Stephen P. King (stephenk1@home.com)
Thu, 09 Sep 1999 17:04:02 -0400
Dar Matti,
Matti Pitkanen wrote:
snip
[SPK]
> > Frieden has pointed out that the Shannon entropy is a "global measure",
> > should we not be conserned with the local measure, since the particular
> > observer's perceptions are restricted to local measures? Or, is this how
> > you define the NMP so that it choses from a global set?
[MP]
> Shannon entropy defined here is local in the sense that it applies
> inside each self, be it DNA triplet or human. Locality
> is always relative to some scale.
This is an example of what Hitoshi calls <glocal>. The localization is
relative to the scale set by the LSs propagator... I was thinking more
of the difference between Shannon and Fisher information as discussed in
Frieden's paper "Lagrangians of physics and the game of
Fisher-information transfer" By Frieden and Soffer. I'll include it in
my package of papers that I am putting together for you...
snip
[MP]
> > > In p-adic context one must defined logarithm appropriately
> > > and this leads to some exotic effects (entanglement without
> > > entanglement entropy).
> > > ***************
[SPK]
> > Umm, that is interesting but I wish you could give an example of
> > "entanglement without entanglement entropy". :-) I am still building my
> > intuitions about p-adics..
> [MP] p-Adic logarith is counter part of p-based
> real logarithm is defined in such a manner that it satisfies
> the usual sum rule
>
> Log_p[SUM(n=n0) x_n p^n ]= n_0.
>
> It is easy to verify that Log_p(xy)= Log_p/x) + Log_p(y).
> This guarantees the addivity of entropy.
How do we consider the case where x and y interact?
> When x is of form
>
> x_0 +x_1p+...with x_=1,2...., or p-1
>
> one has n_0=0 and p-adic logarith vanishes!
>
> This means that p-adic entanglement entropy vanishes if
> p-adic entanglment probabilities p_n have unit p-adic norm.
> For instance, p_n could be of form
>
> p_k = n_k/N
>
> N= SUM(k) n_k
>
> such that n_k and N are not divisible by p.
This is facinating! The p-adicity makes the entanglement entropy
selective to the value of the prime! Thank you for explaining this to
me. I would like to have a concrete example to test my intuitions
against... :-)
[SPK]
> > > I think that the determination of which "subsystem-complement"
> > > pair has
> > > the minimal quantum entropy is given by a tournament of games "played"
> > > between the pairs. The winner of the tournament is the quantum state
> > > that is the most informative. I see the "tournament" as modelable by a
> > > periodic gossiping on graphs formalism.
> > > http://www.cs.wvu.edu/~chif/cs418/1.html
> > >
> > > [MP] This would represent attempt to reduce quantum jump to
> > > classical computation. What makes me sceptic are Bell inequalities
> > > plus my belief that genuine (not completely) free will resides in quantum
> > > jump. Quantum jump is not reducible to process, quantum jump
> > > is the Spirit, the Godly.
[SPK]
> > I agree, tenatively! I am not sure how to derive the Bell Inequalities
> > from the statistics of tournaments, but I am certain that they can be
> > given since the distinction between classical and quantum computation is
> > that the former does not consider ensembles of systems while the later
> > does.
>
> [MP] I think that the unitary time development defining
> large number of parallel computations (N computations for N-dimensional
> system) is the basic difference. Quantum jump halting the computation
> selects one computation. This is what makes quantum computation so
> effective that it could make finding of prime decompositions of integeres
> child's play some day.
Ok, I think that we need to look long and hard at this issue! I have
found several papers on quantum computation; I'll pass them along to you
if you like.
> > > [SPK] The main ideas presupposes that "subsystem-complement" pairs can
> > > communicate with each other. I suspect that this follows some thing like
> > > this: Subsystem A <-> Complement B, Subsystem B <-> Complement A. If the
> > > complement of subsystem A is subsystem B and the complement of subsystem
> > > B the subsystem A, then subsystems A and B have identical entanglement
> > > entropy or information.
> > >
> > > [MP] I have for a long time pondered the problem whether this is
> > > the case and I have been even enthusiastic about this idea: its
> > > indeed suggested by quantum measurement theory.
> > >
> > > The notion of self seems to resolve the question finally:
> > > communication is *not* in question in the sense one might
> > > think. The self containing *both* the subsystem *and* its complement is
> > > the basic experiencer. Not the subsystem or/and its complement.
[SPK]
> > Yes, I believe you are right about that. I needed to test a
> > hypothesis... The key point is that we can model the interaction of
> > observers as resulting in an equilibration in both thermodynamic and
> > information theoretical terms!
> [MP] Average entanglement entropy probably corresponds to thermodynamic
> entropy but I am somewhat cautious here. The point is that
> thermodynamic entropy is concept characterizing ensemble.
> Entanglement entropy characterizes single subsystem: purely
> quantum mechanical concept is in question. No idealizations
> brought in by thermodynamics.
Yes, but the equilibration of temperatures, etc. is a real phenomenon!
The same ensemble approach that props up out thinking of probability
waves and information theory so we must be consistent here! Perhaps we
need to look at this concept more closely!
[MP]
> > > Note: the map m-->M(m) defined by the diagonalized
> > > density matrix maps the states of the subsystem
> > > of self to the states of its complement in self and is
> > > perhaps analogous to *'bi-simulation map'* that Stephen has
> > > been talking.
> > >
> > > Entanglement would define the fundamental bisimulation.
> > > Subsystems of self would simulate each other just at the
> > > moment when they wake-up and reduce quantum entanglement
> > > to zero. When they are selves they do not anymore bisimulate.
> > > This would be sub-conscious bisimulation. Note that
> > > any entangled subsystem of self would unconsciously bisimulate its
> > > complement.
[SPK]
> > Bisimulation, by Peter's definition, captures the notion "underwhat
> > conditions do two systems have the same behaviour". The difficulty that
> > I see the "atomisity" of systems is not necessarily an absolute. We can
> > consider it to be such if we only are considering possible systems that
> > have similar enough subjective measures, e.g. clocking and gauging
> > standards.
> >
> [MP] Entanglement does not define bisimulation in this sense.
> It characterizes only measurement: map of the states of system
> A to those of B.
Round and round we go... What is a measurement?
[SPK]
> > > Now, what is a given pair of subsystems do not have complete
> > > agreements, but do share some information? (I see "information sharing"
> > > as the existence of identical configurations in the configuration space
> > > of each subsystem, following the logic that "identical configurations
> > > encode identical information".) Can we model how, given an initial
> > > common information, a pair of subsystems can evolve such that they
> > > become equivalent? This is what happens in the periodic gossiping
> > > situation, so I suspect that it may be useful.
> > > [MP] This would require precise specification of a model for
> > > interaction. As I mentioned: quantum entanglement defines
> > > a map between states of subsystem and its complement
> > > resembling bisimulation: M(m) simulates m and vice versa.
> > > Schrodinger cat bravely simulates atomic nucleus whose
> > > transition leads to the opening of the bottle of poison.
[SPK]
> > Can we think of this as a process, like Fitini Markopoulou's idea of
> > evolving sets? I see a loose analogy in her thinking and your q-jumps,
> > Matti, in that the set of past events is "updated", but there are a lot
> > of differences. I think that you sould take a hard look at the Category
> > theory approach!
> [MP] Category theory might provide stimulate some ideas. If I only had the
> time to do all those things I want to do! In any case, m-->M(m) does not
> mean that two systems have same behaviour: this is unfortunately the case.
Ok, but can there behaviour be similar enough so that the information
content can be related?
Later,
Stephen
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