The No Cloning Theorem Versus the Second Law of Thermodynamics

arXiv:quant-ph/0202109v3 21 Jan 2004 nfil icsigteiseo hsppra ela o ver for as well avail as his paper and this honesty of issue intellectual suggestions the his discussing for fairly on Peres Asher prof. h er fteirdcblt fQatmInformation Quantum one. classical of the irreducibility to the at Theory of lying Theorem heart states case, the nonorthogonal the non- with the for equivalence two Indistinguishability not its of by is clone implied this to is anyway, that able with Thermodynamics; connection no gate have Quantum to quantum seem would states, a orthogonal building of [8]). [7], Landauer, [6], (by [5], [4], demons [3], Mawxell’s Zurek of and Bennett en- presence thermodynamical in the and algorithmic- tropy the to Wooters of contribution comprehension Dieks, the information’s (by and Mechan- [2]) Theorem [1], Quantum No-Cloning Zurek of Foundation the the ics: at looking of way URL: ∗ ak..0o yPDtei 1] shr extensively here is [10], PHD-thesis analyzed. my of mark7.3.10 violated. be to Principle Second the preventing thermodynamical entropy, semi- the Peres’ to im- Landauer- of contribute membrane that algorithmic-information the permeable the demon, account also Mawxell’s that into on ply take results Zurek’s doesn’t Bennett- it since rect, entropy. a Universe’s in the lowering used of are order in states, way nonorthogonal suitable of distinguish analysis be to to the able absurdum ”magic” on ad some assumed which based membranes, in semi-permeable is engine statement thermodynamical cyclic this a Thermodynamics; of of proof in Law Second his necessary the Indis- is preserving of of states Theorem order the nonorthogonal that for claims tinguishability Peres Asher [1]) book lcrncades [email protected]; address: Electronic n[](swl si h 9 the in as well (as [9] In impossibility the stating theorem, No-cloning The our decades, two last the in changed, have results Two hscnieain led rsne ntere- the in presented already consideration, This cor- not is proof a such anyway, show, will we As http://www.gavrielsegre.com h oCoigTermvru h eodLwo Thermodynami of Law Second the versus Theorem Cloning No The ASnmes 36.a,0.0,03.65.-w , 05.30 , accoun 03.67.-a into numbers: semi-pe PACS take the to of Entropy not Thermodynamical shown the is to Thermodynamics contribution of Law Second se ee’pofta ilto fteN lnn Theorem Cloning No the of violation a that proof Peres’ Asher .INTRODUCTION I. th hpe fhswonderful his of chapter ol togyt thank to strongly would I ; useful y are Segre Gavriel ability aigteinrpouto q21ade.. n obtains one eq.2.2 and equation: eq.2.1 the of product inner the Taking and: de n w upt de uhta hr xs nor- a exist there state that start such edges malized outputs two and edges and rmwihi olw that follows it which from oie e nwrt o’ araepooa sending states proposal possible marriage two of Bob’s one to him answer her codiﬁes then is states, nonorthogonal two a proved. cloning of of impossibility able the device stating Theorem, No-Cloning The rvosycnoddrl that rule concorded previously | ntercie tt h esrmn ecie ythe by described measurement the positive-operator-valued-measure state received the on re ogeswa h ne a sn oerule some using was i Bob index measurement the the f what on guess outcome to the tries on Depending j. t fNnrhgnlSae ttn htif that stating States Nonorthogonal of ity guess. | fAiehsacpe i araeproposal. marriage his accepted has Alice if h odto htBbcnifrAiesase a be may answer Alice’s constraint: infer the can by formalized Bob that condition the ψ ψ ( 2 2 j e su osdraqatmgate quantum a consider us us Let e snwcnie ieetstaini hc Alice which in situation diﬀerent a consider now us Let ewl nwpoeteTermo Indistinguishabil- of Theorem the prove know will We nrdcdteoperators: the Introduced ,where ), > > ∗ | ψ r ootooa tflosta o antinfer cannot Bob that follows it nonorthogonal are 2 means I OCOIGTERMAND THEOREM NO-CLONING II. > mal ebae fPrs engine. Peres’ of membranes rmeable uhthat: such NITNUSAIIYOF INDISTINGUISHABILITY OOTOOA STATES NONORTHOGONAL f ( no U U ˆ ˆ · ψ < h algorithmic-information’s the t ol ml ilto fthe of violation a imply would ersnsterl eue omk the make to uses he rule the represents ) ψ < | | .T nwAiesase,Bbmakes Bob answer, Alice’s know To ). ψ ψ E ˆ 2 1 i i | 1 | E > > ˆ > s | ψ < ψ := i | | | 2 ψ > s > s > i 1 j n w itntvectors distinct two and | : > 2 ψ f X = = ( = 2 j 1 = )= > ψ < | | | ψ ψ ψ i (2.4) 0 = M 1 | 2 1 ˆ ψ { i > 1 > > j 1 † M | M ˆ 1 = ψ ˆ and | | > j ψ ψ 2 j } U 2 1 > ˆ j 2 means , =1 > > (2.6) 2 | ihtoinput two with ψ cs ihoutcome with 2 > | ψ yes wt the (with 1 > | while ψ (2.2) (2.1) (2.3) (2.5) i and 1 > = 2

ˆ I ˆ Since i Ei = it follows that i < ψ1 Ei ψ1 > = expenditure of work. This is the second law of 1; assuming the distinguishability condition| of| eq.2.6 it thermodynamics, and it is undoubtedly true as long as P P follows that < ψ1 Eˆ2 ψ1 > = 0 and thus: we can deal with bodies only in mass, and have no | | power of perceiving or handling the separate molecules Eˆ ψ > = 0 (2.7) of which they are made up. But if we conceive a being 2| 1 whose faculties are so sharpened that he can follow q Let us now suppose ad absurdum that < ψ ψ > = 0. every molecule in its course, such a being, whose 1| 2 6 attributes are still as essentially finite as our own, It follows that there exist a state ψ⊥ > and two complex numbers α and β such that: | would be able to do what is at present impossible to us. For we have seen that the molecules in a vessel full of air at uniform temperature are moving with velocities by ψ2 > = α ψ1 > + β ψ⊥ > (2.8) | | | no means uniform, though the mean velocity of any α 2 + β 2 = 1 (2.9) | | | | great number of them, arbitrary selected, is almost β < 1 (2.10) exactly uniform. Now let us suppose that such a vessel | | < ψ1 ψ⊥ > = 0 (2.11) is divided in two portions, A and B, by a division in | which there is a small hall, and that a being, who can Conseguentially: see the individual molecules, opens and closes this hole so as to allow only the lower ones to pass from B to A. He will see, thus, without expenditure of work, raise the Eˆ ψ > = β Eˆ ψ⊥ > (2.12) 2| 2 2| temperature of B and lower that of A, in contradiction q q and hence: with the second law of thermodynamics”; cited from the last but one section ”Limitation of the Second Law of th 2 Thermodynamics” of the 22 chapter of [11] < ψ Eˆ ψ > = β < ψ⊥ Eˆ ψ⊥ > 2| 2| 2 | | | 2| 2 2 In the 220 years after the publication of Maxwell’s book β < ψ⊥ Eî ψ⊥ > = β < 1 (2.13) ≤ | | | | | | an enormous literature tried to exorcize Maxwell’s demon i X in different ways; an historical review may be found in which contradicts the absurdum hypothesis. the first chapter ”Overview” as well as in the ”Chrono- Beside their apparent diversity, the No-Cloning logical Bibliography with Annotations and Selected Quo- Theorem and the Theorem of Indistinguishability of tations” of the wonderful books edited by Harvey S. Leff Nonorthogonal States may be easily proved to be equiv- and Andrew F. Rex [12] , [13]. alent: All these exorcisms were based on the idea that, to if Bob was able to distinguish the two nonorthogonal accomplish his task, Maxwell’s demon necessarily causes states ψ1 > , ψ2 > Alice used to answer his marriage a thermodynamical-entropy’s raising causing the Second proposal,| he could| clone them by making the measure- Law to be preserved: ment distinguishing them and making at will multiple they anyway strongly differed in identifying the ele- copies of the state Alice had given him. ment of the demon’s dynamical evolution which is nec- Contrary, if the non-orthogonal states ψ1 > , ψ2 > essarily thermodinamically-irreversible: were clonable, Bob could easily distinguish| them| by coming to recent times, most of the Scientific Com- cloning them reapetedly in order of obtaining the states munity (not only of Physics: cfr. e.g. the third chapter n n ψ1 > , ψ2 > whose inner product tends to zero ”Maxwell’s Demons” of [14]) strongly believed in Leon when| Nn | andN are, conseguentially, asymptotically Brillouin’s exorcism [15], identifying such an element in distinguishable→ ∞ by projective measurements. the demon’s information-acquisition’s process. When anyone thought that the ”The-end” script had at last appeared to conclude ”The Exorcist” movie, Charles III. BENNETT’S THEOREM ON MAXWELL’S H. Bennett showed in 1982 [4], [5], [16], basing on the DEMONS IN CLASSICAL THERMODYNAMICS previous work by Rolf Landauer on the Thermodynamics of Computation [3], that: Almost all the greatest physicists of the last two cen- 1. Maxwell’s Demon was still alive owing to the turies has, at some point, fought against one of the deep- nullity of Brillouin’s exorcism: the demon’s ac- est problems of Thermodynamics: Maxwell’s demon. quisition process may be done in a completelly Let us introduce it with Maxwell’s own words: thermodynamically-reversible way ”One of the best extablished facts in thermodynamics is 2. the necessarily-thermodinamically-irreversible ele- that it is impossible in a system enclosed in an envelope ment is instead demon’s information-erasure’s pro- which permits neither change of volume nor passage of cess heat, and in which both the temperature and the pressure are everywhere the same, to produce any The corner-stone of the Themodynamics of Computa- inequality of temperature or of pressure without the tion is Landauer’s Principle: 3

in this framework an arbitrary function is called Let us suppose to make the demon-computer operate logically-reversible if it is injective while it is called n times on n different molecules. thermodynamically-reversible if there exist a physical de- When n grows the demon, with no expenditure of work, vice computing it in a thermodynamically-reversible way; raises the temperature of B and lowers that of A. Landauer’s Principle states the equivalence of logical- But let us now analyze more carefully Clausius’s for- reversibility and thermodynamical-reversibility. mulation of the Second Principle: it states that no ther- An immediate consequence of Landauer’s Principle is modynamical transformation is possible that has as its that the erasure of information is thermodynamically- only result the passage of heat from a body at lower tem- irreversible: perature to a body at higher temperature. to prove it, it is sufficient to observe that to In the above process the passage of heat from A to any logically-irreversible function one may associate a B is not the only result: another result is the storage logically-reversible function different from the original in the demon-computer’s memory of the n-ple of inputs one in that the output is augmented by some of the in- ((s1, v1) , , (sn, vn)). ··· put’s information (usually called garbage); assuming ad To make the passage of heat from A to B to become the absurdum that garbage’s erasure is thermodynamically- only result of the process we could think that the demon, reversible, it would then follow that the original function at the end, erases his memory; but this, as we have seen, would be thermodynamically-reversible too, contradict- cannot be done in a thermodynamically-reversible way: ing the hypothesis. such an erasure causes an increase of entropy that may We can at last introduce Bennett’s exorcism of be proved to be greater than or equal to the entropy- Maxwell’s demon: conceptually Maxwell’s demon may decrease produced by the passage of heat from A to B. be formalized as a computer that: Bennett’s exorcism of Maxwell’s demon, has, anyway, a far reaching conseguence; supposed that the gas is de- 1. gets the input (s, v) from a device measuring both scribed by the thermodynamical ensemble (X , P ), let us the side s from which the molecule arrives and its introduce the Bennett’s entropy of P: velocity SBennett(P ) := H(P )+ I(P ) (3.3) p 2. computes a certain semaphore-function (s, v) p[(s, v)] giving as output a 1 if the molecule must be→ where: left to pass while gives as output a 0 if the molecule H(P ) := < log P > (3.4) must be stopped: − 2 specifically, the semaphore-function may be defined is Shannon’s entropy of the distribution P (i.e. its Gibbs’ through the following Mathematica expression [17]: entropy in thermodynamical language), while:

p[s− , v−] := If[s = Left, If[v vT , 0 , 1] , min x : U(x) = P if x : U(x)= P, ≤ I(P ) := {| | } ∃ If[v > v , 0 , 1]] (3.1) + otherwise. T ( ∞ (3.5) where vT is a fixed threshold velocity is its prefix-algorithmic-information (denoted simply as algorithmic information form here and beyond), 3. gives the output p[(s,v)] to a suitable device that i.e. the length of the shortest program computing it on operates on the molecule in the specified way the fixed Chaitin universal computer U (demanding to [18] for details we recall that a Chaitin universal com- Both the first and the third phases of this process, taking puter is a universal computer with prefix-free halting set into account also the involved devices, may be made in a and the property that, up to an input-independent addi- thermodinamically-reversible way. tive constant, it describes algorithmically any output in As to the second step, anyway, let us observe that a way more concise that any other computer). the semaphore-function p is logically-irreversible and Bennett’s exorcism of Mawxell’s Demon implies Ben- hence, by Landauer’s Principle, also thermodinamically- nett’s Theorem stating that the thermodynamical en- irreversible. tropy of the ensemble (X , P ) is equal to to its Bennett’s As above specified, such a thermodinamically- entropy: irreversibility may be avoided conserving the garbage; let us, precisely, suppose, that the demon-computer Stherm(P ) = SBennett(P ) = H(P ) (3.6) computes the thermodynamically-reversibly-computable 6 functionp ˜: To understand why Bennett’s exorcism implies eq.3.6 let us consider some example:

p˜[s− , v−] := If[s = Left , let us suppose, for simplicity, that the initial equilibrium probability distribution is such that the molecules If[v vT , ((s , v) , 0) , ((s , v) , 1)] , ≤ have one of only two possible velocities vL and vH , re- If[v > vT , ((s , v) , 1) , ((s , v) , 0)]] (3.2) spectively lower and higher than the threshold velocity 4 vT Let us now consider the case in which the initial distribution of molecules’ velocities is unfair: vL < vT < vH (3.7) P (v = vL) =1 α (3.12) Let us start from the case in which: − P (v = vH ) = α (3.13) 1 P (v = vL) = P (v = vT ) = (3.8) 2 where α = 1 . 6 2 The qualitative behaviour of the process is analogous Supposing that the demon memorizes in the cbit xn the value of the semaphore function of the nth molecule to the previously discussed one although the greater is the diﬀerence α 1 the littler is the amount of gas’ that he observes, we have that, at the beginning, the 2 probabilistic| information− | converted into algorithmic string ~xn := x1 xn seems to increase its length in an algorithmically-random··· way, i.e. in a way such that, in- information. formally speaking 1: Now, as we have already stressed, the involved thermodynamical process doesn’t violate the Second Principle of

I(~xn) n (3.9) Thermodynamics since the passage of heat from the low- ≈ temperatures-source A to the high-temperature-source B As the distribution of the molecules becomes more and is not the only result: an other result is the memorization more disuniform, with the slow molecules accumulating in demon’s memory of the sequence xn . { } on the left side and the speed molecules accumulating Such a memorization, that as we have seen is a trans- on the right side (i.e. when the temperature’s diﬀerence fer of information from the gas to the demon as well as among the two sides arises), the probability distribution a transfer of a portion of the overall information of the of xn becomes more and more unfair preferring for xn = Universe from probabilistic to algorithmic form, cor- 0, so that the string ~xn increases its deviation from Borel- responds to an accumulation in algorithmic form of normality. useful-energy (i.e. of energy that may be transformed Such an increasing regularity of xn corresponds to the in work), i.e. in an accumulation of thermodynamical- fact that its algorithmic-information becomes to increase entropy in algorithmic-form that has to be counted in more and more slowly. the Universe’s overall thermodynamical balance prevent- Reasoning in terms of a ﬁnite number N of molecules ing, indeed, the Second Principle to be violated. 2 , after a certain number nord of measurements made by the demon, the system reaches the state in which all the slow molecules are on the left side of the vessel, while all IV. THERMODYNAMICAL ENTROPY, the speed molecules are on the right side; from that point STATISTICAL MECHANICS AND THE further the demon stops every molecule so that: KOLMOGOROVIAN FOUNDATION OF INFORMATION THEORY xn = 0 n>nord (3.10) ∀ At this point, in which the demon has completed its task Despite Richard Feynman’s strongly authoritative of lowering the probabilistic information of the gas acclamation of the Landauer-Bennett’s results on so that such a probabilistic information ceases to de- Maxwell’s Demons (cfr. the section5.1.1 ”Maxwell’s De- mon and the Thermodynamics of Measurement” of [16]) crease, the algorithmic information of the string ~xn ceases to increase: and its appreciation by Nobel prize awarded theoretical physicists such as Murray Gell-Mann (cfr. e.g. the 15th chapter ”Time’s arrows” of [19]), these, and in particular I(~xn) = I(~xnord ) n>nord (3.11) ∀ Bennett’s Theorem, are far from having being accepted The whole process may, conseguentially, be seen as a by the Theoretical Physics’ community. transfer of information from the gas to demon’s memory The objections (implicitely or explicitely) moved to in which an amount of gas’ probabilistic information Bennett’s Theorem are essentially the following: is transferred to the demon as algorithmic information. 1. the Mawxell-demon’s issue simply shows that the Second Law has a statistical validity

2. the action of Mawxell’s demon moves the system 1 the sign ≈ of ”rough-equality” may be formalizied more precisely out of thermodynamical equilibrium: conseguen- by the condition that the l.r.h. grows at least as quicker than the tially the thermodynamical entropy ceases to be r.h.s. so that eq.3.9 may be rigorously considered as as a different defined way of stating Chaitin’s condition limn→∞I(~xn) − n = +∞ 2 This must be considered only as an artifice to clarify the argument, since one has to remember that the thermodynamical limit 3. the interdisciplinary attitude of Algorithmic N → ∞ has to be taken at the end; the qualitative behaviour Physics is not necessary to understand Thermody- here described becomes, in this limit, an asymptotic one namics 5

The first objection, i.e. the claim that the Second Law ratchet and pawl. Then when the shaft tries to jiggle has only a statistical validity, is the κoινη´ as far as the one way, it will not turn, and when it jiggles the other, Mathematical-Physics’ literature is concerned. it will turn. Then the wheel will slowly turn, and Such a claim is, anyway, false: perhaps we might even tie a flea onto a string hanging though Statistical Mechanics (historically pioneered by from a drum on the shaft, and lift the flea! Now let us Maxwell, Thomson and Boltzmann: cfr. e.g. the chap.3- ask if this is possible. According to Carnot’s hypothesis, 7 of [20]) allows to obtain the Equilibrium Thermody- it is impossible. But is we just look at it we see, prima namics of a macroscopic thermodynamical system deriv- facie, that it seems quite possible. So we must look more ing it from a probabilistic description of its underlying closely”; cited from the chapter 46 of [22] microscopic dynamics, Thermodynamics is a perfectly self-consistent physical theory predicting the value and streghtens Jean Perrin’s restatement of the idolatric dynamical evolution of all the thermodynamical observ- claim that the Second Law has only a statistical validity: ables of thermodynamical systems, (generally not in ther- ”But is must be remembered that the brownian modynamical equilibrium), with certainty. This occurs, movement, which is a fact beyond··· dispute, provides an in particular, as to the Second Law of Thermodynamics experimental proof (deduced from the molecular stating that in any thermodynamical cycle of any iso- agitation hypothesis) by which of means Maxwell, Gibbs lated thermodynamical system (generally not in thermo- and Boltzmann robbed Carnot’s Principle of its claim to dynamical equilibrium) one has with certainty that: rank as an absolute truth and reduced it to the mere δQ expression of a very high probability”; cited from the 0 (4.1) 51th section ”The brownian movement and Carnot’s T ≤ I Principle” of [23] where δQ is heat’s amount absorbed by the system while claiming that: T is its temperature. The source of the erroneous claim that Maxwell- ”It is important to keep in mind that here we are demon’s issue simply shows the statistical validity of the somewhat stretching the validity of thermodynamics Second Law may be understood in terms of the following laws: the above machines are very idealized objects, like words by Joel Lebowitz: the daemon. They cannot be realized in any practical way; one can arrange them to perform one cycle, ”The various ensembles commonly used in statistical but what one needs to violate the second law is the··· mechanics are to be thought of as nothing more than possibility of performing as many energy producing mathematical tools for describing behaviour which is cycles as required.Otherwise their existence ”only” practically the same for ”almost all” individual proves that the second law has only a statistical validity, macroscopic systems in the ensemble. While these tools a fact that had been well establishes with the work of can be very useful and some theorems that are proven Boltzmann. In fact an accurate analysis of the actual about them are very beautiful they must not be confused possibility of building walls semipermeable to colloids with the real thing going in a single system. To do that and of exhibiting macroscopic violation of the second is to commit the scientific equivalent of idolatry, i.e. principle runs into grave difficulties: it is not possible to substituting representative images for reality”; cited realize a perpetual motion of the second kind by using from the Introduction of [21] the properties of Brownian motion. It is possible to Such an idolatric attitude for which a thermodynami- obtain a single violation of Carnot’s law (or of a few of cal system is confused with its modellization trough Sta- them) of the type described by Perrin, but as time tistical Mechanics is, indeed, a typical mental attitude elapses and the machine is left running, isolated and of some Mathematical Physicists that had often induced subject to physical laws with no daemon or other even authoritative scientists to assert trivially erroneous extraterrestrial being intervening (or performing work statements of Thermodynamics; this is the case, for ex- accounted for), the violations (i.e. the energy produced ample, of Giovanni Gallavotti’s analysis of brownian mo- per cycle) vanish because the cycle will be necessarily tors that, misunderstanding the celebrated analysis by performed as many times in one direction (apparently Richard Feynman of a ”ratchet and pawl heat engine”: violating Carnot’s principle and producing work) as in the opposite direction (using it). This is explained in an ”Let us try to invent a device which will violate the analyis on Feynman, see [22] chapter 46, where the Second Law of Thermodynamics, that is a gadget which semi-permeable wall is replaced by a wheel with an will generate work from a heat reservoir with everything anchor mechanism, a ”ratchet and a pawl”, allowing it at the same temperature. Let us say we have a box of to rotate only in one direction under the impulses gas at a certain temperature, and inside there is a an communicated by the colloidal particle collisions with the axes with vanes in it. . Because of the bombardments valves of a second wheel rigidly bound to the same axis. of gas molecules on the··· vane, the vane oscillates and Feynman’s analysis is really beatiful, and remarkable as jiggles. All we have to do is to hook into the other end an example of how one can still say something of the axle a wheel which can turn only one way- the interesting on perpetual motion. It also brings important 6 insight into the related so-called ”reversibility paradox” significant entropy reductions possible may be thought of (that microscopic dynamics generates an irreversible as a large memory capacity in which the demon stores macroscopic world).”; cited from the section8.1 the information about the system which he acquires as ”Brownian motion and Einstein’s Theory” of [24] he works reducing the entropy. As soon as the demon’s memory is completelly filled, however, he can i.e. that a brownian motor can violate Carnot’s Law for achieve no further reduction of the Boltzmann··· entropy. a few cycles, a thing that, if it was true (that unfortu- He gains nothing for example, by deliberately forgetting nately this is not the case is shown, for example, in [25]), or erasing some of his stored information in order to would have allowed Gallavotti to definitively resolve the make more memory capacity available; for the erasure energetic problem of the World saving it from the slavery being a setting process, itself increases the entropy by an of oil. amount of at least as great as the entropy decrease made The second objection (implicitely or explicitely) moved possible by the newly available memory capacity”; cited to Bennett’s Theorem, namely that since the action of from [27] Mawxell’s demon moves the system out of thermody- in which, as it has been observed by Harvey S. Leff and namical equilibrium the thermodynamical entropy ceases Andrew F. Rex in the 1th-chapter ”Overview” of their to be defined, is based again on the idolatric attitude wonderful book [12], Olivier Penrose arrived very near to of making confusion between a thermodynamical system the right Bennett’s exorcism, though lacking to make the and its modellization through Statistical Mechanics de- final intellectual step to understand that erasure is the nounced by Lebowitz: fundamental act that saves Mawxell’s demon. the fact that there doesn’t exist a universally accepted Let us now explicitely show how Lebowitz’s remark notion of entropy in Nonequilibrium Statistical Mechan- against idolatry allows to confute the first objection to ics is consequentually seen a synonimous of the false Bennett’s Theorem, namely that the Second Law of statement that the notion of thermodynamical entropy Thermodynamics has only statistical validity: let us ana- of a thermodynamical system not in equilibrium is not lyze, at this purpose, the following pass in which Maxwell defined. himself explains what he wanted to show through the in- Such a confusion appears, for example, in the following troduction of his demon: passage by Gallavotti: ”This is only one of the instances in which conclusions ”One of the key notions in equilibrium statistical which we have drawn from our experience of bodies mechanics is that of entropy; its extension is consisting of an immense number of molecules may be surprisingly difficult, assuming that it really can be found not to be applicable to the more delicate extended. In fact we expect that, in a system that observations and experiments which we may suppose reaches under forcing a stationary state, entropy is made by one who can perceive and handle the individual produced at a costant rate, so that there is no way of molecules which we deal with only in large masses. In defining an entropy value for the system, except perhaps dealing with masses of matter, while we do not perceive by saying that its entropy is . Although one should −∞ the individual molecules, we are compelled to adopt what keep in mind that there is no universally accepted notion I have described as the statistical method of calculation, of entropy in systems out of equilibrium, even when in a and to abandon the strict dynamical method, in which stationary state, we shall take the attitude that in a we follow every motion by the calculus. It would be stationary state only the entropy creation rate is interesting to enquire how far those ideas about the defined: the system entropy decreases indefinitely, but at nature and method of science which have been derived at costant rate. Defining ”entropy” and ”entropy from examples of scientific investigation in which the production” should be considered an open problem”; dynamical method is followed are applicable to our cited from the section9.7 ”Entropy Generation. Time actual knowledge of concrete things, which, as we have Reversibility and Fluctuation Theorem. Experimental seen, is of an essentially statistical nature, because no Tests of the Chaotic Hypothesis” of [24] one has yet discovered any practical method of tracing or in the following passage by Olivier Penrose: the path of a molecule, or of identifying it at different times.”; cited from the last but one section ”Limitation ”Even in thermodynamics, where entropy is defined only of the Second Law of Thermodynamics” of the 22th for equilibrium states, the definition of entropy can chapter of [11] depend on what problem we are interested in and on and the following pass by Thomson (later Lord Kelvin): what experimental techniques are available”; cited from [26] ”’Dissipation of Energy’ follows in nature from the fortuitous concourse of atoms. The lost motivity is that is implicitly a kind of self-criticism as to the follow- essentially not restorable otherwise than by an agency ing analysis of Mawxwell’s-demon: dealing with individual atoms”; cited from [28] ”The large number of distinct observational states that They don’t say that the Second Principle of Thermo- the Maxwell demon must have in order to make dynamics have only a statistical validity, but a differ- 7 ent thing: that the usual link existing between such a and reducing to eq.4.3 in the equilibrium case. principle (that, not falling in the idolatry denounced by Under conditions explicitely formalizable, furthermore, Lebowitz, one have to remember to have an its own valid- the Local Equilibrium Condition, stating that the local ity in Thermodynamics) and Statistical Mechanics have and istantaneous relations between the thermal and me- to be modified as soon as entitities able to handle indi- chanical properties of a physical system are the same as vidual molecules are involved. for a uniform system at equilibrium, holds. In this case Having followed Lebowitz’s advise of not falling into eq.4.5 reduces to: the idolatric attitude of making confusion among a physical thermodynamical system and its modellization stherm(X,t) = stherm[u(X,t) , v(X,t) , nk(X,t)] (4.6) through Statistical Mechanics and, hence, preserving us from the error of confusing the difficulties involved in the i.e.: definition of entropy in Nonequilibrium Statistical Me- N ∂s ∂s ∂s chanics from the difficulties involved in defining entropy ′ ds = ( )v,nk + ( )u,nk + ( )u,v,nk′ ( for k = k) ∂u ∂v ∂nk 6 in Nonequilibrium Thermodynamics, let us briefly recall k=1 these latter: X (4.7) given a thermodynamical system made of N different Consequentially, if the one-parameter family of nonequi- species, the thermodynamical entropy of a thermody- librium thermodynamical states Xt satisfy the Local- namical state X is defined as: Equilibrium Condition, one has that the temperature in

X the point ~x of the system at time t may be simply ex- δQ pressed as: Stherm(X) := (4.2) REV T Z ∂s T (X , ~x, t) = [( ) ]−1 (4.8) where the integral is over a thermodynamically-reversible t ∂u v,nk tranformation starting in a fixed reference thermodynamical state O (to be ultimatively fixed by the Third Law of Returning at last to our Maxwell’s demon, let us observe Thermodynamics requiring that limT →0 Stherm(X) = 0) that its way of taking the whole thermodynamical sys- and ending in the state X. tem out of the thermodynamical equilibrium satisfies the If X is a state of thermodynamical equilibrium the conditions under which the Local-Equilibrium Condition thermodynamical entropy may be expressed as a func- holds. tion of the internal energy U, of the volume V and of the The third objection moved (implicitely or explicitely) number of moles of each contributing specie Nk: to Bennett’s Theorem, namely that the intedisciplinary attitude of Algorithmic Physics is not necessary to un- X equilibrium state derstand Thermodynamics, is certainly the subtler one. ⇒ Stherm(X) = Stherm[U(X) , V (X) , Nk(X)] (4.3) To analyze it, let us observe that the involved thermodynamical system is the compound system Gas + De- If X is not a state of thermodynamical equilibrium, any- mon. way, its thermodynamical entropy cannot be expressed As a mechanical system such a compound system is a anymore as a function of the internal energy U, of the classical dynamical system (Xcompound , Hcompound) with volume V and of the number of moles of each contribut- phase space: ing specie Nk: Xcompound := XGas XDemon (4.9) Xnonequilibrium state ⇒ [ Stherm(X) = Stherm[U(X) , V (X) , Nk(X)] (4.4) and hamiltonian: 6 This fact is often erroneously expressed as the claim that Hcompound := HGas + HDemon + Hinteraction (4.10) thermodynamical entropy is not defined out of equilib- ∞ ∞ rium: this is simply false, since the the operational defi- where HGas C (XGas), HDemon C (XDemon), and ∈ ∞ ∈ Hinteraction C (Xcompound). nition of Stherm(X) through eq.4.2 continues to hold. ∈ Simply one has, denoting with lower case letters the The mechanical description of the whole process is defined by the Hamiltonian flow Tt : Xcompound (intensive) densities of (extensive) quantities, that the → equation eq.4.3 of Classical Thermodynamics must be Xcompound induced by Hamilton’s equation: generalized by its expression in Generalized Thermody- dx namics, having the form [29]: = H , x (4.11) dt { }

stherm(X,t) = stherm[u(X,t) , v(X,t) , nk(X,t) , (IN) associating to any initial state xcompound = u(X,t) , v(X,t) , nk(X,t) , ∇ ∇ ∇ (IN) (IN) 2 2 2 (xGas , xDemon) Xcompound the ﬁnal state u(X,t) , V (X,t) , nk(X,t), ] (4.5) ∈ ∇ ∇ ∇ ··· xOUT := limt→∞ TtxIN . 8

The strategy of Statistical Mechanics would con- The approach underlying Bennett’s Theorem is a par- sist in deriving the macroscopic thermodynamical vari- tial application of the Algorithmic Physics’ approach in ables of the thermodynamical system Gas+Demon as which not the whole hamiltonian ﬂow Tt : Xcompound 7→ properly-deﬁned functions of a suitable statistical ensem- Xcompound is seen as a computational process, but only ble (Xcompound , Pcompound). its restriction as to the Demon Tt XDemon : XDemon | 7→ The ensemble (X , P ) involved in the formulation of XDemon.

Bennett’s Theorem, instead, doesn’t take into account The dynamical evolution of the gas Tt XGas : XGas | 7→ the demon: as we saw in the last section, it is the XGas continues to be described through Mechanics, i.e. equilibrium statistical ensemble (Xgas , Peq ) that Statis- owing to the enormous number of involved degrees of tical Mechanics would associate to the dynamical system freedom, through Statistical Mechanics. (XGas , HGas), the underlying reason for that deriving The third objection to Bennett’s theorem is based on substantially from the Algorithmic Physics’ attitude, as the observation that such an (hybrid) recourse to Algo- we will now explain. rithmic Physics is, at last, completelly avoidable: Algorithmic Physics is, by definition, that discipline why, for particular compound systems, should one to analyzing physical processes looking at them as compu- give up the usual, traditional approach of Statistical Me- tational processes according to the following correspon- chanics to derive the thermodynamical entropy in the dence’s table: usual way from the partition function of a Gibbs’s ensem- PHYSICAL PROCESS COMPUTATIONAL PROCESS ble for the dynamical system (Xcompound , Hcompound)? initial state input And which should be exactly these particular compound systems? dynamical evolution computation A minimal answer to the last question is immediate: final state output those particular compound systems in which XDemon and conseguentially applying the conceptual instru- and HDemon are such to result in the scattering pattern ments of Computation’s Theory. that, looking at the demon with the eyes of Algorith- Essentially owing to the overwhelming ”new age” folk- mic Physics, corresponds to the computational-process of lore by which it has been popularized in the divulga- computing the semaphore function p and making to pass tive literature, the interdisciplinary nature of Algorith- or to reflect the molecule correspondigly as described in mic Physics is looked by many theoretical and mathe- the last section; as we will see in the next section, any- matical physicists with great mistrust; as a consequence, way, such a class of particular compound systems may be also the beautiful and serious insight it has produced, considerably enlarged through a suitable characterization such as the investigations concerning the foundations of the notion of an intelligent system. of Computational Physics (i.e. of the discipline study- As to the former question, namely why for these coming the computer-simulation of physical systems) such as pound systems one should indeed to give up the usual Stephen Wolfram’s notion of computational irreducibility Statistical Mechanics’ approach, the answer is: simply (i.e. the situation in which the faster way of predicting because it is simpler. the final state of a dynamical system of known laws-of The third objection to Bennett’s Theorem is, with this motion is to simulate its whole dynamical evolution and regard, correct: there is no necessity of adopting the hy- to see what happens at the end) or his analyses concern- brid Algorithmic-Physics’approach on which Bennett’s ing the rule of Undecidability in Physics [30] 3 concretized Theorem is based on: by the work of Chris Moore and many others [35], [36], simply, given an arbitrary many-body physical system are ignored. like XGas, whenever its interaction with another physi- dσ cal systems gives rise to a scattering-cross-section dΩ of the particular kind specified above, the usual Statistical Mechanics’ approach, though still perfectly applicable, 3 such as Wolfram’s professional path has been characterized by is not the simpler approach since it doesn’t catch the his departure from academia in order of constituting Wolfram particular structural peculiarity of the analyzed system, Research Inc. producing Mathematica [17], his intellectual path has been characterized by an analogous non-conformism that led structural peculiarity that allows an alternative, more him to develop his seminal ideas culminating into a personal concise, explication that, according to Occam’s Razor, conception of randomness and complexity radically different by have conseguentially to be preferred. that of Algorithmic Information Theory [31] and based on what Such a passage from a purely probabilistic approach he calls the Principle of Computational Equivalence; the inter- to a hybrid mix of two approaches, the probabilistic and relations existing among such a viewpoint and Algorithmic In- formation Theory is analyzed in the first four sections ”Introduc- the algorithmic one, reflects itself in the the link between tion”, ”What Perception and Analysis Do”, ”Defining the No- Thermodynamics and Information Theory: tion of Randomness”, ”Defining Complexity” of the 10th chapter th in terms of the three different approaches to the defi- ”Processes of Perception and Analysis” and in the 12 chapter nition of information introduced by A.N. Kolmogorov in ”The Principle of Computational Equivalence” of [32], in the final section ”Afterthoughts...” of the 7th chapter ”Mathematics his fundamental papers on the Foundation of Informa- in the Third Millenium” of [33] and in the final section ”Final tion Theory [37] such a passage is exactly a passage from remarks” of the 4th part ”Future Work” of [34] an interpretation of thermodynamical entropy in terms 9 of the probabilistic approach alone to an interpretation tically a molecule according to a computed arbitrary bi- of thermodynamical entropy as a hybrid mix of the prob- nary predicate. abilistic and the algorithmic approaches. But one immediately realizes that such a generalization is wrong in that not every chosen semaphore-function corresponds to a resulting behaviour that seems to be V. ZUREK’S THEOREM ON MAXWELL’S intelligent. DEMONS IN QUANTUM THERMODYNAMICS Indeed, Szilard tells us, the particular ”intelligence” of the semaphore-predicate p derives from the fact that the Bennett’s work on Maxwell’s demon has been general- resulting algorithm performed by the demon acts on each ized by Wojciech Hubert Zurek in many respects [6], [7], molecule in order of lowering the probabilistic informa- [8], [38]. tion of the gas: we are tempted to say that it acts in a The first point analyzed by Zurek concerns the charac- clever way exactly since his behaviour seems to be tele- terization of the particular structural peculiarity of the ological, finalized to the objective of taking the gas in a dynamical system (Xcompound , Hcompound) under which more ordered state. the mix probabilistic + algorithmic approach underlying Let us observe, finally, that such an ordering-process Bennett’s Theorem may be applied: made by the demon acts necessarily out of thermodynam- up to this point we have assumed that Maxwell’s de- ical equilibrium, since its ”intelligence” is accomplished mon acts on a particular molecule in a very particular precisely by creating a ”clever” disuniformity in the spa- way: if the molecule arrives from the left side the demon tial distribution of a thermodynamical variable (in this makes it to pass unaltered if and only if its velocity is case the temperature). less or equal to a given threshold-velocity, acting in the We are conseguentially led to the following generaliza- opposite way if the molecule arrives from the right side. tion of Bennett’s theorem: Such a behaviour of the demon, as it was first observed the thermodynamical entropy of a classical many-body by Leo Szilard in his basic 1929’s paper [39], appears as system (Xm.b,Hm.b): a kind of intelligence. preliminary prepared in a state of thermodynam- While, taken too literally, Szilard’s paper was unfor- • ical equilibrium described, in Classical Statistical tunately also the source of many confusionary specu- Mechanics, by the statistical ensemble (X , P ) lations concerning the contribution of the Subject (or m.b eq the Cartesian Cogito in more philosophically palatable in a second time made to interact with an other • terms) to the Object’s thermodynamical entropy, some- physical system (Xint , Hint) that is intelligent: times appealing to the wrong claim that Subject’s measurements are necessary thermodynamically-irreversible may be expressed as: processes (whose falsity, as we have seen, may be directly derived by Landauer’s Principle; in the quantum Stherm(Peq ) = Iprob(Peq) + Ialg(Peq ) (5.1) case we are going to introduce, anyway, it was time be- where I (P ) and I (P ) are, respectively, the fore directly derived by Yakir Aharonov, Peter Bergmann prob eq alg eq probabilistic information and the algorithmic informa- and Joel Lebowitz [40]), it had the great merit of intu- tion of the ensemble (X , P ), where the intelligence- itively suggesting the essential structural peculiarity of m.b eq condition of (X , H ) is defined in the following way: Maxwell’s demon, allowing Zurek to generalize Bennett’s int int Theorem starting from the following questions: dσ 1. the scattering cross-section dΩ , seen from the point of view of Algorithmic Physics, corresponds to a de- 1. which is exactly the kind of intelligence showed by terministic algorithm f acting on a single molecule Maxwell’s demon? 2. in a way that takes the many-body system out of 2. can Bennett’s Theorem to be generalized to sys- thermodynamical equilibrium tems having the same kind of intelligence? 3. reducing its probabilistic information Observing that, according to the way we characterized it in the previous section, the structural peculiarity of Such a definition of an intelligent system is, indeed, Xdemon and Hdemon is to give rise to a scattering cross- a strenghtening of Gell-Mann’s notion of information dσ section dΩ that, from the point of view of Algorithmic gathering and using system (IGUS) defined as a Physics, corresponds to the prescribed algorithm of leav- complex adaptive system able to make observations ing to pass or reflecting elastically a molecule according (cfr. [19] and the 12th section of [41]) that, contrary to to the value of a computed semaphore-binary predicate these notions difficult to formalize, has a precise mathe- p(s,v), one could think that Bennett’s Theorem might matical meaning. be generalized to any situation in which Xdemon and A part from having generalized it to a strongly larger dσ Hdemon give rise to a scattering cross-section dΩ that, class of intelligent systems, Zurek’s main extension of from the point of view of Algorithmic Physics, corre- Bennett’s work concerns its extension to the quantum sponds to an algorithm leaving to pass or reflecting elas- domain: 10

Zurek’s theorem 4, the quantum analogue of Bennett’s 3. reducing its probabilistic information theorem, states that the thermodynamical entropy of a It is important to remark, at this point, that such a defi- quantum many-body system ( m.b, Hˆm.b): H nition of intelligence, applied in the quantum domain, is preliminary prepared in a state of thermodynami- indeed subtler, owing to the entanglement’s phenomenon ˆ • cal equilibrium described, in Quantum Statistical between the quantum many-body system ( m.b, Hm.b) ˆ H Mechanics, by the quantum statistical ensemble and ( int , Hint) having no classical analogue, that is H ( m.b,ρeq) itself used by ( int , Hînt) as to realize its teleological H action. H in a second time made to interact with an other • physical system ( int , Hînt) that is intelligent: H VI. THE THERMODYNAMICAL COST OF may be expressed as: ERASING THE MEMBRANES’ MEMORY OF PERES’ ENGINE Stherm(ρeq) = Iprob(ρeq ) + Ialg (ρeq) (5.2) In the previous sections we have introduced all the in- where Iprob(ρeq ) is the quantum probabilistic infomation gredients required to present the contribution of this pa- of the density operator ρeq, namely its Von Neumann’s entropy: per, namely the confutation of the claim, presented by Asher Peres in [9] as well as in the 9th chapter of his wonderful book [1], that the Theorem of Indistinguisha- Iprob(ρeq ) := Trρeq log ρeq (5.3) − bility for nonorthogonal states is necessary in order of while Ialg (ρeq) is the quantum algorithmic information preserving the Second Law of Thermodynamics. 5 of the ensemble ( m.b,ρeq) , namely: Peres’s argument is based on the assumption that the H thermodynamical-entropy of a quantum system is described by Von Neumann’s entropy, assumption min x : U(x) = ρeq if x : U(x)= ρeq, I(ρeq) := {| | } ∃ that he deeply analyzes explicitly reporting the cele- + otherwise. ( ∞ brated original calculus by which Von Neumann, in the (5.4) section5.2 of [45], computed the thermodynamical en- n i.e. the length of the shortest program computing it tropy of a quantum mixture pi , φi >< φi as if { | |}i=1 on the fixed Chaitin quantum universal computer U, each φi >< φi was a specie of ideal gas enclosed in where the intelligence-condition of the quantum system a large| impenetrable| box, and inferring that the ther- ˆ ( int , Hint) is defined exactly as in the classical case modynamical mixing entropy of the different species is throughH the following conditions: Iprob( i pi φi >< φi ). Peres reviews| Von| Neumann’s procedure in the follow- 1. the scattering cross-section dσ , seen from the point dΩ ing way:P of view of Algorithmic Physics, corresponds to an algorithm f acting on a single molecule ”It also assumes the existence of semipermeable membranes which can be used to perform quantum tests. 2. in a way that takes the many-body system out of These membranes separate orthogonal states thermodynamical equilibrium with perfect efficency. The fundamental problem here is whether it is legitimate to treat quantum states in the same way as varieties of classical ideal gases. This issue was clarified by Einstein in the early days of the 4 I would like to advise the reader that my presentation of Zurek’s ”old” quantum theory as follows: consider an ensemble theorem differs slightly from Zurek’s own ideas for which we of quantum systems, each one enclosed in a large strongly demand to the previously cited original Zurek’s papers 5 Zurek claims that the assumption of the Church-Turing’s Thesis impenetrable box, so as to prevent any interaction eliminates any dependence from the particular universal com- between them. These boxes are enclosed in an even puter U adopted by (or better constituting) the demon. He, in larger container, where they behave as an ideal gas, particular, claims that, by the Church-Turing’s Thesis, it doesn’t because each box is so massive that classical mechanics classical computer quantum computer matter if U is a or a ; is valid for its motion ( ). The container itself has so he substantially claims that Quantum Algorithmic Informa- ··· tion Theory collapses to Classical Algorithmic information The- ideal walls and pistons which may be, according to our ory, an arbitrary assumption for whose discussion I demand to needs, perfectly conducting, or perfectly insulating, or [10]. I will therefore assume, from here and beyond, that the with properties equivalent to those of semipermeable computer U in eq.5.4 is a Universal Quantum Computer. But membranes. The latter are endowed with let then observe that, in this way, one implicitly assumes that the quantum algorithmic information of a quantum state must automatic devices able to peak inside the boxes be defined in terms of classical-descriptions of such a state, as and to test the state of the quantum system enclosed claimed by Svozil [42] and Vitanyi [43], and not in terms of quan- therein.” (from the section9.3 of [1]) tum descriptions as it is claimed by Berthiaume, Van Dam and Laplante [44], an issue, this one, about which I demand once There is a point, anyway, of this review in which, delib- more to [10] erately, Peres moves away from Von Neumann’s original 11 treatment: since the state of the mixture-of-species is com- he doesn’t assume that the membranes separate • pletelly determined by ρ, and not by a particular nonorthogonal states with perfect efficiency as, in- its decomposition, to represent the actual state of stead, Von Neumann does: affairs by or by the Schatten’s decomposition of ρ: E ”Each system s , ,sn is confined in a box K , ,Kn 1 ··· 1 ··· whose walls are impenetrable to all transmission := (ρ−, e− >< e− ) , (ρ , e >< e ) (6.4) E1 { | | + | + +| } effects – which is possible for this system because of the 1 ρ± := (2 √2) (6.5) lack of interaction” (from the section5.2 of [45]) 4 ± The reason why Peres, contrary to Von Neumann, e± > := (1 √2)( 0 > + 1 >) (6.6) | ± | | doesn’t make such an assumption is that, according to him, this would imply a violation of the Second Law of is absolutely equivalent Thermodynamics; his argument is the following: if semi- let us now replace the two ”magic” membranes permeable membranes which unambiguously distinguish • with ordinary membranes able to distinguish only non-orthogonal states were possible, one could use them orthogonal species; since the e− >< e− -specie | | to realize the following cyclic thermodynamical transfor- and the e+ >< e− -specie are orthogonal, the re- mation for a mixture of two species of 1-qubit’s states, versible diffusion| of| the two species separate them, >< 1 > > < < the 0 0 -specie and the 2 ( 0 + 1 )( 0 + 1 ) - with the e+ >< e+ -specie occupying the left-half | | | | 1 | | } | | specie, both with the same concentration 2 of the vessel and the e− >< e− -specie occupying the right-half of the vessel.| | in the initial state the two species occupy two cham- • bers with equal volumes, with the 0 >< 0 - specie finally an isothermal compression takes the system occupying the right-half of the left-half| of the| vessel • in a situation in which the volume and the pressure and the 1 ( 0 > + 1 >)(< 0 + < 1 ) - specie occu- are the same of the initial state; such a compression 2 | | | | } pying the left-half of the right-half of the vessel requires an expenditure of work of:

the ﬁrst step of the process is an isothermal expan- ∆L2 = nT [ρ1 log ρ1 + ρ2 log ρ2] (6.7) • sion by which the 0 >< 0 - specie occupies all the − | | 1 left-half of the vessel while the ( 0 > + 1 >)(< ﬁnally a suitable unitary evolution takes the system 2 | | 0 + < 1 )-specie occupies all the right-half of the • again in the initial state. vessel;| this| expansion supplies an amount of work: The net work made by the engine during the cycle is:

∆L = ∆L1 + ∆L2 > 0 (6.8) ∆L1 = +nT ln 2 (6.1) so that the whole thermodynamical cycle converts the T being the temperature of the reservoir. heat extracted by the reservoir in a positive amount of at this stage the impenetrable partitions separating work of ∆L. • the two species are replaced by the ”magic”-semi- This, according to Peres, violates the Second Principle, permeable membranes having the ability of distin- proving that the ”magic” membranes able to sepa- guish non-orthogonal states; precisely one of them rate nonorthogonal states with perfect efficiency is transparent to the 0 >< 0 -specie and reflect the cannot exist. 1 | | Such a proof, anyway, in not correct, owing to Zurek’s 2 ( 0 > + 1 >)(< 0 + < 1 ) -specie while the other membrane| | has the| opposite| } properties; then, by a theorem; the key point touches the conceptual deep- double frictionless piston, it is possible to bring the ness underlying eq.5.2, whose complete comprehension engine, without expenditure of work or heat trans- requires to explicitly analyze the bug in Von Neumann’s fer, to a state in which all the two species occupy proof that Stherm(ρ) = Iprob(ρ). with the same concentration only the left-hand of The key point is based in the own definition of the vessel, the right-hand of the vessel remaining the semi-permeables membranes of Einstein’s method: empty; we can represent mathematically the state as correctly observed by Peres the semipermeable- of affairs of the system by the following decompo- membranes are endowed with automatic devices sition: able to peak inside the boxes and to test the state. What Peres seems unfortunately not to catch is that 1 1 1 a semi-permeable membrane is then an intelligent sys- := ( , 0 >< 0 ) , ( , ( 0 > + 1 >)(< 0 + < 1 )) E1 { 2 | | 2 2 | | | | } tem operating in the following way: (6.2) 1. gets the input (s,i) from a device measuring both 3 1 the side s from which the φi >< φi -specie arrives 4 4 | | ρ := 1 1 (6.3) and its kind, i.e. the classical information codified 4 4 ! by its label i. 12

2. computes a certain semaphore-function p such that But this must be done, in particular, in the cases of p (s,i) p[(s,i)] giving as output a 0 if the φi >< Peres’-engine: → | φi -specie must be left to pass while gives as output taking into account also the algorithmic-information | a one if the φi >< φi -specie must be stopped of the semi-permeable’s membranes, one sees that it is | | greater than or equal to the universe’s entropy decrease 3. gives the output p[(s,i)] to a suitable device that corresponding to the work made by the engine, so that, operates on the φi >< φi -specie in the speciﬁed by eq.5.2: way | | ∆Stherm 0 (6.9) The argument of Bennett’s exorcism concerning the ne- ≥ cessity of taking into account the algorithmic-information and Peres’ arguments falls down. of the sequences of successive recorded (s,i)’s in the membrane’s memory thus apply.

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