Black Holes, Decoherence and
Objective Reduction, II

Date: Sun, 6 Oct 1996 00:14:28 -0700 (PDT)
From: Lawrence B. Crowell <lcrowell@unm.edu>
 and Andrew Matacz <andrewm@maths.su.oz.au>
To: quantum-d@teleport.com
Subject: Black Holes, Decoherence and Objective Reduction, part II

Black Holes, Decoherence and Objective Reduction, part II

  "The two decoherence mechanisms are similar at the level of the
   ensemble, but completely different when you talk of an individual
   system."
                              A. Matacz

Andrew Matacz, 9/13: Lawrence Crowell has suggested that a successful theory
of quantum biology is much more likely to emerge if we focus on thermal
environmental decoherence mechanisms rather the gravitational mechanism of
Penrose. This argument appears to use the two  mechanisms of decoherence
interchangeably which is dangerous. A common misconception regarding
environmental decoherence is that the environment objectively destroys
quantum coherence. This misconception comes from taking a density operator,
which provides the statistical information required to calculate ensemble
averages, and using it to make statements about actual "happenings" of
individual members of an ensemble. Environmental decoherence explains the
quantum-to-classical transition of an ensemble but this does not allow us
to say that an individual member of the ensemble has objectively undergone
a quantum-to-classical transition. In fact it does not allow us to say
anything about an individual system unless we also specify how the
environment is to be monitored. This leads into the concept of quantum
trajectories used in quantum optics.

This point is explained in detail in an excellent article by H.J  Carmichael
in "Quantum Optics VI" ed. J.D. Harvey and D.F. Walls.

He illustrates the point quite dramatically by showing how a decohered
classical ensemble can be constructed out of individual systems that
show macroscopic quantum coherence.

This is in stark contrast to the gravitational induced reduction mechanism
of Penrose which applies to an individual system and does objectively
describe the quantum-to-classical transition of an individual system.

Lawrence Crowell: The superscattering operator for the Hawking process
essentially converts a pure state into a mixed state.

The Hawking radiation has a blackbody distribution of states.  A system in
a pure state has a density matrix

   rho_{ij} = a_i^*a_j|j>

            = sum_i |a_i|^2 = sum_i(p_i) = 1,

which just expresses the sum over probabilities.  Since the system is in a
pure state then the trace over the square of the density matrix is also
unity

  Tr(rho^2) = sum__{ij}|a_i|^2|a_j|^2 = 1.

A collapse of a wave function is where the off diagonal terms vanish.
this reflects the destruction of interference effects.  Then the trace of
the square of the density matrix is now

  Tr(rho^2) = sum_i(p_i)^2 < 1.

The Von Neumann interpretation of the collapse is that an increase in
entropy

         S  = -k Tr(rho*log(rho))

has resulted.  The measurement process has reduced the total information
that was previously available from the wave function to one certain piece
of information concerned with the measurement.

The situation with the thermal radiance from a black hole is similar. The
vacuum modes of the field has geodesics that propagate into the black hole
and away from it.  Those modes that propagate into the hole are analytic
across the horizon at r = 2M, but those that propagate away are not. This
leads to a Green's function that is cyclic in time t ---> t + i(8pi/M).
This curious feature leads to the superscattering operator that fails to
preserve pure states. (This results in an information erasure in the scatter
of of quanta off the black hole.) This in the simplest of terms gives the
relationship between the collapse of a wave function and the decoherent
nature of radiation emitted from a black hole.

In both cases [environment decoherence and scattering off of a black hole]
you have a thermal macroscopic system that is constructed from individual
systems that are quantum statistical mixtures.

RS: Wasn't Andrew holding that the Penrose-Hameroff reduction is more
fundamental than any kind of environmental decoherence, black holes or no,
since their mechanism explicitly reduces the wavefunction to an actual
outcome instead of just to a classical density matrix?

LC: Andrew's problem is a subtle, maybe valid one. He raises the objection
of equating decoherence of a single system with macroscopic decoherence.
For the decoherence of a single system you are left with a system in a
mixed state with a probability that is not the modulus squared of a
quantum amplitude.  If you have an ensemble of these systems then this
probability, often a Boltzmann distribution, then describes the
statistical occurrence of each of each of these systems in a certain
state.  I am invoking the ergodic hypothesis in saying this, even though
there are many that disagree with these notions.  I acknowledge that
there are other subtleties to this sort of issue, but I'll stick to this
for now.  If the measurement apparatus is regarded as a system with
a temperature, then this is a reasonable model for the occurrence of a
measurement record.  There is no theory that is going to say which of the
particular eigenstates the measurement apparatus is going to place the
quantum system into.

Much the same occurs with the decoherence of a wave function that scatter
off a black hole.  The system emerges as a statistical mixture with a
Boltzmann probability for a certain outcome.  Again the black hole emits
a gaggle of photons and the astronomer is left with a measurement of a
black body radiation spectrum.

AM: The Penrose objective decoherence is really nothing more than the
well known Ghiradi, Rimini, Weber spontaneous localization theory.
This theory is a modification of QM that spontaneously localizes an
individual system. Because quantum theory is fundamentally changed
you can never get the same physics by appealing to environmental
decoherence. See for eg. PRA vol 51, pg 4404. The theory had little
effect on small systems but has a big effect on meso/macro systems.
Penrose simply claims that gravity is the cause of this spontaneous
localization. Whether or not you believe gravity is the cause doesn't
really matter. The point is many physicists philosophical view of
the world (eg Einstein, Penrose, Leggett) requires them to change QM
in such a way to give us an objective classical world. This is a
legitimate viewpoint that cannot be argued away. Arguing that gravity
can't play a role in biophysics misses the real point.

The thermal nature of Hawking radiation is an environmentally induced
effect. The Hawking vacuum is a pure state but you have particles going
into the black hole which are unobservable and must be coarse-grained.
This leads to the thermal nature of Hawking radiation. The key point
is the coarse-graining of the unobservable sector of your theory. Just
like you coarse grain out the environment of quantum brownian motion.
Its this coarse-graining which is the physical origin of decoherence
in both cases.

RS: You're saying that the Penrose mechanism is the familiar GRW
stochastic objective reduction, to be distinguished from any thermal
decoherence whether or not by black holes (that being an "environmentally
induced effect" arising within the theory). Is there any underlying
connection between these two pictures of collapse?

AM: What they have in common is that they both give decoherence.
They both lead to non-unitary dynamics and this is always described
via a master equation. The key point however is that a master
equation describes an ensemble only, ie an averaged dynamics over
many individual systems. It would be very nice to be able to
describe the quantum dynamics of an individual system. You can
do this by finding a stochastic schrodinger equation (SSE) which
when averaged over the classical noise driving the equation gives
back the master equation. Its the quantum version of going from
a fokker planck eqn to a langevin eqn in classical statistical
physics. The problem is a given master equation does not map to
a unique SSE. Each different SSE describes a very  different
quantum dynamics for an individual system. Environmental decoherence
in regular quantum mechanics can only be described by a master
eqn. Quantum mechanics does not allow you to go to SSE description
for an individual system for the reason above. The article I described
earlier by Carmicheal describes this. The GRW, Penrose decoherence
mechanisms are more fundamental because they postulate a unique SSE
for the dynamics of an individual system. Of course you could average
over this eqn and get a master eqn that may well be the same as your
favorite environmental decoherence master eqn. The two decoherence
mechanisms are similar at the level of the ensemble, but completely
different when you talk of an individual system.