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A Quantum Mechanical Maxwellian Demon A Quantum Mechanical Maxwellian Demon Meir Hemmo1 and Orly Shenker2 1. Introduction J. C. Maxwell devised his so-called Demon in 1867 to show that the Second Law of thermodynamics cannot be universally true if classical mechanics is universally true. Maxwell’s Demon is a way of demonstrating that the laws of mechanics are compatible with microstates and Hamiltonians that lead to an evolution which violates the Second Law of thermodynamics by transferring heat from a cold gas to a hot one without investing work.3 That is, the Demon is not a practical proposal for constructing a device that would violate the law of approach to equilibrium, but rather a statement to the effect that the Second Law of thermodynamics cannot be universally true if the laws of mechanics are universally true. Since then, many attempts have been made to disprove Maxwell’s intuition. David Albert, in his book Time and Chance (2000, Ch. 5), used a phase space argument to demonstrate that a classical mechanical macroscopic evolution that satisfies Liouville’s theorem can be entropy decreasing. This is the beginning of a proof that Maxwell’s Demon is compatible with classical mechanics. However, Albert’s proof was incomplete since he did not complete the thermodynamic cycle. The missing link to complete the cycle required proving that erasing the Demon’s memory is not dissipative, that is one needs to show that the so-called Landauer-Bennett thesis is mistaken. In our (2010, 2011, 2012, 2013, 2016) we provided this missing link. By this we have shown that classical mechanics does not rule out a dynamical evolution that takes all the points in an initial macrostate of the universe to macrostates with a smaller Lebesgue measure, i.e. with lower total entropies. The situation just described is in the context of classical mechanics. There is a vast literature (see Leff and Rex 2003) on the question of Maxwell’s Demon also in the 1 Department of Philosophy, University of Haifa, Haifa 31905, Israel; email: [email protected] Program in History and Philosophy of Science, The Hebrew University of Jerusalem, Jerusalem 91905, Israel; email: [email protected]. 3 In the context of the question of the Demon, one needs to distinguish between the Second Law and the law of approach to equilibrium, which underlies it (see Brown and Uffink 2001). 2 1 A Quantum Mechanical Maxwellian Demon context of quantum mechanics. Most of it is based on the classical Landauer-Bennett thesis which we have disproved (see the discussion by Earman and Norton 1999 of Zurek 1984). In his Time and Chance (2000, Ch. 7) Albert proposes an approach to statistical mechanics in which the only kind of probabilistic statements are those derived from the micro-dynamics. His proposal replaces standard quantum mechanics with the dynamics proposed by Ghirardi, Rimini and Weber (1986) (see Bell 1987), which postulates that for any initial quantum state0 of a system S, the state will evolve for some time t (probabilistically determined by the GRW temporal constant and size of the system) according to the Schroedinger equation to another quantum state 1, and then the evolved state will collapse spontaneously into a third state 2, where 2 is a Gaussian superposition of positions centered around point X, with probability fixed by the amplitude of X in 1. Although the GRW spontaneous localizations are in position, one can as usual also talk about other observables. Suppose that the eigenvalues {ai} of the observable A are such that eigenvalue a1 corresponds to a farfrom-equilibrium state of the system, a2 is a little closer to equilibrium, and a3 corresponds to equilibrium. For every quantum state the probabilities for the {ai} eigenvalues are given by the Born rule. If, given a certain Hamiltonian H (discussed below) (t0) has a high probability for a1 while (t1) (t1> t0) has a low probability for a1 and higher ones for a2 and a3, then we say that (t0) is a thermodynamic quantum state relative to H (Albert uses the term “thermodynamically normal”). If, given the same dynamical evolution, (t1) gives a higher probability to a1, we shall say that (t0) is an anti-thermodynamic quantum state relative to H. Since in the above description of states that are “thermodynamic relative to a Hamiltonian” no GRW collapses are involved, but only Schroedinger evolutions under a given Hamiltonian H, and since the Schroedinger equation is time-reversal invariant, then if there are thermodynamic quantum states relative to a given H, there are also antithermodynamic quantum states relative to the same H. In his proposal Albert adds the following dynamical hypothesis to the GRW dynamics which we divide into two parts: (a) There exists a Hamiltonian H such that every initial (t0) has high Born probability to collapse under the GRW dynamics to another 2 A Quantum Mechanical Maxwellian Demon quantum state (t1) which is thermodynamic relative to H, regardless of whether or not (t0) itself was thermodynamic relative to H. (b) H is the actual Hamiltonian that governs thermodynamic evolutions. The term “exists” in part (a) above means that such a Hamiltonian is (by hypothesis) possible according to fundamental physics, that it is compatible with fundamental physics to say that there could be such a Hamiltonian. Albert provides no proof of either (a) or (b), and his plausibility argument for their conjunction is based on the fact that observed systems are actually thermodynamic. It seems to us that it would be as compatible with the fundamental physics, as far as Albert’s argument is concerned, if, for example, the actual Hamiltonian of a given system would entail high probability for anti-thermodynamic evolutions, so that, for example, the system would be a Maxwellian Demon. Thus Albert’s proposal is not a proof but only a conjecture that there are no Demons (see also Sklar 2015; Uffink 2002; Callender 2016.) It is an achievement of Albert’s approach that whichever Hamiltonian turns out to be the case in our world the probabilities for thermodynamic (or anti-thermodynamic) behavior are only the Born probabilities. This means that the probabilities for thermodynamic (or anti-thermodynamic) behavior involve no ignorance over quantum “initial” states. All the other theorems concerning thermodynamic behavior in both classical and quantum statistical mechanics are valid only for most of the initial quantum states (given a measure; see Shenker 2017). This also holds for our own proposals in Hemmo and Shenker (2001, 2003, 2005) in which the initial conditions lead to environmental decoherence, and according to prevalent decoherence models these results hold at best for “most” quantum states of the universe, given the right measures. Albert’s dynamical hypothesis entails that a Maxwellian Demon is strictly impossible. In this sense, Albert’s proposal concerning the quantum mechanical foundations of statistical mechanics seems, at first sight, to be a way to exorcise the Demon, the same Demon he re-introduced to classical physics in the argument we mentioned above. In this paper we propose an alternative dynamical hypothesis called Maxwell’s Demon according to which the Second Law in its mechanical probabilistic version is false (even in a GRW world). We shall prove that our hypothesis is compatible with 3 A Quantum Mechanical Maxwellian Demon quantum mechanics. Of course the question of which Hamiltonian is actually the case in our world, Maxwell’s Demon or Albert’s dynamical hypothesis, is a question of fact, which we don’t address here. Our point rather is that given everything we know from experience and from theory the answer to this question is as yet open. It turns out that the situation in quantum statistical mechanics is the same as in classical statistical mechanics: Maxwellian Demons are compatible with the fundamental theory, and there is even no proof that the actual dynamics in our world is not Demonic. Our discussion is in the framework of standard quantum mechanics, by which we mean quantum mechanics in von Neumann’s (1932) formulation, with no hidden or extra variables in addition to the quantum state and without the projection postulate in measurement (we set aside questions concerning the physical interpretation of such a theory). In addition we shall formulate our argument within a quantum theory with the projection postulate (and with no hidden variables, again, setting aside questions of interpretation). 2. A Quantum Mechanical Demon We consider a thought experiment along the lines of Szilard’s and Bennett’s particlein-a-box, in a quantum mechanical context (see our 2011 for a classical analysis). Since we are interested in the question of whether or not thermodynamics is consistent with quantum mechanics as a matter of principle, we consider the experiment in a highly idealized framework, disregarding practical questions (some of which we shall mention along the way).4 Consider the setup in the Figure. At t 0 a particle is placed in a box. At t1 a partition is inserted exactly at the center of the box so that the particle is trapped in the left-hand side L or the right-hand side R of the box. At t 2 a measurement of the location of the particle, left or right, is carried out and the outcome of the measurement – 0 or 1, 4 We do not address the question of whether a single particle in a box is a thermodynamic system. Since the arguments in the literature concerning the entropic cost of measurement and erasure have been given in this setup, this is the setup we consider. 4 A Quantum Mechanical Maxwellian Demon respectively – is registered in the memory state of the measuring device. At t 3 the partition is replaced by a piston (in accordance with the measurement outcome), which is subsequently pushed by the particle at t 4 . The piston is coupled to a weight located outside the box which is raised during the expansion. At t 5 the particle is again free to move throughout the box and the weight is at its maximal height. At t 6 the memory of the device is erased and returns to its initial standard state. The particle returns to its initial energy state by receiving from the environment the energy it lost to the weight. The cycle of operation is thus closed. By this last statement we mean that everything returns to its initial state except for the energy transfer from the heat bath (environment) to the weight (see our 2010). 5 A Quantum Mechanical Maxwellian Demon We will now show that according to quantum mechanics this setup can be a Maxwellian Demon. We start by assuming a quantum mechanical dynamics without the projection postulate, i.e., a dynamics that satisfies the Schrödinger equation at all times. We will subsequently consider the projection postulate and comment on the GRW dynamics in our set up. At t 0 the quantum state of the entire setup is the following: (2.1) Y(0) = x 0 where x 0 p p S m down w e0 e, is the initial state of the particle in a one-dimensional box; S initial standard (ready) state of the measuring device; down w m is the is the initial state of the weight which is positioned at some initial height we denote by down; and e0 e is the initial state of the environment, which we assume does not interact with the particle or the weight (or the partition). In particular, this means that the quantum state of the particle and the partition do not undergo a decoherence interaction with the environment. The initial energy state of the particle x 0 p is some superposition of energy eigenstates which depend on the width a of the box where the amplitudes in x 0 p give a definite expectation value x 0 for the energy of the particle. We assume for simplicity that the box is an infinite potential well so that all the energy eigenstates at the initial time have a node in the center of the box, and the quantum mechanical probability of finding the particle exactly in the center of the box is zero. In general the quantum state of the particle x 0 p will be a superposition of energy eigenstates of the form: npx -(E n / )t , e a with an eigenvalue x0 p En = = sin n 2h 2 , 8ma 2 where n is an even number and m is the mass of the particle. In standard quantum mechanics such a superposition is not interpreted as expressing ignorance about the 6 A Quantum Mechanical Maxwellian Demon energy eigenstates of the particle. It is the fine-grained quantum-mechanical microstate of the particle and not a macrostate in any standard sense. However, our preparation applies to any such superposition and in this sense it is a macroscopic preparation in the usual sense of the term in classical statistical mechanics (see Appendix). S m is the standard ready state of the measuring device which is an eigenstate of the so-called pointer observable in this setup. For simplicity we take a spin-1 particle to represent the measuring device with S m = -1 in the z-direction. The two other spin- 1 eigenstates, 0 and 1 in the z-direction, correspond to the two possible outcomes of the measurement. Later we consider pointer states that are only approximately orthogonal. At t1 we insert a partition in the center of the box, where the wavefunction is zero, so that the expectation value for the energy remains completely unaltered. This is highly idealized, to be sure, creating many technical questions. For example, given the quantum mechanical uncertainty relations, one cannot insert the partition exactly at a point since in this case its momentum would be infinitely undetermined. To avoid this, one may assume that the partition and consequently also the particle are fairly massive, but that nonetheless they are kept in isolation from the environment. Alternatively, if there is a certain amount of decoherence, we may assume that the degrees of freedom in the environment are controllable in the subsequent stages of our experiment. Of course if the partition is massive, the idealization of zero width may also be problematic and consequently the wavefunction may change when the partition is inserted. What we need is a way of inserting a partition in a way that will not change the expectation values of the energy of the particle. In this sense we assume here that the effects of the above issues are negligible and we continue with our idealization. We are not concerned here with the experiment’s feasibility, but rather with its consistency with thermodynamics. We therefore take it that ideally at t1 immediately after the insertion of the partition in the center of the box the quantum state of the setup becomes: 7 A Quantum Mechanical Maxwellian Demon (2.2) Y(1) = x1 p S m down w e0 e, but the expectation value of the energy of the particle is unaltered x 0 = x1 (On each side of the partition the superposition involves both odd and even eigenstates, but the width of the box is now a/2.) Although the expectation value for the particle’s energy does not change in this interaction, the wavefunction of the particle now becomes a superposition of two components: (2.3) x1 p = where L p and R 1 2 ( L p + R p ), p are (in general) superpositions of the energy states of a particle in a box of width a/2, where for L for R p p the position x varies over the range [0, a/2] and x varies over the range [a/2 ,a], while the equal amplitudes are a consequence of our idealization that the partition is placed exactly in the center of the box. At t 2 a measurement of the coarse-grained position observable X of the particle is carried out, corresponding to whether the particle is located in the left- or right-hand side of the box. The quantum state immediately after the measurement, as described by the Schrödinger equation is: (2.4) Y(2) = 1 2 (L where the states 0 m p 0 m + R p 1 m ) down and 1 m w e0 e , are the recording (pointer) states of the device, which are one-to-one correlated (respectively) with the locations of the particle given by the energy states L p and R p. The overall quantum state of the setup is superposed, i.e. there is no collapse in the measurement. We do not consider the question of in what sense the reduced state of the pointer state (which is quantum mechanically mixed) records a determinate outcome of the measurement. The measurement at time t 2 increases the von Neumann entropy of the device m and of the particle (since both enter a mixed state). But this is irrelevant for the issue of 8 A Quantum Mechanical Maxwellian Demon Maxwell’s Demon in this setup since as we will see, the particle and the device will return to their separate pure states at time t 5, when their von Neumann entropy will decrease to its initial zero value.5 At t 3 the partition is replaced by the piston (in both components of the superposition Y(2) ), which is coupled to the weight. But we assume for simplicity that the quantum state of the setup remains the same, Y(2) = Y(3) . Here we are considering the partition and the piston to be external constraints, as is customary in the literature. Taking the constraints as part of the system would create technical problems, although these are solvable. At t 4 the piston is pushed (to the left or to the right) by the particle. The quantum state of the setup at t 4 has the form: (2.5) Y(4) = ( 1 L(w(t)) 2 p 0 m ) + R(u(t)) p 1 m y(t) w e0 e , where the energy states L(w(t)) and R(u(t)) change with the changing width of the box, w(t) and u(t), respectively. The expectation value of the particle’s energy x(t) p is given by x(t) = 1 ( L(w(t)) + R(u(t))) 2 and decreases continuously over time in the interval from t 4 up to t 6. The energy state of the weight y(t) w changes accordingly, so that y ranges from the down to the up position, and the expectation value of the energy of the weight increases accordingly. Conservation of energy, which is stated in terms of expectation values in quantum mechanics according to Ehrenfest’s theorem, implies that (2.6) x(t) p + y(t) w = x0 p. 5 For the reason why the identification of von Neumann entropy with thermodynamic entropy fails, see our (2006). For an alternative understanding of the von Neumann entropy, see Shenker (2017). 9 A Quantum Mechanical Maxwellian Demon Since the weight is positioned in a gravitational field, higher energy expectation values for the weight are coupled with higher positions of the weight. This means that the quantum-mechanical transfer of energy increases the probability that the weight will be found at higher positions upon measurement. During the expansion of the particle as described by (2.5) the correlations between the memory states of the device and the energy states of the particle are gradually lost.6 The energy states L(w(t)) p and R(u(t)) p are such that at any time t the overall probability of finding the particle in the left- or right-hand side of the box is invariably ½. To be sure, the conditional probability of finding the particle in the right (left) hand side, given that the device is in the state 0 m ( 1 m ) at t 3, is almost zero, but it increases with time and becomes ½ at t 5, since the correlations between the memory state of the device and the location of the particle are gradually lost. At t 5 the quantum state of the setup is given by: (2.7) Y(5) = x5 p 1 2 ( 0 m + 1 m ) up w e0 e , where x 5 p is the energy state of a particle in a one-dimensional box of width a with amplitudes which match the decrease in the expectation value of the energy, x5 p = x0 p - up w , and up w corresponds to the up-state of the weight in which the expectation value of the energy is higher. The measuring device remains in the superposed state 1 2 ( 0 m + 1 m ) , but the particle and the device are now decoupled (and disentangled). To complete the cycle of operation7 we need to return the particle to its initial energy state and erase the memory of the measuring device. However, closing the cycle does not mean also returning the environment into its initial state. What we need is to erase 6 Since the time evolution of the total state is reversible, of course there must be some record of the memory state of the device for each of the two components of the superposition. For example, it may be in the state of the weight or of the pulleys. 7 Albert (2000, Ch. 5) argues that closing the cycle of operation is irrelevant to the issue of the Demon. We show in our (2011) how the cycle of operation can be closed in the appropriate sense, but we do not require the environment to return to its initial state (see Section 5). 10 A Quantum Mechanical Maxwellian Demon any trace in the universe from which one could read off the outcome of the measurement (see our 2010) (and evolve the measuring device back to its ready state). We can use the memory in the pointer states to erase traces that remain due to the difference in the motion of the piston to the right or to the left. When this is done, the quantum state of the setup will have the same form as in (2.7), in which we have already omitted (for simplicity) any traces of the outcome of the measurement of the left or right motion of the piston in any degree of freedom of the universe other than the pointer state of the device itself. We can now also return the quantum state of the device 1 2 ( 0 m + 1 m ) to its initial state S m. Since the device is no longer entangled with the particle (and we assumed that it is isolated from the environment), these two states are non-orthogonal pure states, and so there is a quantum-mechanical Hamiltonian that will do that. If the memory states of the device are not strictly orthogonal, there is a nonzero probability that the particle will be on the right side of the box and at the same time we shall release the piston from the right side, and in this case the weight clearly will not rise. But nothing in quantum mechanics stands in the way of making this probability as small as we wish. We believe that by this we have established our task (although, perhaps, not with certainty, but with near-certainty). In particular, as can be seen in equation (2.6), we have transferred energy from the particle to the weight with no further changes in the universe. We now wish to return the particle to its initial energy state by taking energy from the environment e. For this purpose we let the particle interact with e in such a way that the particle will certainly return to its initial energy state x 0 p . Since we assume that the environment is initially in a pure state, possibly very complex but nevertheless pure rather than a mixture, there is a deterministic evolution that will yield precisely this energy transfer, no matter how complicated it may be. In this interaction the expectation value of the energy of e decreases by the same amount as the expectation value of the energy of the particle (or ultimately the weight) increases, e0 e - e1 e = up w - down w as required by conservation of energy (where e1 e is the expectation value of the environment’s energy after the interaction with the 11 A Quantum Mechanical Maxwellian Demon particle). Strictly speaking, the energy of the environment is finite.8 To be sure, the reverse evolution is also possible – namely that the particle will transfer energy to the environment – but we assume that the initial conditions in our setup match the first course of evolution.9 The expectation value of the weight’s energy is greater than in its initial state. The cycle is completed and the setup is ready to go once again. Let us now consider the framework of standard quantum mechanics, say in von Neumann's formulation, with the projection postulate in measurement. Applying the projection postulate at t 2 , the quantum state is either: (2.8) Yc (2) = L down w e0 e Yc (2) = R p 1 m down w e0 e p 0 m or with probability ½, and at t 5 (before the interaction of the particle with e) it is either (2.9) Yc (5) = x 5 p 0 m up w e0 e or Yc (5) = x 5 p 1 m up w e0 e. The unitary Schrödinger equation cannot map by the same Hamiltonian the final states 0 m and 1 m of the device m to S m . But we can erase the memory of the device m non-unitarily in von Neumann's way (see von Neumann 1932, Ch. 5) by measuring on m at t 6 an observable , which is incompatible (in the above sense) with the pointer observable. Subsequently, to bring the device back to its standard state, we measure the pointer observable on m until the outcome corresponding to 8 For a discussion of the limitations of the idealization of an infinite heat bath, see our (2012, Ch. 1). There is absolutely no reason to think that the entropy of the environment increases due to the interaction with the particle. For example, the von Neumann entropy does not change, since the environment evolves from one pure state to another.. 9 12 A Quantum Mechanical Maxwellian Demon S m is obtained, at which case the cycle is completed. The erasure of the memory does not require additional degrees of freedom in the environment of the setup, and in particular involves no environmental dissipation. Alternatively, we can map at t 6 the quantum state of the composite system m+e as follows: (2.10) where e1 e and e2 e are eigenstates of some observable Q of e and the arrow represents the Schrödinger evolution. Since the evolution in (2.10) is deterministic, the final states are still correlated with the device's memory states, and so at t 6 there are memories in e of the outcome of the measurement at t 2. But we can now measure on e an observable Q' which is maximally incompatible with Q in the sense that the quantum-mechanical probability distribution (as given by the modulus square of the amplitudes) of the different values of Q given any eigenstate of Q' is uniform. This means that the measurement of Q' on the device m erases with certainty any traces in the environment e of the memory state of the device at any earlier time (see the next section).10 A simpler way to restore the memory state in a world with a projection postulate is by replacing the memory states 0 m and 1 m by memory states that are only approximately orthogonal (this would be the case in, e.g. the GRW theory), in which case we can map the memory states back to the standard memory state S m by the same Hamiltonian. By applying the above Hamiltonian for erasing the memory states of the device we completed the thermodynamic cycle. Often in the literature it is argued that the 10 If erasure can be only probabilistic, the measurement of any observable of the environment which does not commute with Q would be enough. 13 A Quantum Mechanical Maxwellian Demon erasure of the memory states is accompanied by dissipation, according to the Landauer-Bennett thesis. We have disproved this thesis in the classical case in our (2010, 2012, 2013). In the above Demonic evolution we demonstrated that a Maxwellian Demon is compatible with quantum mechanics. The physical details of all the above erasure evolutions do not seem to require any dissipation. Discussions of Maxwell’s Demon in the quantum mechanical context rely on the disproved Landauer-Bennett thesis from the classical theory and there are no independent sound arguments in support of this thesis that are genuinely quantum mechanical (see Earman and Norton 1999). 3. The Connection between Measurement and Erasure The discussion of Maxwell’s Demon in the literature focuses on measurement and erasure. In the classical case, because of the determinism of the dynamics, the notions of both measurement and erasure must be understood as referring to a macroscopic evolution, namely evolutions of sets of microstates (see our 2010, 2011, 2012, 2013). In quantum mechanics these two notions can be understood microscopically and are intertwined. Consider first measurement and erasure in no-collapse quantum-mechanical theories. As we saw, the measurement and erasure evolutions of the measuring device m are given by the following sequence: (2.11) S ® 12 ( 0 0 + 1 1 ) ® 1 2 ( 0 + 1 )® S , where the first arrow stands for the measurement, the second for the disentanglement, and the third for the erasure. Since the evolution results in no collapse of the wavefunction after the measurement, the measurement transforms the device m from a pure to a mixed (i.e. improper) state, the disentanglement evolves m from a mixed to a pure superposition of the two memory states, and the erasure transforms m from this superposition to the standard state. The entire evolution described by (2.11) is possible since it is brought about by the Schrödinger evolution of the global wavefunction Y(t) , which is time-reversible. Notice that although the von Neumann 14 A Quantum Mechanical Maxwellian Demon entropy increases by the first arrow, it decreases to zero by the second arrow, and therefore it has no effect on the total entropy change in the Demonic evolution. It might seem that the third arrow in (2.11) does not actually erase the outcome of the measurement since the Schrödinger evolution is reversible. But this idea is mistaken for the following reasons. If one posits that the third arrow in evolution (2.11) does not bring about an erasure just because the dynamics is reversible, then by the same argument the determinism of the dynamics entails that the first arrow in (2.11) does not bring about a measurement with a definite outcome. Any account in which one obtains a determinate outcome at the end of the measurement, would also be an account in which the outcome is erased at the end of the erasure.11 Moreover, whether or not the evolution in (2.11) describes measurement and erasure is immaterial to the issue at stake: we have a reversible quantum evolution in which energy is transferred to the particle in such a way that at the end of the experiment the weight is lifted, and the particle and everything else in the world returns to its initial state. This is why such an evolution is a Maxwellian Demon as we explained above. In standard quantum mechanics with collapse in measurement, the notions of measurement and erasure are two aspects of the same physical process. We can represent these two evolutions as follows. (2.12) As we saw above, the left or right location of the particle is measured by coupling the memory states 0 and 1 of the device m to the states of the particle, and then collapsing the memory onto one of them. The memory of the outcome of the measurement is erased in the standard theory by measuring the observable of the 11 We skip the details of how this is accounted for in for example Bohm's and Everett's theories. Let us only say that in Bohm’s theory there is no microscopic erasure because of the determinism of the velocity equation for the trajectories. In Everett-style theories there might be a microscopic erasure depending on what one says about the evolution of memories when the branches of (2.11) re-interfere. 15 A Quantum Mechanical Maxwellian Demon measuring device, which is quantum-mechanically incompatible with the pointer (or memory) observable (or alternatively by measuring Q’ of the environment).12 In a sense, in standard quantum mechanics (with or without collapse), the measurement of any observable that does not commute with the memory observable is an erasure, since after such a measurement it is impossible to retrodict the preerasure state of the memory system from the post-erasure state. In a collapse theory the situation is even worse: after a quantum erasure one cannot even retrodict which observables had definite values in the past. In our setup, since is maximally incompatible with the pointer (or memory) observable, even if we assume that there is a memory of the identity of the pointer observable, one cannot retrodict even probabilistically which memory state 0 or 1 , and therefore which outcome of the measurement existed at t 2. 4. Maxwell’s Demon The question associated with Maxwell’s Demon is not only whether transitions that lead to anti-thermodynamic behavior are consistent with (quantum or classical) dynamics, but also whether there are cases in which such transitions are highly probable. In this paper we showed that a quantum mechanical evolution, which includes measurement, erasure, and transfer of energy from the environment to a particle, and results in lifting a weight, is possible. This evolution just is antithermodynamic. As we said in the Introduction, Albert proposes a reconstruction of statistical mechanics based on the GRW theory. In his reconstruction Albert puts forward the hypothesis that the Hamiltonian of the world is such that every quantum state (regardless of whether or not it is thermodynamic) has high probability to jump on thermodynamic trajectories. In our set up we have given an example of a quantum mechanical Hamiltonian called Maxwell’s Demon in which the GRW jumps will not avoid anti-thermodynamic evolutions..Nothing in quantum mechanics (nor in the GRW theory) rules out this Hamiltonian. The question of which is the case in our 12 This analysis applies mutatis mutandis to the GRW collapse theory. 16 A Quantum Mechanical Maxwellian Demon world is open. Appendix: Quantum Macrostates How can one prepare a macroscopic state according to quantum mechanics? We take it that the quantum mechanical counterparts of thermodynamic magnitudes are eigenvalues of certain observables. Suppose that we prepare a collection of systems by measuring on each of them the observable M, applying the projection postulate, and collecting only those with eigenvalue M0 (with the corresponding eigenstate or eigensubspace). Then (in the standard theory with the projection postulate, except in special cases described below) the resulting quantum state is known with complete accuracy: it is a pure state. Unlike the classical case of preparing a system by measuring a macrovariable on it, which leaves us with ignorance concerning the other aspects of the microstates, here the information about the quantum state is complete, not partial. No ignorance remains. The only case (in standard quantum mechanics) where one can prepare a system with a certain eigenvalue and nevertheless remain with a set of quantum microstates compatible with the measured eigenvalue is the measurement of degenerate observables in special circumstances. In the general case, upon measurement of the observable A on some quantum state |>, if the outcome is eigenvalue an with degree of degeneracy gn, then according to the projection postulate, the final state is a (normalized) superposition of all the gn eigenvectors of A associated with an. But if the initial quantum state |> is itself one of the eigenvectors associated with an then the final state will remain unchanged and will not become a superposition of all the eigenvectors of an. This has the following consequence. Suppose that before A was measured, a non-degenerate observable B was measured non-selectively, and then a suitable Hamiltonian was applied on the resulting proper mixture, such that each eigenstates bn of B in the mixture evolved to an eigenstate of the degenerate eigenvalue an of A. In this case the state of affairs just prior to the measurement of A can be described in terms of a proper mixture of the eigenstates associated with an and this proper mixture will carry over to the state after the measurement of A, with the same (normalized) probability distribution. The result is that by the end of the measurement process we know that the quantum state is a definite pure quantum 17 A Quantum Mechanical Maxwellian Demon eigenstate of an, but we do not know which it is – much like in a classical preparation. This is the only way to understand a preparation of macrostate, i.e. a set of microstates in standard quantum statistical mechanics. In non standard theories, such as GRW or Bohmian mechanics, the problem does not arise: the observer is often ignorant of the actual state of affairs, due to the spontaneous localization of the wave function in the first case and the inaccessibility of the positions of the Bohmian particles in the second case. 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