Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 10 Sep 1999 08:48:15 +0300 (EET DST)
Dear Stephen,
[SPK]
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.
[SPK] 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...
[MP] OK.
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.
[SPK] How do we consider the case where x and y interact?
[MP]Entropy is additive only when x and y are selves. Interaction means
generation of entanglement in general and x and y form larger then one
must consider subsystem pairs of x and y. The new self is x+y, the
'irreducible whole' formed by x and y. By the way, in this manner probably
irreducible wholes are formed as mental images composed of smaller
mental images. The quantum jump decomposing x and y to
separate selves would perhaps corresponds to formation of association:
Grandma-apple, to use classical example.
> 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.
[SPK] 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... :-)
[MP] This is what occurs. Remember that entropy characterizes the
consciousness of subsystem characterized by prime p. p roughly measures
the intelligence of subsystem and sqrt(p) characterizes
its size. Upper bound for entropy and thus
for entanglement information gain in quantum jump is p*logp and
increases with p.
Note that vanishing entanglement entropy corresponds to large real
entropy. Note also that systems for which S=0 for all subsystem
complement pairs are good candidates asymptotic fixed points for evolution
by quantum jumps, enlightened p-adic Buddhas which cannot become more
enlightened!
I forgot to give an example. A good example is two-level system with
identical entanglement probabilities.
p_1=p_2=1/2.
a) In p>2-adic case entropy vanishes: 2 and hence also 1/2
has p-adic norm 1 for all p>2:
S=0.
b) In 2-adic case S_2= 1/2 log_2(1/2) +1/2log(1/2)= -1. It is
formally negative but as *2-adic* number. The real image
of S_2^R= (-1)_R= (1+2^1+2^2+2^3+....)= 1+1/2+1/4+... = 1/(1-1/2)= 2.
This is in based on 2-based logarithm and one must multiply by
log_2 to get the entropy in standard units:
S^R=2*log(2). In real case one would have S^R= log(2) so that
2-adic world gives larger entropy.
[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.
[SPK] 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.
[MP]
OK. I try to find time to read. I am working now busily with updating
last chapters of 'TGD inspired theory of cs..' and cannot promise
anything.
> > > [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.
[SPK] 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] You are right: note however that the concept temperature is concept
which is based on idealizations (extremely successful): 'thermal
entanglement' is only a particular case of entanglement: p_n =g_n
exp(-E_n/T).
Entanglement entropy involves no statistical assumptions, such as thermal
equilibrium, and precisely this makes entangnlement entropy interesting
concept from the point of view of consciousness theory which is
about nonstatistical aspects of qm, about quantum jump.
[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.
[SPK] Round and round we go... What is a measurement?
[MP]
a) Quantum jump to eigenstate of density matrix, one of thes m:s,
is measurement. In real context.
b) In p-adic context measurementis something more general: quantum
jump to an entangled state with vanishing entanglement entropy which
is *eigenstate of density matrix in generalized sense*: that
is density matrix is proportional to identity matrix: all eigenvalues
in subspace are same. In real context this kind of quantum jumps
are not allowed by strong NMP but are allowed by measurement
postulate alone.
If density matrix has some eigenspaces for which p_k= p
such that
Log_p(p_k)=0
then quantum jump can also occur to entangled state in this
kind of subspace such that entanglement entropy stays zero: final
state is S=0 entangled state. This is true if the dimension N
of final state degenerate space is not divisible by p:
p_p(final)= 1/N so that Log_p(1/N)=0.
But this is what only enlightened Buddhas can do and, at least on my side,
we can safely forget this possibility! One could wonder whether
quantum computer technological applications might lurk behind.
I am afraid that enlightened Buddhas are not interested in achieving
Computer Like State since it would mean Final Enslavement.
[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.
[SPK] Ok, but can there behaviour be similar enough so that the
information
content can be related?
[MP] One of the paradoxal things is that entanglement entropy is
only potential information gained in quantum jump when illusions
disappear. Just as Eastern thinkers teach.
The final state has no information nor any entanglement entropy!
There are however infinite number of other types of information
and information measures for them. I think however that
information in the sense that we are accustomed to use it requires
the concept of self.
Best,
MP
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