[Fis] Entropy, Information and Quantum Coherence

Dear Koichiro and colleagues,
I agree that it was a difficult task for scientists like Shannon to
discern the connection between entropy and information. You write,
Koichiro, that �it may be extremely difficult to say something decent
about the relationship between entropy and information as of now�. But,
I believe that relationship is, since Jaynes work (E. T. Jaynes, Phys.
Rev. 10 6, 6, 620-630, 1957), almost entirely explicated.
As I�ve continued to argue in this forum, all information is physical,
Shannon�s equation for entropy (information) depicts the identical
property of a physical system as does thermodynamic entropy, an entropy
change is always equal and opposite to the information change (would
Shu-Kun disagree with that?), and finally, as Jaynes explained, the
Shannon formula prescribes the amount of information which will erase
the uncertainty (entropy) in the probability distribution for any
thermodynamic system.
Previously, I�ve maintained that the identity of the mathematical form
of von Neumann�s quantum entropy and Shannon�s prescription (often
called the von Neumann-Shannon formula) is no coincidence. Shannon tells
us that information is just removal of the uncertainty of the outcome of
a measurement (Bell Sys. Tech. J., 28, 3, p. 379, 1948). But, that�s
exactly what von Neumann described some sixteen years before Shannon.
The quantum wave function predicts the probable outcome of the next
measurement of a system, and the von Neumann formula gives the
uncertainty in that measurement outcome.
In my view, the principle unresolved questions about information and
entropy pertain to time-dependent information where no heat flows
through the system, to the fundamental problem for computer information
storage regarding the entropy cost of writing and erasing a sequence of
information bits, and to all those problems with non-equilibrium
thermodynamic systems. Are there other, additional areas of remaining
concern?
You write about Maxwell�s demon and the Szilard engine, Koichiro.
Maxwell suggested his demon, not as an example of failure of the Second
Law in special circumstances, but as a demonstration that the Second Law
is statistical in origin. And, Szilard�s engine (Z. Phys. 53, 1929, p.
840) is probably the best model for understanding why at least one
information bit is required to reduce the entropy of a confined ideal gas.
Szilard suggested that his engine could not violate the Second Law
because the information acquired by his �demon� observer would also
increase that demon�s entropy. Zurek (in Frontiers of Nonequilibrium
Statistical Physics, Plenum, New York, 1984, p. 151) analyzed the engine
quantum mechanically, and claimed to have affirmed Szilard�s suggestion.
Many other physicists have reached essentially the same conclusion.
But, in a recent publication (Found. Phys. Lett., 16, 1, 2003, p. 41 and
reprised on the web in Entropy) I�ve demonstrated that the measurement
necessary for engine operation actually LOWERS the entropy of the
measurement apparatus (the �demon�), and I listed several fatal mistakes
in Zurek�s analysis. More generally, Earmann and Norton (Stud. Hist.
Phil. Mod. Phys., 30, 1, 1999, pp. 1-40) reviewed many of the attempted
information-based exorcisms of Maxwell�s demon, including Zurek�s, and
found they all lacked credibility. They conclude, as Maxwell did, that
the �demon� verifies the statistical character of the Second Law; it
doesn�t violate that law.
You suggest, Koichiro, that the demon could be �a very strange folk
contacting two worlds, very hot and very cold, at the same time....� And
that the Maxwell demon �can take advantage of quantum coherence as a
means of living in the two worlds, hot and cold, simultaneously.� But,
quantum coherence is just the state of any physical system (usually a
compound system of elementary constituents, like the S-state of two
polarized photons) which survives between measurements (information
transfers) of the system. The wave function is reduced by measurement
(Zurek�s decoherence), and that coherent state is annihilated, while
another is created. So, I suppose your proposed �demon�, Koichiro, could
contact both a hot and cold reservoir, just like any classical
non-equilibrium thermodynamic device.
You write that �Forming a quantum coherence effectively at a low
temperature and letting the once cohered decohere effectively at a high
temperature is the way of living on the part of the shrewd Maxwell
demon.� This just means that the �demon� would make one measurement at
the low temperature and then another at high temperature.
But, a �demon� is conceived of as a device that might somehow lower the
entropy of a thermodynamic system without increasing its own entropy or
doing work. So, to approximate plausibility, such a device cannot,
itself, be a heat engine operating between different temperatures. (It
is no surprise, after all, that an engine can do work which compresses
an ideal gas, say, lowering its entropy.) It must be in thermal
equilibrium with the thermodynamic system, and is usually portrayed as a
measuring device which detects particular properties of the constituents
of that thermodynamic system.
I�m unaware of any physically conceivable device (no violations of
existing physical laws) with a believable claim to challenge the Second
Law. Sheehan�s extensive menagerie of devices, for example, were all
spawned from a relatively obvious approximation error in his initial
publication.
Thanks for your very interesting and galvanizing postings.
Cordially,

Michael Devereux

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Received on Wed Jun 2 08:22:01 2004