Re: Conservation of information (long)

David Bowman (
Thu, 19 Aug 1999 19:56:20 -0400 (EDT)

Regarding Brian Harper's comments & question:
>Well, I had always taken the conservation of information to be
>a bold and daring hypothesis. Lest people misunderstand, bold and
>daring hypotheses are good things is science :).

I with both these comments.

>After all the
>discussion surrounding this hypothesis I was really floored to
>read the following in <Science>:
>#"Basic principles of physics teach that information in
.#the universe is preserved: If you had perfect knowledge
>#of the present, you could, in theory at least, reconstruct
>#the past and predict the future. (Such perfect knowledge
>#is impossible in practice, of course.) Suppose you threw
>#an encyclopedia into a fire, for example; if you had perfect
>#knowledge of the radiation emitted and the ensuing motions
>#of all the atoms and molecules, you could, with infinite
>#attention to the details, reconstruct the knowledge inside
>#the encyclopedia. Physicists refer to their equations as
>#'unitary'--that is, they preserve information."
>#-- Gary Taubes <The Holographic Universe>,
># Science 285:515, 23 July 1999.
>Quite a remarkable statement. I guess the first thing to
>occur to me is that what little I know about info theory
>is in the context either of communication theory (Shannon
>info theory) or complexity theory (algorithmic info theory).
>I was aware that physicists also use the word information,
>perhaps they mean something different by the term? Any
>physicists like to clarify? David Bowman?

Although I have not read the article in question (other than the part
provided by Brian's quote above), the above quote sounds like it is not a
main thesis, but is a setup for another shoe to drop.

First, the statement is a little misleading in that it conflates a
reconstruction of an initial physical state of some complicated object
with a reconstruction of the information contained in or coded for by
that state, and gives scant attention to any supplementary colateral
information required to carry out the reconstruction process, which is
nessessarily effectively infinite in practice. The article hides this
necessary quasi-omnicience in the phrase 'perfect knowledge', and,
to its credit, *admits* that its possession is impossible "in practice".

Second, The article misleadingly conflates the definition of the concept
of quantum unitarity with the property of being information preserving.
Physicists know what the author is saying here, but it is sloppy. In
quantum mechanics (and mathematical analysis as well) unitarity actually
denotes a property of a linear operator whereby its inverse is also
identical to its Hermitian adjoint (as a complex generalization of the
idea of an orthogonal transformation in a real-valued space). If a
linear operator is unitary then its action is a norm-preserving map on
the Hilbert space on which it acts. This is just a fancy abstract
generalization (appropriate to the infinite dimensional abstract Hilbert
space of quantum states of a dynamical system) of the simple idea of a
rotation in ordinary space. If we rotate all of R^3 space about some
arbitrary axis by some arbitrary angle (or, equivalently, rigidly rotate
the coordinate axes used to label the points in that space), then all the
points, vectors and other geometric objects in that space rotate
together and retain their same mutual relationships with respect to each
other. No part of the space is distorted by stretching, shrinking,
differentially shearing, etc. The *whole space* rotates *in tact*. This
means after the rotation the space is still identical to how it was
before the rotation, except for its overall orientation.

In quantum mechanics the 'evolution operator' that takes an arbitrary
initial quantum state and transforms it into the state it will
dynamically evolve into at some fixed later time *is* an *unitary*
operator on the Hilbert space of quantum states. IOW,
translating/transporting a system's quantum state into the future (or
into the past for that matter) is accomplished by a generalization of the
idea of a rotation of the space of quantum states which contains the
system's particular current state. It should be emphasized that this
Hilbert space does *not* encode the system's quantum state as a
particular 'point' in the space, (as, for instance, a classical state is
encoded in the phase space of classical dynamics), but it is a
*1-dimensional complex-valued subspace* of the whole Hilbert space, (i.e.
it is a 'line' through the origin). The Hilbert space of all conceivable
quantum states is an infinite dimensional-space where each
(complex-valued) "line" through the origin in that space represents a
different quantum state. As time goes on the space rotates in some
complicated way and each of the lines through the origin rotate into
other lines as the quantum state changes with time into different states.

This process is analogous to how in classical mechanics the system's
state is represented by a 'point' in phase space, and the Liouville
evolution of the system is represented by the Hamiltonian flow of system
points in phase space from place to place as time goes on. For such a
Hamiltonian system all the points in the phase space (each representing a
possible legitimate state of the system) flow as an 'incompressible
fluid' as time goes on. The points always retain the same "density" and
never get compressed together, nor do they locally expand becoming more
rarified. All that happens is that the points move in a complicated way
(on lower dimensional hypersurfaces of constant conserved quantities such
as energy or angular momentum, etc.) along local flows that locally may
only shear relative to other neighboring points, yet preserving their

In classical Hamiltonian dynamics there is the Liouville theorem that
insures that regions of phase space keep from contracting or expanding
under the action of the Hamiltonain flow. For such a dynamical system
the sum of the Lyaponev exponents adds up to zero. This means, among
other things, that a purely Hamiltonain system cannot be fully chaotic
(with all positive Lyaponev exponents in all directions of phase space).
Any sensitive dependence on initial conditions that the system displays
where two initially neighboring points diverge exponentially fast from
each other in some directions in phase space is compensated for by those
points exponentially converging toward each other in *other* directions
of phase space. (The Lyaponev exponents measure the rate of exponential
separation or convergence in some direction of neighboring points
depending on whether or not they are positive or negative.) It is
Liouville's theorem which preserves size of the local volume elements in
phase space that is the reason for this behavior (i.e the behavior of the
flow locally diverging in some directions and converging in others). For
such a system a perfect knowledge of the location of a given point, in
principle, allows the determination of all the other points on the
dynamical orbit of that point in phase space (whose ordered sequence
gives the dynamical evolution of the initial state in time). So each
initial state of the system (defined by all dynamical variables connected
with the fire and the encyclopedia) is connected by a unique sequence of
subsequent states whose final value represents a given configuration of
the atoms of the waste gases, soot, and released kinetic energy of
oxidized molecules. Perfect knowledge or exact specification of any
point on the sequence uniquely determines all the other points on the

In a quantum mechanical description a similar phenomenon occurs. Here
each state is a different line through the origin in an infinite
dimensional Hilbert space. As time goes on the whole hedgehog pattern
of all possible such lines rigidly rotates in a very complicated way
under the action of the unitary evolution operator. Each initial
line/state rotates in its own special way into other lines as time goes
on. The overall rotation pattern is rigid and is such that none of the
lines rotate on top of each other. The analog of the flow in phase space
being a volume conserving incompressible flow for a classical system
(which prevents any two differently moving phase points from ending up on
top of or crossing each other) is, in the quantum system, the relative
mutual orientation preserving pattern for the infinite hedgehog pattern
of states as they all rotate together in complicated ways in time. The
unitarity of the evolution operator guanantees that each initial line
(orientated in its particular way in phase space) will undergo a *unique*
continuous sequence of rotated versions such that the knowledge or exact
specification of the orientation of any line in the sequence will
uniquely determine all the other lines in the continuous sequence. The
particular pattern generated by a given initial quantum state is found by
solving the time-dependent Schrodinger equation for that state (i.e. wave
function). BTW, each conceivable normalized quantum wave function
represents a particular unit vector in Hilbert space whose direction
identifies which line through the origin happens to represent the
system's state, and the different values of the wave function for
different values of its argument(s) are just the different coordinate
components of this unit vector referred to (i.e. projected against) the
various axes of some standard coordinate basis (e.g. position space,
momentum space, etc.) for the Hilbert space. The wave function just
tells us which line in Hilbert space happens to be the system's state.
The solution of the Schrodinger equation for an arbitrary initial state
represents the unitary evolution operator.

In principle, *both* the quantum system and the classical system possess
Laplacian *determinism* in the sense that a perfect specification of any
initial state perfectly determines a unique past/future history sequence
of states continuously connected to it. As long as measurements (or
measurment-like interactions of the system with its environment) are not
made, quantum mechanics is *just as* deterministic as classical
mechanics. What gives QM its randomness and its indeterminism is not its
(unitary) dynamics for the unmonitored, initially set up, closed system,
but the *interaction* of the system with an outside measuring apparatus
when the system is disturbed. This is because a measurement must be
made with an apparatus which is not a part of the system itself and
which probes it in some way which will yield some of the system's
properties, and this yielding is often the result of a random choice
when the system's state does not have a unique value for the quantity to
be measured, and the measurment act forces the system to (randomly)
choose a unique value for the measured property for the system to
possess. The measurement process also neccessitates a randomization of
the potential values of other properties which are not measured and which
are not compatible with the kind of property being measured.

In practice, *both* classical mechanics, and quantum mechanics, are
indeterminant in operation (over a sufficiently long time), because in
both cases the system's determinism depends *crucially* on the system
being closed to *all* outside influences no matter how small. In the
real world no system--especially those with any signifiant degree of
complexity, can be sufficiently isolated from its environment well
enough so that all possible outside influences don't mess up
deterministic sequences of states after enough time elapses. In the
quantum case the sequence typically gets disturbed almost immediately
after the initial state is prepared when the system's environment
effectively performs measurment-like disturbances on it, thereby causing
its subsequent behavior to be completely unpredictable. In the classical
case the deterministic sequence effectively becomes unpredictable after
any infinitesimal disturbance from the outside is amplified by the
system's positively exponentially growing Lyaponev flow in phase space to
the point where the disturbed and the undisturbed trajectories go off to
completely different regions of phase space.

>More interesting would be if anyone would like to defend
>that statement.

I won't defend it. But the author does admit his scenario is impossible
in practice.

>A bad sign is that the author talks about
>the knowledge inside the encyclopedia. The best case scenario
>would be that with "infinite attention to the details" one
>might be able to reconstruct the physical arrangement of
>ink on paper. But this physical arrangement does not itself
>reflect knowledge.

It *does* reflect the knowledge of the atomic arrangements of the ink
and paper atoms, but it does *not* reflect any knowledge of the *meaning*
of the information (coded in a symbolized form at the macroscopic level)
that we associate with the meaning of the contents of the articles in the

>But even this ideal case I would say is
>still impossible. It is reminiscent of Laplace's dream,
>which I had thought that most everyone now recognizes as
>a pipe dream. Am I wrong about this?

I believe you are correct. It *is* a pipe dream that cannot be carried
out in practice, or even in principle. In the classical case it is
impossible in principle to initially prepare a classical state of the
system and to perfectly isolate it with perfectly infinite precision in
a finite amount of time. In the quantum case it is impossible in
principle to prepare a complete set of mutually commuting observables
with perfect precision (get the initial state properly prepared), and to
perfectly isolate the system from all outside influences--especially in a
finite amount of time. Because the inital state is not really perfectly
precisely defined (in the classical case) it is not really represented by
a point in phase space but by a small finite region which, under the
influence of the Hamiltonian flow, is sheared to such a great extent that
in a short amount of time, typically, different parts of the initial
region end up in widely scattered places in phase space. Besides this,
the lack of perfect isolation from the outside means that the subsequent
dynamical evolution of the state of the system is not uniquely
determined. Even if the system started out represented by a point in
phase space its flow with time results in its trajectory broadening out
into a ever wider tube (rather than remaining an infinitely thin curved
path) as different possible disturbances from the outside give an ever
wider range of subsequent possible future paths for the system to follow.
The effects of the external disturbances on the flow is to blur the
stream lines with an ever greater amount of bluring as time goes on.
Similar effects happen in the quantum case, except the problem is much
worse. It is *much* harder for even a moderately complicated system to
narrow down the range of possiblities of the initial state to 1, and it
is *much* harder for the effects of outside influences to be suppressed
to the point where they will not mess up subsequent development of the
system's future state with time.

David Bowman