Thursday, August 25, 2016

Cloning of maximally negentropic states is possible: DNA replication as cloning of this kind of states?

In Facebook discussion with Bruno Marchal and Stephen King the notion of quantum cloning as copying of quantum state popped up and I ended up to ask about approximate cloning and got a nice link about which more below. From Wikipedia one learns some interesting facts cloning. No-cloning theorem states that the cloning of all states by unitary time evolution of the tensor product system is not possible. It is however possible clone orthogonal basis of states. Does this have some deep meaning?

As a response to my question I got a link to an article of Lamourex et al showing that cloning of entanglement - to be distinguished from the cloning of quantum state - is not possible in the general case. Separability - the absence of entanglement - is not preserved. Approximate cloning generates necessarily some entanglement in this case, and the authors give a lower bound for the remaining entanglement in case of an unentangled state pair.

The cloning of maximally entangled state is however possible. What makes this so interesting is that maximally negentropic entanglement for rational entanglement probabilities in TGD framework corresponds to maximal entanglement - entanglement probabilities form a matrix proportional to unit matrix- and just this entanglement is favored by Negentropy Maximization Principle . Could maximal entanglement be involved with say DNA replication? Could maximal negentropic entanglement for algebraic extensions of rationals allow cloning so that DNA entanglement negentropy could be larger than entanglement entropy?

What about entanglement probabilities in algebraic extension of rationals? In this case real number based entanglement entropy is not maximal since entanglement probablities are different. What can one say about p-adic entanglement negentropies: are they still maximal under some reasonable conditions? The logarithms involved depend on p-adic norms of probabilities and this is in the generic case just inverse of the power of p. Number theoretical universality suggests that entanglement probabilities are of form

Pi= ai/N

with ∑ ai= N with algebraic numbers ai not involving natural numbers and thus having unit p-adic norm.

With this assumption p-adic norms of Pi reduce to those of 1/N as for maximal rational entanglement. If this is the case the p-adic negentropy equals to log(pk) if pk divides N. The total negentropy equals to log(N) and is maximal and has the same value as for rational probabilities equal to 1/N.

The real entanglement entropy is now however smaller than log(N), which would mean that p-adic negentropy is larger than the real entropy as conjectured earlier (see this). For rational entanglement probabilities the generation of entanglement negentropy - conscious information during evolution - would be accompanied by a generation of equal entanglement entropy measuring the ignorance about what the negentropically entangled states representing selves are.

This conforms with the observation of Jeremy England that living matter is entropy producer (for TGD inspired commentary see this). For algebraic extensions of rationals this entropy could be however smaller than the total negentropy. Second law follows as a shadow of NMP if the real entanglement entropy corresponds to the thermodynamical entropy. Algebraic evolution would allow to generate conscious information faster than the environment is polluted, one might concretize! The higher the dimension of the algebraic extension rationals, the larger the difference could be and the future of the Universe might be brighter than one might expect by just looking around! Very consolating! One should however show that the above described situation can be realized as NMP strongly suggests before opening a bottle of champaigne;-).

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.


At 4:51 PM, Blogger Stephen Paul King said...

My definition of Intelligence is this: the capacity of a system to tend to increase the number of its (macroscopically distinguishable) possible future states. Note that this implied that intelligence acts against the tendency of thermodynamic entropy, which is to decrease such states.

At 8:45 PM, Anonymous said...

Distinguishability looks rather sensible notion. One an wonder how the difference of negentropy (sum of negative p-adic entnglement entropies) and real entropy relates to this. The higher the algebraic complexity of algebraic extension, the larger this number. In any case, macroscopically distinguishable would be replaced with "distinguishable in given measurement resolution". algebraic extension of rationals. Monadic imbedding space/space-time surface consists of monads labelled by distinguished points with coordinates in algebraic extension of rationals realizes the finite resolution. Monads are however local continua, like charts of manifold: this both in real and various p-adic senses making possible to talk about solutions of field equations. What is remarkable that for rational points, the notion of p-adic angle (phase as root of unity is totally lost so that one could say that cognition is absent. Therefore N(p-adic)-S(real) could be a good measure for cognitive information. One must be however very cautious here.


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