p and o/p gates• No information may be lost
Information is also defined by:• Discontinuities & Points of Non-Analyticity
• It is also distributed/contextual and localised, so has Holographic characteristics
Definition & Properties of InformationDefinition & Properties of InformationDiscrete (Shannon) Information lni i
i
I
i p x x
ln lnI p x x p x x p x p x dx dx
lndiffI p x p x dx
However the divergence is ‘constant’ for all space (x), but when considering transfer of information from A to B, we are interested in differences in information at A and B. Hence the differential (a.c.) information is given by:
Maxwell’s Equations are an example of the Cauchy-Riemann equationsMaxwell’s EquationsMaxwell’s Equations
D dDH J
dt
dBE
dt 0B
3-D
dA dE
dx dy dE dA
dx dy
, , ,F x t E x t iA x t
The Electric and Magnetic fields form an analytic function in space-timeThe Electric and Magnetic fields form an analytic function in space-timeAlternative explanation for Wave-Particle Duality for lightAlternative explanation for Wave-Particle Duality for light
1/c /Z
A HZy ct
z x ict x iy
Speed of Light Impedance of Medium
Ohm’s LawTime is the imaginary axis (c.f. Space-time continuum)
d dEH
dx dt
dE dH
dx dt
0J
D EB H
d EdH
dx dt
d HdE
dx dt
1-DAssume no currents & charges(e.g. dielectric medium)
All roots of the denominator polynomial must be in the upper half-plane e.g. Trig functions do not satisfy this - hence cannot be physical information-bearing signals
Hurwitz Polynomials
Paley-Wiener Theorem (also from Causality)
Some Criteria for Physical FunctionsSome Criteria for Physical Functions
Hence, function must be bounded both in space and in time, and tend to zero at infinity. e.g. Gaussian function does not obey Paley-Wiener, and it is not causal (bounded).
Information = Sum of ResiduesInformation = Sum of Residues
Hence, a holomorphic function (entire across the z-plane) has no points of non-analyticityand so contains no differential information. It cannot be used for information transfer.
Likewise, a function which allows analytic continuation from x to x0 contains no informationbetween those points, so that no information is transferred between x and x0.
f t
0t t
“Superluminal” pulse
The “superluminal” pulse allowing analytic continuation from t to t0
transfers no information between these points, since R=0,so that zero information transfer takes place, let alone “superluminal”
information transfer.
Theoretically, “zero” information can be superluminally transferred!! (But that’s not saying very much!)
Information is contained in the points of non-analyticity, and discontinuities etc.
Information & EntropyInformation & EntropyInformation is contained in points of non-analyticity, points of discontinuity etc.Hence, from Sommerfeld/Brillouin information cannot travel faster than c through a medium
Information is inimical to adiabaticity, since “slow moving” and “smooth” conditions do not apply to points of non-analyticity (poles) or discontinuities.
Entropy must increase when information is transferred.
Due to the effects of diffraction (space), or dispersion (time) all information signals will tend to reduce in magnitude, or be absorbed when travelling through space. This leads to information loss (i.e. entropy increase, c.f. Brillouin) or reduction in SNR (also equivalent to entropy increase.)
Passive Media
This is in agreement with the impossibility of noiseless amplification:• Noise must always increase after amplification (3dB minimum optical Noise Figure)• “No-cloning Theorem” also states that perfect (noiseless) duplication of quantum states is impossible.
Active Media
Either way, information transfer is accompanied by an increase in entropy
Dynamic Formulation of Landauer’s PrincipleDynamic Formulation of Landauer’s Principle
• Erasure of information requires energy
Static Case (Conventional form of Landauer’s Principle)
• Entropy increases with erasure of information (Brillouin: S = -I )
Dynamic Case
• Transfer of information is associated with an increase in entropy
• Transfer of information must require energy
• Transfer of information from A to B is equivalent to:• Erasure of Information at A (Landauer’s Principle states this requires energy)• Re-creation of Information at B (Landauer’s Principle doesn’t require energy
• contain infinitely redundant information• zero differential information• can’t be used to transfer information
• Information is associated with points of non-analyticity and discontinuity • Inimical to adiabaticity• Transferring information from A to B requires a change in entropy
• This is in accordance with:• dispersion & diffraction (signals tend to degrade when moved in space)• impossibility of noiseless amplification (3dB minumum optical noise figure)• quantum no-cloning theorem
• Information cannot propagate faster than the speed of light in vacuum c• superluminal information transfer is impossible
• Physical computers are finite in size• Information is moved/shuffled around during a computation• Computation must dissipate energy, and cannot be reversible.