Thermodynamic Constraints on Quantum Mechanics
Jack Sarfatti
"Complexity, Entropy and the Physics of Information", edited by W.H. Zurek,
Santa Fe Institute (Addison-Wesley, ISBN 0-201-51509-1/QC39.S48 1991).
This may turn out to be one of the most important papers that has ever been
written in physics. It warrants your close attention because Peres says:
"The advent of quantum mechanics solved one of the outstanding puzzles of
classical thermodynamics, Gibbs paradox. COnversely, thermodynamics imposes
severe constraints on the axioms of quantum theory. The second law of
thermodynamics would be violated if it were possible to distinguish non-
orthogonal states, if Schrodinger's equation were nonlinear, or if a single
quantum could be cloned. All of these impossibilities are related to each
other: Non-orthogonal states could easily be distinguished if single quanta
could be cloned, but this is forbidden by the no-cloning theorem, which in
turn follows from the linearity of Schrodinger's equation. The key to the
above claims is the equivalence of the von-Neumann-Shannon entropy to the
ordinary entropy of thermodynamics. The proof of this equivalence ...
relies on the introduction of a mock Hilbert space with fictitious degrees
of freedom ... However, this proof ... assumes the validity of of
Hamiltonian dynamics (in order to derive the existence of thermal
equilibrium) and this point is suspicious .... Thus, the final conclusion
... is that if the integrity of the axiomatic structure of quantum theory
is not respected, then every aspect of the theory has to be considered ab-
initio."
Peres assumes the Gibbs ensemble "namely an infinite set of conceptual
replicas of that object, all identically prepared. Only then can we give a
meaning to the notion of probability." He also assumes "that all
information about the preparation of such an ensemble can be represented by
a Hermitian matrix p, satisfying Tr[p] = 1, called the density matrix." He
further assumes the standard lore that observables A are also Hermitian
operators and that the expectation value over the Gibbs ensemble is
<A> = Tr[Ap]. He then makes an important argument: "These rules have a
remarkable consequence. Given two different preparations represented by
matrices p1 and p2, one can prescribe another preparation p by the
following recipe: Let a random process have probability x to 'succeed' and
probability (1 - x) to 'fail'. In case of success, prepare the quantum
system according to p1. In case of failure, prepare it according to p2.
This process results in a p given by
p = xp1 + (1 - x)p2
.....
<A> = x Tr[Ap1] + (1 - x) Tr[Ap2] = Tr[Ap]
What I find truly amazing in this result is that once p is given, it
contains all the available information and it is impossible to reconstruct
from it p1 and p2! For example, if we prepare a large number of polarized
photons and if we toss a coin to decide, with equal probabilities, whether
the next photon to be prepared will have a vertical or horizontal linear
polarization, or, in a different experimental setup, we likewise randomly
decide whether each photon will have right-handed or left-handed circular
polarization, we get in both cases the same
1/2 0
p =
0 1/2
... If this were not true, EPR correlations would allow instantaneous
transfer of information to distant observers in violation of relativistic
causality."
Peres then goes on to show that a new theory beyond standard quantum
mechanics that permitted this kind of causality-violating quantum
connection communication would also allow us to beat the second law of
thermodynamics. That is, in such an alternative reality, Maxwell's demon
could do the job. In particular, it would be possible to extract energy
from the zero point fluctuations of the vacuum. Thus, Peres writes, (e.g.,
quantum multiplexing in a quantum optical computer):
"Another example would be to prepare photons having, with equal
probabilities, linear vertical polarization or circular right-handed
polarization. An observer requested to guess what was the preparation of a
particular photon, under the best conditions allowed by quantum theory,
would be able to give the answer with certainty in only 29.3% of cases. It
will be shown below that a 'superobserver' who could always give an
unambiguous answer would also be able to extract an infinite amount of work
from an isothermal reservoir."
Peres then fixes Einstein's argument for the entropy of a quantum ensemble
using an interaction that "allows us to distinguish different eigenvalues
of Hermitian operators, which correspond to orthogonal states of the
quantum system." He then adds: "Let us suppose now that some other type of
interaction would allow us to distinguish non-orthogonal states." What
about Glauber coherent states and squeezed states? "This would have
momentous consequences; for example, a certain type of EPR correlation
could be used to transfer information instantaneously. I shall now show
that this could also be used to convert into work an unlimited amount of
heat extracted from an isothermal reservoir."
Note the purpose of this sequence of posts is to introduce students to some
of the more subtle, hard to understand, pitfalls in the Santa Fe Institure
papers inspired by John Archibald Wheeler's Vision that, perhaps, some may
find too "diffuse". However, I feel it is worth doing because seldom has
there been one collection of papers that present so many new ideas on the
foundations of physics that are at the cutting edge of research today.
Please correct any errors I may (and will) make, so that we all may learn.
Peres gives a gedankenexperiment to "beat the second law of
thermodynamics". I found Peres's discussion a little too concise.
Start with n photons half polarized vertically (i.e.,|)and half polarized
at 45 deg. (i.e., / )
a _____________________________________________________
| | | | |
| | | | |
| V | n/2 | | n/2 / | V |
| | | | |
|____________|_____________|____________|____________|
1 2 3
V means vacuum. 1 is piston, 2 is barrier,3 is piston. Vertical polarized
photons (i.e., |) between 1 & 2. 45 degree polarized photons between 2 & 3.
Step 1 isothermal expansion doubling volumes. The photons (treated as
ideal gas) do work
nkTlog2
the log 2 is because the volumes have doubled for each compartment. (Use
units in which log is to base 2 so that log2 -> 1 below.)
b _____________________________________________________
| | |
| | |
| n/2 | | n/2 / |
| | |
|_________________________|__________________________|
1 2 3
Imagine we can make a Maxwell Demon filter that can discriminate between
non-orthogonal spin states | and / . (This would be no problem for
orthogonal spin states like | and - , for example.). It is not immediately
obvious that such a filter cannot exist. Thus, filter D transmits / but
reflects |. Replace wall 2 by D.
b' ____________________________________________________
| / |
| / |
| n/2 | / n/2 / |
| / |
|________________________/__________________________|
1 D 3
Now slowly move piston 3 over to 2.
c _____________________________________________________
| /|
| /|
| n/2 | + n/2 / /|
| /|
|_________________________/|__________________________
1 D3
The | photons are not affected since they bounce back from D. The /
photons are moved through D from the right to the left compartment with no
net change in volume. There is no net compression so that this step can be
done with negligible work and heat transfer from an external engine. Note
also that the total volume occupied by all photons is sam in state c as it
was in initial state a. a -> c has mixed the initially spatially separated
photon gas non-orthogonal polarization species. It is important to bear in
mind that this step with the Maxwell Demon filter is not only isothermal at
constant T but is also adiabatic at constant entropy S.
Peres writes:."we thereby obtain a mixture of the two polarization states.
Its density matrix is
1 0 1/2 1/2
p = (1/2)[ + ]
0 0 1/2 1/2
0.75 0.25
=
0.25 0.75
The eigenvalues of p are 0.854 (corresponding to photons polarized at 22.5
degrees from the vertical) and 0.146 (for the opposite polarization)."
c' _____________________________________________________
| /|
| /|
| 0.854n / + 0.146n \ /|
| /|
|_________________________/|_________________________
1 D3
Therefore, state c is physically indistinguishable from state c', where now
we change meaning of notation symbols so that "/" now means 22.5 degree
polarized photons (not the 45 deg photons as above) and "\" means the
orthogonal 112.5 degree polarized photons.
The next step is the tricky one that is, perhaps, not easy to follow in
Peres's original presentation. First put an ordinary 22.5 degree (/) filter
next to piston 1 that will reflect the 112.5 deg (\) photons. Replace the
D filter by an ordinary 112.5 deg. filter that reflects the 22.5 deg.
photons.
c'' _____________________________________________________
| / \ |
| / \ |
| / 0.854n / + 0.146n \ \ |
| / \ |
|_/_________________________\ |_______________________
1(/) (\)3
Move piston 1 to right by distance x. Also move piston 3 to right by the
same distance so that there is no net change in the total volume. Keep the
position of the (\)filter fixed, but move the (/) filter right up to (\).
d ______________________________________________________
| /\ |
| /\ |
| 0.854n / /\ 0.146n \ |
x | /\ x |
__________|______________________/\ __________|_______
The new orthogonal polarization species are now spatially separated. The
total volume and temperature of the n photons in state d is same as in the
initial state a so that we have completed the cycle in the thermodynamic
sense. However, there has been an isothermal compression of the photon
gas orthogonal polarization species. Only now has the entropy changed.
The initial specific entropy was one bit per photon, because for each
initial species
s(a) = - [(1/2)log(1/2) + (1/2)log(1/2)] = 1 bit/photon (base 2)
The new specific entropy is
s(d) = - [0.146 log 0.146 + 0.854 log 0.854] = 0.415 bit/photon
Hence, the work that must be supplied by the external engine do go from c''
to d is
-nkTs(d)
The net useful energy gain in this isothermal closed thermodynamic cycle
allowed by the Maxwell Demon filter that discriminates non-orthogonal
polarization states of photons is
nkT[s(a) - s(d)] = nkT[1 - 0.416] = 0.584nkT
which would violate the classical version of second law of thermodynamics.
Hence the title of Peres's paper "Thermodynamic Constraints on Quantum
Axioms"
Peres closes with a remark that nonlinearities in Schrodinger's equation
can beast the classical second law and also permit quantum connection
communication. Such a nonlinearity has been published by Nobel Laureate
Steven Weinberg "Particle States as Realizations (Linear and Nonlinear) of
Space-time Symmetries" Nucl. Physics B (Proc.Suppl.) 6, (1989) 67-75.
Apparently the N. Rosen of Einstein, Podolsky and Rosen published a
nonlinear quantum mechanics in "On Waves and Particles" J. Elisha Mitchell
Sci. Soc. 61 (1945) pp67-73. If anyone can find that obscure reference I
would like a copy.
Peres makes the confusing remark:
"A nonlinear Schrodinger equation deos not violate the superposition
principle in its weak form. The latter merely asserts that the pure states
of a physical system can be represented by rays in a complex linear space.
This principle does not demand that the time evolution of the rays obey a
linear equation."
By a "nonlinear" equation Peres means that if
f(0) -> f(t)
g(0) -> g(t)
f(0) + g(0) -> f(t) + g(t) is false
but
f(0) + g(0) -> some pure state
Peres then proves that this type of nonlinearity will beat the second law
of thermodynamics "if the other postulates of quantum mechanics are kept
intact" - in particular keep "two states can be divided by a semipermeable
membrane wall if they are orthogonal". Read the original for the details.
Peres does show that orthogonal states must remain orthogonal or else the
second law will be violated. Therefore, my quantum connection
communication scheme (in non-standard nonlinear extention of QM) will also
violate the second law. I am prepared to consider quantum violations of
the second law which may be strictly true only in the classical limit -
it's really a matter for experiment to decide. Steven Weinberg's model
will, according to Peres also violate the second law. Peres shows that the
only way to preserve the second law of thermodynamics is to preserve all
inner products in Hilbert space. That is, the unitarity of time evolution
in quantum mechanics is required to preserve the second law of
thermodynamics as well as to prevent quantum connection communication.
This means that Schrodinger's equation must be linear. But what about the
nonlinearities in the Hartree-Fock equations of many electron atoms. Can
any one clear up my confusion here - is it that from the point of view of
second quantization we can have nonlinear products of creation and
destruction operators which still act linearly in the Fock space? However,
I am still bothered because Peres is talking first quantization not second
quantization. So perhaps someone can clarify this.
Peres concludes:
"The advent of quantum mechanics solved one of the outstanding puzzles of
classical thermodynamics, Gibbs paradox. COnversely, thermodynamics imposes
severe constraints on the axioms of quantum theory. The second law of
thermodynamics would be violated if it were possible to distinguish non-
orthogonal states, if Schrodinger's equation were nonlinear, or if a single
quantum could be cloned. All of these impossibilities are related to each
other: Non-orthogonal states could easily be distinguished if single quanta
could be cloned, but this is forbidden by the no-cloning theorem, which in
turn follows from the linearity of Schrodinger's equation. The key to the
above claims is the equivalence of the von-Neumann-Shannon entropy to the
ordinary entropy of thermodynamics. The proof of this equivalence ...
relies on the introduction of a mock Hilbert space with fictitious degrees
of freedom ... However, this proof ... assumes the validity of of
Hamiltonian dynamics (in order to derive the existence of thermal
equilibrium) and this point is suspicious .... Thus, the final conclusion
... is that if the integrity of the axiomatic structure of quantum theory
is not respected, then every aspect of the theory has to be considered ab-
initio."
However, unless I am mistaken, Zurek showed (same volume as Peres's paper)
that the von-Neumann-Shannon entropy is not the ordinary entropy of
thermodynamics because the ordinary entropy or the physical entropy "the
quantity which allows for the formulation of thermodynamics from the
viewpoint of the observer" (Zurek) is the von-Neumann-Shannon entropy plus
the algorithmic complexity.
A second point is that Peres's proof needs thermal equilibrium. What
happens far off equilbrium in the Prigogine sense?
Third, what about the first-quantization non-linearities in the Hartree-
Fock equations of many-electron atoms?
Fourth - what about the breakdown in Hamiltonian methods pointed to by
Feynman and lately by Gell-Mann and Hartle who apparently suggest that the
Lagrangian "histories" version of quantum mechanics is not equivalent to
the Hamiltonian version but includes it as a special case?
The intimate connection between quantum connection communication and
beating the second law of thermodynamics is extremely important. Rather
than beating it, one might say there is a quantum loophole in the second
law which in its familiar context is a classical law. For example, if you
build a Carnot engine with a hot negative temperature and a cold positive
temperature, heat flows from both hot and cold and is converted entirely
into work.
FINIS