### About quantum cognition

The talks in the conference Towards Science of Consciousness 2015 held in Helsinki produced several pleasant surprises, which stimulated more precise views about TGD inspired theory of consciousness. Some of the pleasant surprises were related to quantum cognition. It is a pity that I lost most of the opening talk of Harald Atmanspacher (see this).

The general idea is to look whether one could take the formalism of quantum theory and look whether it might allow to construct testable formal models of cognition. Quantum superposition, entanglement, and non-commutativity, are the most obvious notions to be considered. The problems related to quantum measurement are however present also now and relate to the basic questions about consciousness.

- For instance, non-commutativity of observables could relate to the order effects in cognitive measurements. Also the failure of classical probability to which Bell inequalities relate could have testable quantum cognitive counterpart. This requires that one should be able to speak about the analogs of quantization axis for spin in cognition. Representation of Boolean logic statements as tensor product of qubits would resolve the problem and in TGD framework fermionic Fock state basis defines a Boolean algebra: fermions would be interpretation as quantum correlates of Boolean cognition.

- The idea about cognitive entanglement described by density matrix was considered and the change of the state basis was suggested to have interpretation as a change of perspective. Here I was a little bit puzzled since the speakers seemed to assume that density matrix rather than only its eigenvalues has an independent meaning. This probably reflects my own assumption that density matrix is always assignable to a system and its complement regarded as subsystems of large system in pure state. The states are purifiable - as one says. This holds true in TGD but not in the general case.

- The possibility that quantum approach might allow to describe this breaking of uniqueness in terms of entanglement - or more precisely in terms of density matrix, which in TGD framework can be diagonalized and in cognitive state function reduction reduces in the generic case to a 1-D density matrix for one of the meanings. The situation would resemble that in hemispheric rivalry or for illusions in which two percepts appear as alternatives. One must be of course very cautious with this kind of models: the spoken and written language do not obey strict rules. I must however admit that I failed to get the gist of the arguments completely.

- One builds composite words from simpler ones. The proposed rule in classical linguistics is that the composites are describable as unique functions of the building bricks. The building brick words can however have several meanings and meaning is fixed only after one tells to which category the concept to which the world refers belongs. Therefore also the composite word can have several meanings.

- If the words have several meanings, they belong to at least n=2 two categories. The category associated with the word is like spin n=2 and one can formally treat the words as spins, kind of cognitive qubits. The category-word pairs - cognitive spins- serve building bricks for 2 composite worlds analogous to two-spin systems.

- A possible connection with Bell's inequalities emerges from the idea that if word can belong to two categories it can be regarded as analogous to spin with two values. If superpositions of same word with different meanings make sense, the analogs for the choice of spin quantization axis and measurement of spin in particular quantization direction make sense. A weaker condition is that the superpositions make sense only for the representations of the words. In TGD framework the representations would be in terms of fermionic Fock states defining quantum Boolean algebra.

- Consider first a situation in which one has two spin measurement apparatus A and B with given spin quantization axis and A' and B' with different spin quantization axis. One can construct correlation functions for the products of spins s
_{1}and s_{2}defined as outcomes of measurements A and A' and s_{3}and s_{4}defined as outcomes of B and B'. One obtains pairs 13, 14, 23, 24.

- Bell inequalities give a criterion for the possibility to model the system classically. One begins from 4 CHSH inequalities follow as averages of inequalities holding for individual measurement always (example: -2≤ s
_{1}s_{3}+ s_{1}s_{4}+s_{2}s_{3}- s_{2}s_{4}≤ 2) outcomes by*assuming*classical probability concept implying that the probability distributions for s_{i}s_{j}are simply marginal distributions for a probability distribution P(s_{1},2_{2},s_{3},s_{4}). CHSH inequalities are necessary conditions for the classical behavior. Fine's theorem states that these conditions are also sufficient. Bell inequalities follow from these and can be broken for quantum probabilities.

- Does this make sense in the case of cognitive spins? Are superpositions of meanings really possible? Are conscious meanings really analogous to Schrödinger cats? Or should one distinguish between meaning and cognitive representation? Experienced meanings are conscious experiences and consciousness identified as state function reduction makes the world look classical in standard quantum measurement theory. I allow the reader to decide but represent TGD view below.

- Consider first a situation in which one has two spin measurement apparatus A and B with given spin quantization axis and A' and B' with different spin quantization axis. One can construct correlation functions for the products of spins s

- In TGD quantum measurement is measurement of density matrix defining the universal observable leading to its eigenstate (or eigen space when NE is present in final state) meaning that degenerate eigenvalues of the density matrix are allowed). In the generic case the state basis is unique as eigenstates basis of density matrix and cognitive measurement leads to a classical state.

If the density matrix has degenerate eigenvalues situation changes since state function can take place to a sub-space instead of a ray of the state space. In this sub-space there is no preferred basis. Maybe "enlightened" states of consciousness could be identified as this kind of states carrying negentropy (number theoretic Shannon entropy is negative for them and these states are fundamental for TGD inspired theory of consciousness. Note that p-adic negentropy is well-defined also for rational (or even algebraic) entanglement probabilities but the condition that quantum measurement leads to an eigenstate of density matrix allows only projector as a density matrix for the outcome of the state function reduction. In any case, in TGD Universe the outcome of quantum measurement could be enlightened Schrödinger cat which is as much dead as olive.

Entangled states could represent concepts or rules as superpositions of their instances consisting of pairs of states. For NE generated in state function reduction density matrix would be a projector so that these pairs would appear with identical probabilities. The entanglement matrix would be unitary. This is interesting since unitary entanglement appears also in quantum computation. One can consider also the representation of associations in terms of entanglement - possibly negentropic one.

- Mathematician inside me is impatiently raising his hand: it clearly wants to add something. The restriction to a particular extension of rationals - a central piece of the number theoretical vision about quantum TGD - implies that density matrix need not allow diagonalization. In eigen state basis one would have has algebraic extension defined by the characteristic polynomial of the density matrix and its roots define the needed extension which could be quite well larger than the original extension. This would make state stable against state function reduction.

If this entanglement is algebraic, one can assign to it a negative number theoretic entropy. This negentropic entanglement is stable against NMP unless the algebraic extension associated with the parameters characterizing the parameters of string world sheets and partonic surfaces defining space-time genes is allowed to become larger in a state function reduction to the opposite boundary of CD generating re-incarnated self and producing eigenstates involving algebraic numbers in a larger algebraic extension of rationals. Could this kind of extension be an eureka experience meaning a step forwards in cognitive evolution?

If this picture makes sense, one would have both the unitary NE with a density matrix, which is projector and the algebraic NE with eigen values and NE for which the eigenstates of density matrix outside the algebraic extension associated with the space-time genes. Note that the unitary entanglement is "meditative" in the sense that any state basis is possible and therefore in this state of consciousness it is not possible to make distinctions. This strongly brings in mind koans of Zen buddhism. The more general algebraic entanglement could represent abstractions as rules in which the state pairs in the superposition represent the various instances of the rule.

- Can one really have superposition of meanings in TGD framework where Boolean cognitive spin is represented as fermion number (1,0), spin, or weak isospin in TGD, and fermion Fock state basis defines quantum Boolean algebra.

In the case of fermion number the superselection rule demanding that state is eigenstate of fermion number implies that cognitive spin has unique quantization axis.

For the weak isopin symmetry breaking occurs and superpositions of states with different em charges (weak isospins) are not possible. Remarkably, the condition that spinor modes have a well-defined em charge implies in the generic case their localization to string world sheets at which classical W fields carrying em charge vanish. This is essential also for the strong form of holography, and one can say that cognitive representations are 2-dimensional and cognition resides at string world sheets and their intersections with partonic 2-surfaces. Electroweak quantum cognitive spin would have a unique quantization axes?

But what about ordinary spin? Does the presence of Kähle magnetic field at flux tubes select a unique quantization direction for cognitive spin as ordinary spin so that it is not possible to experience superposition of meanings? Or could the rotational invariance of meaning mean SU(2) gauge invariance allowing to rotate given spin to a fixed direction by performing SU(2) gauge transformation affecting the gauge potential?

- A rather concrete linguistic analogy from TGD inspired biology relates to the representation of DNA, mRNA, amino-acids, and even tRNA in terms of dark proton triplets. One can decompose ordinary genetic codons to letters but dark genetic codons represented by entangled states of 3 linearly order quarks and do not allow reduction to sequence of letters. It is interesting that some eastern written languages have words as basic symbols whereas western written languages tend to have as basic units letters having no meaning as such. Could Eastern cognition and languages be more holistic in this rather concrete sense?

For a summary of earlier postings see Links to the latest progress in TGD.

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