Information, Entropy and Reversible Computation

1. Information, Entropy and Reversible Computation Michael C. Parker1, and Stuart D. Walker2 1:Fujitsu Laboratories of Europe Columba House, Adastral Park, Ipswich IP5 3RE, U.K. m.hisatomi@fle.fujitsu .com , m.parker@fle.fujitsu.com 2: University of Essex Dept. of Electronics Systems Engineering, Wivenhoe Park, Colchester, Essex, CO4 3SQ, U.K.

2. Introduction Any meaningful discussion of Information immediately opens a large Pandora’s Box of questions, and requires a startlingly wide knowledge of physics! • 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 • Miscellaneous Physics (and Meta-Physics) • Diffraction & Dispersion • Noise, e.g. Amplified Spontaneous Emission (ASE) • Causality • Holograms

3. Historical Background to Information • Entropy Boltzmann (1872) • Information content Shannon (1948) • Information=negentropy Brillouin (1956) • ‘Information is Physical’ Landauer (1962) • ‘Everything is Information’ Wheeler (1998) Boltzmann Shannon Brillouin Landauer Wheeler

4. ‘Fast’ Information Transfer ? “Superluminal” Light Nature, vol.424, p.638, 7th August 2003 Wang et al., Nature, vol.406, p.277, 20th Jul’00

5. Information obeys Physical Laws • Information/signals cannot propagate faster than c Einstein (1905) • Wavefront travels at c through any medium Sommerfeld/Brillouin (1914) The signal is always slower than c Sommerfeld Brillouin

6. Landauer’s Principle 1 0 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 1 0 • Erasure of information requires energy, e.g. setting a computer register to zero Information I = High Input Energy Required 0 0 0 0 0 0 0 0 0 0 0 0 Information I = Low • 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.

7. Reversible & Irreversible Computing A A A C NOT C C AND XOR B B A C A B C A B C 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 • Erasure or Loss of information requires energy • Conventional Logic, e.g. AND, XOR gates • tends to ‘lose’ information • dissipates energy • irreversible i/p 1 bit o/p 1 bit i/p 2 bits o/p 1 bit i/p 2 bits o/p 1 bit IRREVERSIBLE IRREVERSIBLE REVERSIBLE

8. Reversible Computing - Toffoli Gate A B C C’ 0 0 0 0 0 1 0 0 A 1 0 0 0 B A A 1 1 0 1 B C’ AND B A=1 C=0 B=1 C’=C NOT A B C C’ A B C C 0 0 1 1 0 0 0 i/p 3 bits o/p 3 bits C C’ i/p 3 bits o/p 3 bits 0 1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 1 1 0 REVERSIBLE 1 1 1 REVERSIBLE 1 0 • Reversible Logic requires equal number of i/p and o/p gates • No information may be lost A A’=A A’ B B’=B B’ TOFFOLI C C’ C’=C, unless A=B=1=> C’=C

9. Definition & Properties of Information Differential (Continuous) Information Diverges to infinity as a.c. d.c. 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: • Information is also defined by: • Discontinuities & Points of Non-Analyticity • It is also distributed/contextual and localised, so has Holographic characteristics Discrete (Shannon) Information

10. Analytic/Holographic Functions e.g. A function is Analytic if it obeys the Cauchy-Riemann equations

11. Wave-like Delocalised Particle-like Localised Properties of Analytic/Holographic Functions • Nature of Holograms • Complete image is represented over entire space • Shattered ‘particles’ each carry a representation of the overall image Denis Gabor • Complementary Characteristics • Wave-Particle Duality

12. Holographic Functions: Analytic Continuation I know what f (z) is - I’m standing there! Q: What is f (z0) ? Q: What is f (z1) ? • Holographic/Analytic functions have a Wave-Particle Duality • It is a wave function • fully delocalised • Also completely defined from any single point • all ‘information’ is contained at any single point, anywhere in space. • fully localised

13. Has ‘Holographic’ Properties Analytic Continuation Integral Form of Cauchy-Riemann Equations Dispersion Relations (Causality) e.g. refractive index Summary of Analytic-Holographic Functions A function is Analytic if it obeys the Cauchy-Riemann equations

14. Causality & Fourier Transforms Complex frequency plane real frequency axis imaginary frequency axis [s-1] [is-1] obeys the Cauchy-Riemann equations: Fourier Transform of a causal function (i.e. bounded in time) is an analytic function[1] [1] J.S. Toll, “Causality and the dispersion relation: logical foundations”, Phys Rev, 104(6), p.1760-70, 1956

15. Fourier Transform of a causal function (i.e. bounded in time) is an analytic function Far-field Fresnel-Kirchhoff Diffraction Integral ‘Complex’ space plane [m] real space axis imaginary space axis [im] obeys the Cauchy-Riemann equations: x X G(X) g(x) Far-field (Fraunhofer) diffraction yields an analytic diffraction pattern Young’s Slits Interference Pattern is Holographic R  FT of a bounded function (e.g. bounded in space) is also an analytic function:

16. Maxwell’s Equations 3-D 1-D Assume no currents & charges (e.g. dielectric medium) Speed of Light Impedance of Medium Time is the imaginary axis (c.f. Space-time continuum) Ohm’s Law The Electric and Magnetic fields form an analytic function in space-time Alternative explanation for Wave-Particle Duality for light Maxwell’s Equations are an example of the Cauchy-Riemann equations

17. Analytic Continuation “Superluminal” pulse “Leading Edge” of pulse contains complete information about the rest of the pulse shape Analytically-continued “superluminal” pulse Q: Is information “transferred” when analytic continuation takes place?

18. Holomorphic (Holographic) Functions Meromorphic Functions Analytic function with discrete points of non-analyticity, e.g. a pole in the z-plane Analytic function which is entire across the z-plane i.e. does not have any points of non-analyticity Analytic function which is entire across the z-plane i.e. does not have any points of non-analyticity Closed contour integral is zero i.e. sum of Residues is zero Closed contour integral is equal to the sum of the Residues Contour Integration Cauchy-Riemann Equations

19. Some Criteria for Physical Functions Paley-Wiener Theorem (also from Causality) Square integrability 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). Hurwitz Polynomials 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 These theorems are well-known from filter theory.

20. Analytic solution to a wave equation r & s are purely real functions is the Analytic Continuation Operator, such that Complex Conjugate (does not commute) g & h , u & v are purely real functions, respectively obey C-R equations plane How much information is in a Holographic Function?

21. Hence, a holomorphic function (entire across the z-plane) has no points of non-analyticity and 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 information between those points, so that no information is transferred between x and x0. The “superluminal” pulse allowing analytic continuation from t to t0 transfers no information between these points, since SR=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!) “Superluminal” pulse Information = Sum of Residues Information is contained in the points of non-analyticity, and discontinuities etc.

22. Information & Entropy Active 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. Passive Media 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.) Either way, information transfer is accompanied by an increase in entropy Information 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.

23. Static Case (Conventional form of Landauer’s Principle) • Erasure of information requires energy • Entropy increases with erasure of information (Brillouin: DS = -DI ) Dynamic Case • Transfer of information must require energy • Transfer of information is associated with an increase in entropy • 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 for this) Dynamic Formulation of Landauer’s Principle

24. A smaller computer will be more energy efficient. Reversible Computation? Computation tends to ‘shuffle’ information around: • A physically-realisable computer must have a finite size. • If ‘shuffling’ information is over a finite space, information is being moved about. • Energy must therefore be dissipated, and entropy increase. Bottom Line: Reversible computation is only true for an infinitely-small computer

25. Conclusions • Perfectly holographic analytic functions • 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.