# [time 725] Entanglement defines the fundamental bi-simulation?

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 9 Sep 1999 08:36:54 +0300 (EET DST)

Hi Stephen et al,
*****

[SPK] Hi All,

I am trying to understand Matti's Strong NMP idea. He gives a
definition:

"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'

|A,B > = SUM(m) c_m |m>|M(m)>

m is state of subsystem and M(m) is state of its complement.

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.

Entanglement entropy is the Shannon entropy associated with the
probabolities p_m:

S=-SUM(m) -p_m log(p_m)

Here it is!

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

also, this ties to Bill's event horizon thinking! :)

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

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] 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
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.

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]
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.

Best,
MP

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