maxwell’s demon in a quantum set uplecturers.haifa.ac.il/he/hcc/mhemmo/documents/quantu… ·...

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

context of quantum mechanics. Most of it is based on the classical Landauer-Bennett

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] 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).

1


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 far-

from-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 anti-

thermodynamic 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

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

2


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

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

3


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 particle-

in-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 t0 a particle is placed in a box. At t 1 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,

respectively – is registered in the memory state of the measuring device. At t3 the

partition is replaced by a piston (in accordance with the measurement outcome),

which is subsequently pushed by the particle at t4 . The piston is coupled to a weight 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


located outside the box which is raised during the expansion. At t5 the particle is

again free to move throughout the box and the weight is at its maximal height. At t6

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).

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

5


times. We will subsequently consider the projection postulate and comment on the

GRW dynamics in our set up.

At t0 the quantum state of the entire setup is the following:

(2.1) |Y (0 )⟩=|x0 ⟩ p|S ⟩m|down ⟩w|e0 ⟩e ,

where |x0 ⟩ p is the initial state of the particle in a one-dimensional box; |S ⟩m is the

initial standard (ready) state of the measuring device; |down ⟩w 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|x0 ⟩ p is some superposition of energy eigenstates

which depend on the width a of the box where the amplitudes in |x0⟩ p give a definite

expectation value⟨ x0 ⟩ 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 |x0 ⟩ p will be a superposition of energy eigenstates of the form:

|x0 ⟩ p=sin npxa

e−(En /h )t

,

with an eigenvalue

En=n2h2

8 ma2 ,

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

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

6


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 t 1 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 t 1 immediately after the insertion of the partition in

the center of the box the quantum state of the setup becomes:

(2.2) |Y (1) ⟩=|x1 ⟩p|S ⟩m|down ⟩w|e0 ⟩e ,

7


but the expectation value of the energy of the particle is unaltered ⟨ x0 ⟩=⟨ 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=1√2

(|L ⟩p+|R ⟩p ),

where |L ⟩ p and |R ⟩ p are (in general) superpositions of the energy states of a particle

in a box of width a/2, where for|L ⟩ p the position x varies over the range [0, a/2] and

for |R ⟩ p 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) |Ψ (2 )⟩= 1√2

(|L ⟩ p|0 ⟩m+|R ⟩ p|1 ⟩m )|down ⟩w|e0 ⟩e ,

where the states |0 ⟩m and |1 ⟩m 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


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 t5 , when their von Neumann entropy will

decrease to its initial zero value.5

At t3 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 t4 the piston is pushed (to the left or to the right) by the particle. The quantum

state of the setup at t4 has the form:

(2.5) |Ψ ( 4 )⟩= 1

√2 (|L( w( t ))⟩ 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 )⟩=⟨ 12 ( L(w ( t ))+R(u( t )))⟩

and decreases continuously over time in the interval from t4 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


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 ))⟩ pand |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 t5 , 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) |Ψ (5 ) ⟩=|x5 ⟩p1√2

(|0 ⟩m+|1 ⟩m)|up ⟩w|e0 ⟩e ,

where |x5 ⟩ 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


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 |x0 ⟩ 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


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)

|Y c (2 )⟩=|L⟩ p|0 ⟩m|down ⟩w|e0 ⟩e or

|Y c (2 ) ⟩=|R ⟩p|1⟩m|down⟩w|e0 ⟩e

with probability ½, and at t5 (before the interaction of the particle with e) it is either

(2.9)

|Y c (5 ) ⟩=|x5 ⟩p|0 ⟩m|up ⟩w|e0 ⟩e

or

|Y c (5 ) ⟩=|x5 ⟩ 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 λ , 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). 9 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..

12


|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 t6 the quantum state of the composite system m+e as

follows:

(2.10)

|0 ⟩m|e0 ⟩e →|S ⟩m|e1 ⟩e|1 ⟩m|e0 ⟩e→|S ⟩m|e2 ⟩e

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 t6 there

are memories in e of the outcome of the measurement at t2 . 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 ⟩mby 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 ⟩mby 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


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


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) |S ⟩→ 12 (|0 ⟩⟨ 0|+|1 ⟩ ⟨1|)→ 1

√2(|0 ⟩+|1 ⟩ )→|S ⟩

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 λ of the

measuring device, which is quantum-mechanically incompatible with the pointer (or

memory) observable (or alternatively by measuring Q’ of the environment).12

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.12 This analysis applies mutatis mutandis to the GRW collapse theory.

15


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 pre-erasure

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 λ 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 t2 .

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 anti-

thermodynamic.

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

world is open.

Appendix: Quantum Macrostates

16


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

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

17


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.

Acknowledgment

This research was supported by the Israel Science Foundation, grant number 713/10,

and the German-Israel Foundation, grant number 1054/2009.

References

Albert, D. (2000) Time and Chance, Cambridge, MA: Harvard University Press.

Bell, J. S. (1987) “Are There Quantum Jumps?” in J. S. Bell (1987) Speakableand

Unspeakable in Quantum Mechanics. Cambridge: Cambridge University Press, pp.

201–212.

Brown, H. and Uffink, J. (2001) “The Origins of Time-Asymmetry in

Thermodynamics: The Minus First Law,” Studies in History and Philosophy of

Modern Physics, 32(4), 525-538.

Callender, C. (2016) “Thermodynamic Asymmetry in Time,” The Stanford

Encyclopedia of Philosophy (Winter 2016 edition), Edward N. Zalta (ed.), URL =

<https://plato.stanford.edu/archives/win2016/entries/time-thermo/>.

Earman J. and Norton J. (1999) “Exorcist XIV: The Wrath of Maxwell’s Demon. Part

II. From Szilard to Landauer and Beyond,” Studies in History and Philosophy of

Modern Physics 30(1), 1-40.

Ghirardi, G., Rimini, A. and Weber, T. (1986) “Unified Dynamics for Microscopic

and Macroscopic Systems,” Physical Review, D 34, 470-479.

Goldstein S., Lebowitz, J., Mastrodonato, C., Tumulka, R. and Zanghi, N. (2010)

“Normal Typicality and von Neumann’s Quantum Ergodic Theorem,” Proceedings of

the Royal Society A 466 (2123), 3203-3224.

18


Hemmo, M. and Shenker, O. (2001) “Can We Explain Thermodynamics by Quantum

Decoherence?” Studies in the History and Philosophy of Modern Physics, 32, 555-

568.

Hemmo, M. and Shenker, O. (2003) “Quantum Decoherence and the Approach to

Equilibrium,” Philosophy of Science 70, 330-358.

Hemmo, M. and Shenker, O. (2005) “Quantum Decoherence and the Approach to

Equilibrium (II),” Studies in the History and Philosophy of Modern Physics 36, 626-

648.

Hemmo, M., and Shenker, O. (2006), “Von Neumann Entropy Does Not Correspond

to Thermodynamic Entropy,” Philosophy of Science 73(2), 153-174.

Hemmo, M. and Shenker, O. (2010) “Maxwell's Demon,” The Journal of Philosophy

107(8), 389-411.

Hemmo, M. and Shenker, O. (2011) “Szilard's Perpetuum Mobile,” Philosophy of

Science 78 (2), 264-283.

Hemmo, M. and Shenker, O. (2012) The Road to Maxwell’s Demon, Cambridge:

Cambridge University Press.

Hemmo, M. and Shenker, O. (2013) “Entropy and Computation: The Landauer-

Bennett Thesis Reexamined,” Entropy 15: 3387-3401.

Hemmo, M. and Shenker, O. (2016) “Maxwell’s Demon,” Oxford Online Handbook

(Oxford University Press) Available at:

http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199935314.001.0001/

oxfordhb-9780199935314-e-63?rskey=plUl7T&result=1

Leff, H. and Rex, A. (2003) Maxwell’s Demon 2: Entropy, Classical and Quantum

Information, Computing. Bristol: Institute of Physics Publishing.

19


Shenker, O. (2017) “Quantum Foundations of Statistical Mechanics and

Thermodynamics,” A. Wilson and E. Knox (eds.) Companion to the Philosophy of

Physics, Routledge, forthcoming

Sklar, L. (2015) “Philosophy of Statistical Mechanics,” The Stanford Encyclopedia of

Philosophy (Fall 2015 Edition), Edward N. Zalta (ed.), URL =

<https://plato.stanford.edu/archives/fall2015/entries/statphys-statmech/>.

Uffink (2002) “Essay review of Time and Chance,” Studies in History and Philosophy

of Modern Physics 33, 555-563.

Zurek, W. H. (1984) “Maxwell’s Demon, Szilard’s Engine and Quantum

Measurement,” in Leff and Rex (2003), pp. 179-189.

20

maxwell’s demon in a quantum set uplecturers.haifa.ac.il/he/hcc/mhemmo/documents/quantu… ·...

Documents