Stephen P. King (stephenk1@home.com)
Fri, 28 May 1999 11:28:21 -0400
Dear Matti,
Getting back to your post.
Matti Pitkanen wrote:
> > > On Wed, 26 May 1999, Stephen P. King wrote:
> > Is there a function type relation between the entanglement entropy and
> > the number of "particles that are entangled"?
>
> [MP] Only the number of entangled states matters.
> If N states are entangled then maximum entropy is log(N). This follows
> directly from the definition of entanglement entropy. One can diagonilize
> density matrix and entropy in this basis is SUM p(n)log(p(n)), just
> the Shannon entropy: p(n) are eigevalues of the density matrix. Maximum
> entanglement entropy corresponds to p(n)= 1/N.
I need to get a better handle on this math. I am reading my information
theory and statistics books... :) Umm, are the terms modified for
situations were choices are not independent of ordering as in the case
of a series of observations of non-commutative measurements?
> > > Rather paradoxically, entanglement entropy would represent
> > > the potential information gain, and is *indeed a property of the
> > > initial state*! This was something which I have not realized
> > > earlier. Information would characterize the relationship between
> > > subsystem and complement. On the other hand subsystem and complement
> > > correspond to measured system and measurer, sender and receiver
> > > of information in some sense. Could one define quantum information as
> > > quantum entanglement between sender and receiver? Information
> > > would be identical with alertness/attentiveness!
> >
> > I can see that this function increases with the distance from the
> > observer, as in the case
> > were more and more particles would have interacted, and thus be
> > entangled, the further away one observes phenomena occurring. This would
> > also translate to time!
>
> [MP] Entanglement need *not* necessarily decrease with distance: entangled
> particles can travel arbitrary far form each other without any change
> in entanglement coefficients. This leads to the idea about
> telepathy made possible by entanglement.
The entangledness of particles, to me, seems to be what composes them
into a "subwhole" that is a Local System, and the notions of distance
and speed to not apply in the usual sense. I was talking in the section
above about the "outside" of the LS, e.g. "what is observed" by an LS.
LSs may have have "parts" that are identical to those of another
adjoint (?) LS and so we could say that they are causally entangled. I
need to discuss this idea more with Hitoshi, as it is his area.
snip
> [MP] Information concept is paradoxical and allows several
> interpretations.
>
> a) One could say that entanglement entropy measures
> amount of illusions, the uncertainy of subsystem about its own state:
> entanglement entropy would measure disinformation.
> In quantum jump subsystem is enlightened and goes to entropy zero state.
>
> b) On the other hand, one could associate to subsystem information as
> *entanglement negentropy gain*= entanglement entropy in quantum jump and
> with this interpretation entanglement entropy becomes information.
> In order to get enlightned one muust first generate illusions,
> the more illusions one has managed to generate, the better are changes
> for enlightment!
>
> Hopeless!!
Don't give up yet! :) It could be that you just need to go a bit
further! What is finite knowledge but "illusion" given that Absolute
knowledge is impossible?! But with a large repertoire of "illusions"
bounded truths can be communicated! More detail:
We can quantify the measure of "the uncertainly of subsystem about its
own state" as an intrinsic error e and show that it can be reduced
proportional to the number of simulations that can be performed by the
system to "check for error". In the limit, and only there, does the
error e-> 0 and the number N of trials go to infinity. This follow the
notion that only if we look at all possible predictions of a theory can
be be absolutely sure that it is an Absolute truth.
It seems that the usual probability theory assumes that the infinite
number of trials is available "in finite time" or it is just assumed to
exist a priori to construct the ensemble or infinite length time series.
This is a pure idealization that must be looked at with caution.
> [As a matter fact, 'englightened' S=0 subsystems can perform
> passive quantum jumps]
Ok, but this is independent of communications/interactions between such
subsystems, is it not?
snip
[MP]
> Yes. I consider quite seriously the possibility that cognitive spacetime
> sheets are the memes, living systems at the highler level of cognitive
> hierarhcy. Able to replicate and fill the Earth.
Umm, since I am distinguishing information from matter as categorically
different (dual in Pratt's CHu space sense), I would say that memes are
an example of "cognitive spacetime sheets" not the other way around. I
am trying to be more general... :)
[MP]
> > > c) Sensory perception as a quantum measurement?
> > >
> > > The interpretation of quantum jump as quantum measurement
> > > suggests that the information content of conscious
> > > experience is represented by the values of quantum numbers
> > > for the selected state. Information gain would correspond to the
> > > reduction of ignorance about quantum numbers.
[SPK]
> > Is this "reduction of ignorance" balance by a consumption of free
> > energy? The Maxwell Demon stuff comes to mind...
> > http://members.home.net/stephenk1/Outlaw/maxdemon.html
>
> [MP] Entanglement entropy characterizes single
> subsystem. Thermal entropy is statistical quantity: perhaps average
> of entanglement entropy for a large number of identical subsystems
> (I have been too lazy to really clear this to myself).
> Concepts of free energy and thermal entropy involve assumptions
> and statistical modelling of system. Entanglement entropy involves
> no such assumption and is more fundamental quantity.
Right, this correlates with what I said above about LSs as being
"subsystems were all parts are entangled". Thermal entropy is
temperature and relates to energy and black body radiation and all sorts
of phenomena that are essential aspects of QM! :) So it would help all
of us to investigate further! :)
The fact that it is a statistical quantity is the reason that we need
to discuss how to deal with statistics within our models.
snip
> > > i) The original motivation for the term 'color group' was that
> > > the Lie-algebra of the 'color group' SU(3) codes the algebra of
> > > color vision. Various values of color quantum numbers (isospin and
> > > hypercharge) would correspond to different colors (all colors can be
> > > formed by mixing two basic colours in suitable proportions) and the
> > > intensity of color would correspond to color Casimir operator, which is
> > > counterpart of angular momentum squared. Black is obtained, when
> > > Casimir goes to zero (no color rotation).
> >
> > Could you elaborate on this "Casimir operator"?
> >
> It is generalization of angular momentum squared. Angular momentum
> squared is J.J, where J is angular momentum vector understood
> as operator whose components are infinitesimal generators of rotations
> around various axis. Eigenvalues are J(J+1), J=0,1,2,....
> In case of color Casimir operator you have 8 component 'angular
> momentum'. Eigenvalues are also now discrete, I do not remember
> the precise formula. Color Casimir operator
> corresponds to Laplacian for CP_2 just like J^2 corresponds to Laplacian
> for CP_1= S^2. The simplest Hamiltonian for rotation rigid body is
> J^2 and same is true in case of color rotations.
Could we have a discussion about operators in general? The use of
infinitesimals is still problematic since we they are unobservable in
principle...
[SPK]
> > So would we say that the particular "experienced" colors are selected
> > by the "incoming light" or by a selective "game" played by the rods and
> > cones against the spectra of the incoming light? See Frieden's
> > discussion of "EPI as a game of knowledge acquisition" pg. 79-82, 275 of
> > his book.
>
> [MP] One might perhaps say that incoming light kicks rods and cones
> (both?) to color rotational motion and isospin and hypercharge depend
> on the wavelength of light. Our color experience would be kind of average
> over contributions of all rods and cones.
I am trying to see both sides, the "incoming light" and the "rods and
cones" as opposing players in the information acquisition game... I
believe that it is a mistake to assume that either one is a priori
definite in its properties.
> Strong NMP requires maximum negentropy gain in quantum jump and this
> favours simultaneous quantum jump for a large number of rods and cones.
> They must be quantum entangled mutually of their conscious
> experience are to integrate to single conscious experience.
Ok, but light has to be emitted by something and that ever the emitter
is, it is gaining information itself. The light would be the
"information channel".
> [Synchronous firing of neurons, syncronous assemly and disassembly of
> microtubules, perhaps also coordinated biochemical reactions
> could all be understood as consequence of strong NMP
> requiring simultaneous quantum jumps to maximize negentropy gain.]
I am reading more about this in Stuart Hammeroff's essay in The
Geometric Universe, S. A. Huggett et al eds. This is very interesting
stuff that I have been following for a while...
> > > iii) Classical gauge electromagnetic fields are accompanied by
> > > classical color fields and visual perception is known to generate
> > > electric field in retina. Therefore observed color experience
> > > would be the collective experience resulting from the simultaneous
> > > measurement of color rotational states of rods and cones.
> > >
> > > This picture seems to encourage the identification of information
> > > content of sensory experience as measured quantum numbers (but perhaps
> > > this is wishful thinking on my side(;-)).
> >
> > I see no conflicts so far. ;) But, again, such :identifications" are
> > intrinsic to particular finite "relationships between subsystem and
> > complement" and are thus not "absolute". I think this due to the very
> > nature of "knowability", e.g. it is not possible to "know" an infinite
> > amount of information...
> >
> [MP]
> Here our beliefs differ. I am sceptic about the concept of absolute
> information. I believe that information is thoroughly 'relativistic
> concept' because it is related to communication/measurement. I would
> like to see information in the relationship between teacher and
> student. If there is alertness, there is information in the
> relationship and strong NMP might allow the transformation of this
> information to conscious information in quantum jump. Information as
> alertness/as catchiness of potential conscious experience.
I fail to see our difference... :) As I explained elsewhere, I too
believe that "information is thoroughly 'relativistic concept' because
it is related to communication/measurement". :) I just think that we
need to consider its categorical status carefully. I am working on a
small papers about my ideas on Information-Matter duality, I am waiting
on Chalmer's book...
snip
> > Umm, this "sequence of passive quantum jumps", what determines the
> > number? This is "persistence"?
>
> [MP]
> The subsequent quantum jumps performed by other subsystems are bound to
> lead to a situation in which entanglement is generated between subsystem
> and complement. Then the perception of red no more continues.
> One could characterize persistence or duration of conscious experience by
> this number of quantum jumps.
OK. :)
snip
> > Two points: 1) How can we think of copying information in a
> > multi-observer quantum world? 2) what are the key differences and
> > similarities between "cognitive spacetime sheets" and "material
> > spacetime sheets". They look to me to be examples of Pratt's Mind and
> > Body as described in http://boole.stanford.edu/chuguide.html#ratmech
> > !!!!!! :) with the caveat that they have "reverse" dynamics...
> >
>
> [MP] The copying of classical information could be more or less copying
> of cognitive spacetime sheets. It would be counterpart of DNA
> replication/cell replication at the level of cognitive spacetime sheets:
> DNA replication is indeed geometric replication of 3-surfaces in TGD
> framework.
That would make sense but it looks like there has to be a separateness
between the "material" upon which information can be encoded by means of
ink spots, localized magnetizations, neuron connections, etc. There is
also the matter of a decoding scheme to transform the material encoding
to another form.... so that it can be communicated. I am not saying that
information is a substance, like the old theory of heat, but that it has
a categorical existence that is different from that of matter... The
fact that the "same" information can be encoded and transformed between
many different material substances is the evidence that I start with to
form this hypothesis. And Descartes' reasoning... :)
> The first possibility is that replication is driven by purely
> classical dynamics. I have tried to understand how the absolute
> minimization of Kahler action could somehow lead to replication but in
> vain.
>
> An alternative possibility is that quantum jumps lead to
> states containing replicas: this would be quantum self-organization.
> The 3-surfacse containing several more or less identical cognitive
> spacetime sheets would have especially large negative Kaehler action and
> would be favoured by the vacuum functional of the theory: quantum jumps
> would sooner or later lead to this kind of situation. Precisely like
> spin glass enters at the bottom of especially deep energy valleys.
> Note that in hydrodynamical self-organization structures consisting of
> identical basic units are created.
I think that part of quantum jumping is "replication". I believe that
we need to extend our notions. I like that Hitoshi's model gives us the
needed extension by relativizing and quantizing individual observers.
You see, the original Classical world was one were there is only one
observer, God, who can observe *all* of the subsets of the Universe with
arbitrary precision without disturbing them in any way. We now know that
this is a severe idealization and that we must construct our models such
that we are always considering the observables of finite subjects with
limited resolvability. This is also why I say that Oracles can not be
consulted in finite time to get "True" information...
> Cognitive spacetime sheets are almost vacuums, probably very few particles
> of ordinary matter. Cognitive spacetime sheets have finite TIME
> duration: this means that a little amount of energy flows to cognitive
> spacetime sheet from material spacetime sheet when it 'starts'. When
> cognitive spacetime sheet ends this energy flows back to the material
> spacetime sheet. This is like putting adding measurement instrument
> to electronic circuit: some current flows through circuit and provides
> 'cognitive representation' of circuit.
> If you want to grasp spacetime sheet concept instantaneously, I
> *strongly* (yes!) recommend the two-dimensional illustrations on my
> homepage. I have again and again found in various discussions that all
> my attempts to explain it verbally lead only
> to wrong associations. Still one stubborn attempt: You have certainly
> seen the Einstein-Rosen bridges in General Relativity books. Now you taken
> finite piece of plane (say disk) and glue it to larger disk by this kind
> of bridge. You can repeat this procedure. Same in higher dimensions.
I need to discuss this idea separately later...
> > Umm, perhaps intelligence is a measure of the range of different
> > meaningful questions that can be asked... or the number of different
> > simulations that can be performed given the observer's knowledge, this
> > would correspond to your suggestion...
>
> You are right: number of cognitive spacetime sheets describes the ability
> to simulate classically.
Ok, well continue... :)
Onward to the Unknown!
Stephen
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