On Jul 24, 2:25=A0pm, C Pedro wrote:
Penrose=92s argument is actually linked with his criticism about
decoherence, and about the nature of mixed-states.

His observations sound to me a lot like a major point I have with
those who make the following distinction:
* Mixed state as unit-norm weighted sum of pure states
vs.
* Mixed state as a (actual, but unknown) pure state (with our
ignorance of its actual identity represented by non-zero "probability
coefficients" attached to each possible pure state, the mixed state
can be).

Those who ascribe to the first interpretation treat the traditional
reduction process as
Pure state --(measurement)-- Mixed state --(reduction)-- Pure state.

If, however, you ascribe to the second interpretation, then there are
only TWO steps in reduction:
Pure state --(measurement)-- Mixed state =3D pure/but/unknown state.

Reduction is then literally all in the head -- a psychological mirage
not corresponding to anything in the objective world (other than the
objective world describing the physics of what's going on in your
brain when you learn the identity of the pure state the mixed state
actually is).

If one makes a distinction between a "mixed" vs. "pure-but-unknown"
state, then there is the obligation of pointing out an objective
physical distinction between the two -- an empirical test that would
allow us to distinguish between
0.5 S1 + 0.5 S2
vs.
S1 with probability 0.5 & S2 with probability 0.5.

Otherwise, an application of Occam's Razor could be used to tag the
distinction as superfluous, to identify the false distinction as a
case of Disflation (the opposite of Conflation), and collapse the 3
steps into 2. Then, we're no longer talking about evolution from pure
to mixed states, but simply stochastic jumps between pure states.

In fact, working backwards: this leads to the notion that though
"Projection" may not be derivable from "Evolution", because of the non-
unitary nature of the reduction process; perhaps "Evolution" can be
derived from "Projection" as a kind of large-number continuum limit of
discrete "Projection" events! That is: maybe there's a theorem
analogous to that in statistical distribution theory in which the
Schroedinger equation plays the role analogous to that of the Gaussian
distribution.

None of the above is to say that the whole enterprise of mixed states
becomes superfluous, in itself! Any boundary you draw in the space-
time manifold between "region comprising system", vs. "region
comprising detector/observer", vs. "external region" automatically
entails entanglement entropy associated with the boundaries of cut-off
and mixed states. These are "improper mixtures".

In fact, one could conceivably take this to an extreme point -- there
is no universal state space, and resort to such cut-offs is
fundamentally necessary. That is, in effect: the universe is an open
system. An infinitely complex nesting of subsystem within subsystem,
no universal covering of it all.

The simplest, and most clear, way this can happen is if the space-time
manifold, in its entirety, is simply not globally hyperbolic and
admits no quantum formalism at all for the entire manifold. Then
you're forced to resort to "locally hyperbolic" submanifolds (that is,
compact 4-dimensional regions that -- as manifolds in their own right
-- as globally hyperbolic, but are do not have a causally convex
embedding into the entire manifold).