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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
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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 state0 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
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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
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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.
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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).
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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
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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:
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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
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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
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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
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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.
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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.
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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
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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.
Acknowledgment
This research was supported by the Israel Science Foundation, grant number 713/10,
and the German-Israel Foundation, grant number 1054/2009.
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