[time 730] Re: [time 725] Entanglement defines the fundamental bi-simulation?

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 9 Sep 1999 17:45:11 +0300 (EET DST)

On Thu, 9 Sep 1999, Stephen P. King wrote:

> Dear Matti et al,
> Matti Pitkanen wrote:
> > Hi Stephen et al,
> > *****
> >
> > [SPK] Hi All,
> >
> > I am trying to understand Matti's Strong NMP idea. He gives a
> > definition:
> >
> > From: http://members.home.net/stephenk1/Outlaw/MattiQMind.htm
> >
> > "Strong NMP says that in given quantum state=quantum history the
> > quantum jump corresponds to a subsystem-complement pair for which the
> > *entanglement entropy reduction in quantum jump is maximal*.
> >
> > The first interpretation coming in mind is that the conscious
> > experiences is such that the information gain is maximal. Perhaps a more
> > natural interpretation is that entanglement entropy tells how
> > interesting, 'catchy', the conscious experience is and only the most
> > interesting experience is actually experienced."
> >
> > Now, we first need to understand what "entanglement entropy" is!
> > Does it have to do with "quantum entropy" re: "value of quantum entropy
> > gives you the upper limit for how much information you can recover from a
> > quantum particle or collections of them."
> >
> > [MP] Consider entangled state
> >
> > |A,B > = SUM(m,M) c_mM |m>|M>
> >
> > m and M refer to states of subsystem and its complement inside self.
> > One can always find state basis m and M such that entanglement
> > coefficients
> > c_mM are 'diagonal'
> [SPK]
> Are you using the sum SUM(m,M) because it is necessary to consider all
> possible subsystem - complement pairs?
> > |A,B > = SUM(m) c_m |m>|M(m)>
> >
> > m is state of subsystem and M(m) is state of its complement.
> How does the particular information content of the pairs enter into |A,
> B>, or is this precluded by taking the ensemble approach, e.g. the
> "SUM(m, M)"?
> > Density matrix represents the state of subsystem and is diagonal
> > in the basis just defined. Its diagonal elements are just
> > *entanglement probabilities*
> >
> > P_mn = p_m delta(m,n) = |c_m|^2 delta (m,n)
> >
> > telling the probability for that quantum jump occurs to state m
> > in measurement of density matrix.
> [SPK]
> This is the dirac delta function? How do we model composition?
> > Entanglement entropy is the Shannon entropy associated with the
> > probabolities p_m:
> >
> > S=-SUM(m) -p_m log(p_m)
> 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?

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.

> > Here it is!
> >
> > In p-adic context one must defined logarithm appropriately
> > and this leads to some exotic effects (entanglement without
> > entanglement entropy).
> > ***************
> 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.

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]
> snip
> > 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.
> 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] 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.
> 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.

> > 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.
> 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]
> > 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.
> >
> > The problem that I have is that I do not know how to show this
> > mathematically! Can you help me? :-)
> >

> > [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.
> 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.



This archive was generated by hypermail 2.0b3 on Sat Oct 16 1999 - 00:36:40 JST