Quantum cognition

Quantum cognition is an emerging field which applies the mathematical formalism of quantum theory to model cognitive phenomena such as information processing by the human brain, language, decision making, human memory, concepts and conceptual reasoning, human judgment, and perception.[1][2][3][4] The field clearly distinguishes itself from the quantum mind as it is not reliant on the hypothesis that there is something micro-physical quantum mechanical about the brain. Quantum cognition is based on the quantum-like paradigm[5][6] or generalized quantum paradigm[7] or quantum structure paradigm[8] that information processing by complex systems such as the brain, taking into account contextual dependence of information and probabilistic reasoning, can be mathematically described in the framework of quantum information and quantum probability theory.

Quantum cognition uses the mathematical formalism of quantum theory to inspire and formalize models of cognition that aim to be an advance over models based on traditional classical probability theory. The field focuses on modeling phenomena in cognitive science that have resisted traditional techniques or where traditional models seem to have reached a barrier (e.g., human memory),[9] and modeling preferences in decision theory that seem paradoxical from a traditional rational point of view (e.g., preference reversals).[10] Since the use of a quantum-theoretic framework is for modeling purposes, the identification of quantum structures in cognitive phenomena does not presuppose the existence of microscopic quantum processes in the human brain.[11]

Main subjects of research
Quantum-like models of information processing ("quantum-like brain")

The brain is definitely a macroscopic physical system operating on the scales (of time, space, temperature) which differ crucially from the corresponding quantum scales. (The macroscopic quantum physical phenomena, such as the Bose-Einstein condensate, are also characterized by the special conditions which are definitely not fulfilled in the brain.) In particular, the brain’s temperature is simply too high to be able to perform the real quantum information processing, i.e., to use the quantum carriers of information such as photons, ions, electrons. As is commonly accepted in brain science, the basic unit of information processing is a neuron. It is clear that a neuron cannot be in the superposition of two states: firing and non-firing. Hence, it cannot produce superposition playing the basic role in the quantum information processing. Superpositions of mental states are created by complex networks of neurons (and these are classical neural networks). Quantum cognition community states that the activity of such neural networks can produce effects formally described as interference (of probabilities) and entanglement. In principle, the community does not try to create the concrete models of quantum (-like) representation of information in the brain.[12]

The quantum cognition project is based on the observation that various cognitive phenomena are more adequately described by quantum information theory and quantum probability than by the corresponding classical theories (see examples below). Thus the quantum formalism is considered an operational formalism that describes nonclassical processing of probabilistic data. Recent derivations of the complete quantum formalism from simple operational principles for representation of information support the foundations of quantum cognition. The subjective probability viewpoint on quantum probability developed by C. Fuchs and his collaborators also supports the quantum cognition approach, especially using quantum probabilities to describe the process of decision making.[13]

Although at the moment we cannot present the concrete neurophysiological mechanisms of creation of the quantum-like representation of information in the brain,[14] we can present general informational considerations supporting the idea that information processing in the brain matches with quantum information and probability. Here, contextuality is the key word, see the monograph of Khrennikov for detailed representation of this viewpoint.[1] Quantum mechanics is fundamentally contextual.[15] Quantum systems do not have objective properties which can be defined independently of measurement context. (As was pointed by N. Bohr, the whole experimental arrangement must be taken into account.) Contextuality implies existence of incompatible mental variables, violation of the classical law of total probability and (constructive and destructive) interference effects. Thus the quantum cognition approach can be considered as an attempt to formalize contextuality of mental processes by using the mathematical apparatus of quantum mechanics.
Decision making

Suppose a person is given an opportunity to play two rounds of the following gamble: a coin toss will determine whether the subject wins $200 or loses $100. Suppose the subject has decided to play the first round, and does so. Some subjects are then given the result (win or lose) of the first round, while other subjects are not yet given any information about the results. The experimenter then asks whether the subject wishes to play the second round. Performing this experiment with real subjects gives the following results:

When subjects believe they won the first round, the majority of subjects choose to play again on the second round.
When subjects believe they lost the first round, the majority of subjects choose to play again on the second round.

Given these two separate choices, according to the sure thing principle of rational decision theory, they should also play the second round even if they don't know or think about the outcome of the first round.[16] But, experimentally, when subjects are not told the results of the first round, the majority of them decline to play a second round.[17] This finding violates the law of total probability, yet it can be explained as a quantum interference effect in a manner similar to the explanation for the results from double-slit experiment in quantum physics.[2][18][19] Similar violations of the sure-thing principle are seen in empirical studies of the Prisoner's Dilemma and have likewise been modeled in terms of quantum interference.[20]

The above deviations from classical rational expectations in agents’ decisions under uncertainty produce well known paradoxes in behavioral economics, that is, the Allais, Ellsberg and Machina paradoxes.[21][22][23] These deviations can be explained if one assumes that the overall conceptual landscape influences the subject's choice in a neither predictable nor controllable way. A decision process is thus an intrinsically contextual process, hence it cannot be modeled in a single Kolmogorovian probability space, which justifies the employment of quantum probability models in decision theory. More explicitly, the paradoxical situations above can be represented in a unified Hilbert space formalism where human behavior under uncertainty is explained in terms of genuine quantum aspects, namely, superposition, interference, contextuality and incompatibility.[24][25][26][19]

Considering automated decision making, quantum decision trees have different structure compared to classical decision trees. Data can be analyzed to see if a quantum decision tree model fits the data better.[27]
Human probability judgments

Quantum probability provides a new way to explain human probability judgment errors including the conjunction and disjunction errors.[28] A conjunction error occurs when a person judges the probability of a likely event L and an unlikely event U to be greater than the unlikely event U; a disjunction error occurs when a person judges the probability of a likely event L to be greater than the probability of the likely event L or an unlikely event U. Quantum probability theory is a generalization of Bayesian probability theory because it is based on a set of von Neumann axioms that relax some of the classic Kolmogorov axioms.[29] The quantum model introduces a new fundamental concept to cognition—the compatibility versus incompatibility of questions and the effect this can have on the sequential order of judgments. Quantum probability provides a simple account of conjunction and disjunction errors as well as many other findings such as order effects on probability judgments.[30][31][32]

The liar paradox - The contextual influence of a human subject on the truth behavior of a cognitive entity is explicitly exhibited by the so-called liar paradox, that is, the truth value of a sentence like "this sentence is false". One can show that the true-false state of this paradox is represented in a complex Hilbert space, while the typical oscillations between true and false are dynamically described by the Schrödinger equation.[33][34]
Knowledge representation

Concepts are basic cognitive phenomena, which provide the content for inference, explanation, and language understanding. Cognitive psychology has researched different approaches for understanding concepts including exemplars, prototypes, and neural networks, and different fundamental problems have been identified, such as the experimentally tested non classical behavior for the conjunction and disjunction of concepts, more specifically the Pet-Fish problem or guppy effect,[35] and the overextension and underextension of typicality and membership weight for conjunction and disjunction.[36][37] By and large, quantum cognition has drawn on quantum theory in three ways to model concepts.

Exploit the contextuality of quantum theory to account for the contextuality of concepts in cognition and language and the phenomenon of emergent properties when concepts combine[11][38][39][40][41]
Use quantum entanglement to model the semantics of concept combinations in a non-decompositional way, and to account for the emergent properties/associates/inferences in relation to concept combinations[42]
Use quantum superposition to account for the emergence of a new concept when concepts are combined, and as a consequence put forward an explanatory model for the Pet-Fish problem situation, and the overextension and underextension of membership weights for the conjunction and disjunction of concepts.[30][38][39]

The large amount of data collected by Hampton[36][37] on the combination of two concepts can be modeled in a specific quantum-theoretic framework in Fock space where the observed deviations from classical set (fuzzy set) theory, the above-mentioned over- and under- extension of membership weights, are explained in terms of contextual interactions, superposition, interference, entanglement and emergence.[30][43][44][45] And, more, a cognitive test on a specific concept combination has been performed which directly reveals, through the violation of Bell's inequalities, quantum entanglement between the component concepts.[46][47]
Human memory

The hypothesis that there may be something quantum-like about the human mental function was put forward with the quantum entanglement formula which attempted to model the effect that when a word's associative network is activated during study in memory experiment, it behaves like a quantum-entangled system.[9] Models of cognitive agents and memory based on quantum collectives have been proposed by Subhash Kak.[48][49] But he also points to specific problems of limits on observation and control of these memories due to fundamental logical reasons.[50]
Semantic analysis and information retrieval

The research in (iv) had a deep impact on the understanding and initial development of a formalism to obtain semantic information when dealing with concepts, their combinations and variable contexts in a corpus of unstructured documents. This conundrum of natural language processing (NLP) and information retrieval (IR) on the web – and data bases in general – can be addressed using the mathematical formalism of quantum theory. As basic steps, (a) K. Van Rijsbergen introduced a quantum structure approach to IR,[51] (b) Widdows and Peters utilised a quantum logical negation for a concrete search system,[41][52] and Aerts and Czachor identified quantum structure in semantic space theories, such as latent semantic analysis.[53] Since then, the employment of techniques and procedures induced from the mathematical formalisms of quantum theory – Hilbert space, quantum logic and probability, non-commutative algebras, etc. – in fields such as IR and NLP, has produced significant results.[54]
Human perception

Bi-stable perceptual phenomena is a fascinating topic in the area of perception. If a stimulus has an ambiguous interpretation, such as a Necker cube, the interpretation tends to oscillate across time. Quantum models have been developed to predict the time period between oscillations and how these periods change with frequency of measurement.[55] Quantum theory and an appropriate model have been developed by Elio Conte to account for interference effects obtained with measurements of ambiguous figures.[56][57][58][59]
Gestalt perception

There are apparent similarities between Gestalt perception and quantum theory. In an article discussing the application of Gestalt to chemistry, Anton Amann writes: "Quantum mechanics does not explain Gestalt perception, of course, but in quantum mechanics and Gestalt psychology there exist almost isomorphic conceptions and problems:

Similarly as with the Gestalt concept, the shape of a quantum object does not a priori exist but it depends on the interaction of this quantum object with the environment (for example: an observer or a measurement apparatus).
Quantum mechanics and Gestalt perception are organized in a holistic way. Subentities do not necessarily exist in a distinct, individual sense.
In quantum mechanics and Gestalt perception objects have to be created by elimination of holistic correlations with the 'rest of the world'."[60]

Each of the points mentioned in the above text in a simplified manner (Below explanations correlate respectively with the above-mentioned points):

As an object in quantum physics doesn't have any shape until and unless it interacts with its environment; Objects according to Gestalt perspective do not hold much of a meaning individually as they do when there is a "group" of them or when they are present in an environment.
Both in quantum mechanics and Gestalt perception, the objects must be studied as a whole rather than finding properties of individual components and interpolating the whole object.
In Gestalt concept creation of a new object from another previously existing object means that the previously existing object now becomes a sub entity of the new object, and hence "elimination of holistic correlations" occurs. Similarly a new quantum object made from a previously existing object means that the previously existing object looses its holistic view.

Amann comments: "The structural similarities between Gestalt perception and quantum mechanics are on a level of a parable, but even parables can teach us something, for example, that quantum mechanics is more than just production of numerical results or that the Gestalt concept is more than just a silly idea, incompatible with atomistic conceptions."[60]
Quantum-like models of cognition in economics and finance

The assumption that information processing by the agents of the market follows the laws of quantum information theory and quantum probability was actively explored by many authors, e.g., E. Haven, O. Choustova, A. Khrennikov, see the book of E. Haven and A. Khrennikov,[61] for detailed bibliography. We can mention, e.g., the Bohmian model of dynamics of prices of shares in which the quantum(-like) potential is generated by expectations of agents of the financial market and, hence, it has the mental nature. This approach can be used to model real financial data, see the book of E. Haven and A. Khrennikov (2012).
Application of theory of open quantum systems to decision making and "cell's cognition"

An isolated quantum system is an idealized theoretical entity. In reality interactions with environment have to be taken into account. This is the subject of theory of open quantum systems. Cognition is also fundamentally contextual. The brain is a kind of (self-)observer which makes context dependent decisions. Mental environment plays a crucial role in information processing. Therefore, it is natural to apply theory of open quantum systems to describe the process of decision making as the result of quantum-like dynamics of the mental state of a system interacting with an environment. The description of the process of decision making is mathematically equivalent to the description of the process of decoherence. This idea was explored in a series of works of the multidisciplinary group of researchers at Tokyo University of Science.[62][63]

Since in the quantum-like approach the formalism of quantum mechanics is considered as a purely operational formalism, it can be applied to the description of information processing by any biological system, i.e., not only by human beings.

Operationally it is very convenient to consider e.g. a cell as a kind of decision maker processing information in the quantum information framework. This idea was explored in a series of papers of the Swedish-Japanese research group using the methods of theory of open quantum systems: genes expressions were modeled as decision making in the process of interaction with environment.[64]
History

Here is a short history of applying the formalisms of quantum theory to topics in psychology. Ideas for applying quantum formalisms to cognition first appeared in the 1990s by Diederik Aerts and his collaborators Jan Broekaert, Sonja Smets and Liane Gabora, by Harald Atmanspacher, Robert Bordley, and Andrei Khrennikov. A special issue on Quantum Cognition and Decision appeared in the Journal of Mathematical Psychology (2009, vol 53.), which planted a flag for the field. A few books related to quantum cognition have been published including those by Khrennikov (2004, 2010), Ivancivic and Ivancivic (2010), Busemeyer and Bruza (2012), E. Conte (2012). The first Quantum Interaction workshop was held at Stanford in 2007 organized by Peter Bruza, William Lawless, C. J. van Rijsbergen, and Don Sofge as part of the 2007 AAAI Spring Symposium Series. This was followed by workshops at Oxford in 2008, Saarbrücken in 2009, at the 2010 AAAI Fall Symposium Series held in Washington, D.C., 2011 in Aberdeen, 2012 in Paris, and 2013 in Leicester. Tutorials also were presented annually beginning in 2007 until 2013 at the annual meeting of the Cognitive Science Society. A Special Issue on Quantum models of Cognition appeared in 2013 in the journal Topics in Cognitive Science.
Related theories

It was suggested by theoretical physicists David Bohm and Basil Hiley that mind and matter both emerge from an "implicate order".[65] Bohm and Hiley's approach to mind and matter is supported by philosopher Paavo Pylkkänen.[66] Pylkkänen underlines "unpredictable, uncontrollable, indivisible and non-logical" features of conscious thought and draws parallels to a philosophical movement some call "post-phenomenology", in particular to Pauli Pylkkö's notion of the "aconceptual experience", an unstructured, unarticulated and pre-logical experience.[67]

The mathematical techniques of both Conte's group and Hiley's group involve the use of Clifford algebras. These algebras account for "non-commutativity" of thought processes (for an example, see: noncommutative operations in everyday life).

However, an area that needs to be investigated is the concept lateralised brain functioning. Some studies in marketing have related lateral influences on cognition and emotion in processing of attachment related stimuli.
See also

Electromagnetic theories of consciousness
Holonomic brain theory
Quantum Bayesianism
Quantum logic
Quantum neural network
NeuroQuantology
Orchestrated objective reduction

References

Khrennikov, A. (2010). Ubiquitous Quantum Structure: from Psychology to Finances. Springer. ISBN 978-3-642-42495-3.
Busemeyer, J.; Bruza, P. (2012). Quantum Models of Cognition and Decision. Cambridge: Cambridge University Press. ISBN 978-1-107-01199-1.
Pothos, E. M.; Busemeyer, J. R. (2013). "Can quantum probability provide a new direction for cognitive modeling". Behavioral and Brain Sciences. 36: 255–274. doi:10.1017/S0140525X12001525.
Wang, Z.; Busemeyer, J. R.; Atmanspacher, H.; Pothos, E. M. (2013). "The potential of using quantum theory to build models of cognition". Topics in Cognitive Science. 5 (4): 672–688. doi:10.1111/tops.12043.
Khrennikov, A. (2006). "Quantum-like brain: 'Interference of minds'". Biosystems. 84 (3): 225–241. doi:10.1016/j.biosystems.2005.11.005.
Khrennikov, A. (2004). Information Dynamics in Cognitive, Psychological, Social, and Anomalous Phenomena. Fundamental Theories of Physics. 138. Kluwer. ISBN 1-4020-1868-1.
Atmanspacher, H.; Römer, H.; Walach, H. (2002). "Weak quantum theory: Complementarity and entanglement in physics and beyond". Foundations of Physics. 32 (3): 379–406. doi:10.1023/A:1014809312397.
Aerts, D.; Aerts, S. (1994). "Applications of quantum statistics in psychological studies of decision processes". Foundations of Science. 1: 85–97.
Bruza, P.; Kitto, K.; Nelson, D.; McEvoy, C. (2009). "Is there something quantum-like about the human mental lexicon?". Journal of Mathematical Psychology. 53 (5): 362–377. doi:10.1016/j.jmp.2009.04.004.
Lambert Mogiliansky, A.; Zamir, S.; Zwirn, H. (2009). "Type indeterminacy: A model of the KT (Kahneman–Tversky)-man". Journal of Mathematical Psychology. 53 (5): 349–361.arXiv:physics/0604166. doi:10.1016/j.jmp.2009.01.001.
de Barros, J. A.; Suppes, P. (2009). "Quantum mechanics, interference, and the brain". Journal of Mathematical Psychology. 53 (5): 306–313. doi:10.1016/j.jmp.2009.03.005.
Khrennikov, A. (2008). "The Quantum-Like Brain on the Cognitive and Subcognitive Time Scales". Journal of Consciousness Studies. 15 (7): 39–77. ISSN 1355-8250.
Caves, C. M.; Fuchs, C. A.; Schack, R. (2002). "Quantum probabilities as Bayesian probabilities". Physical Review A. 65 (2). 022305.arXiv:quant-ph/0106133. doi:10.1103/PhysRevA.65.022305.
Van den Noort, Maurits; Lim, Sabina; Bosch, Peggy (26 December 2016). "On the need to unify neuroscience and physics". Neuroimmunology and Neuroinflammation. 3 (12): 271. doi:10.20517/2347-8659.2016.55.
Khrennikov, A. (2009). Contextual Approach to Quantum Formalism. Fundamental Theories of Physics. 160. Springer. ISBN 978-1-4020-9592-4.
Savage, L. J. (1954). The Foundations of Statistics. John Wiley & Sons.
Tversky, A.; Shafir, E. (1992). "The disjunction effect in choice under uncertainty". Psychological Science. 3 (5): 305–309. doi:10.1111/j.1467-9280.1992.tb00678.x.
Pothos, E. M.; Busemeyer, J. R. (2009). "A quantum probability explanation for violations of 'rational' decision theory". Proceedings of the Royal Society. B: Biological Sciences. 276 (1665): 2171–2178. doi:10.1098/rspb.2009.0121. PMC 2677606.
Yukalov, V. I.; Sornette, D. (21 February 2010). "Decision theory with prospect interference and entanglement" (PDF). Theory and Decision. 70 (3): 283–328. doi:10.1007/s11238-010-9202-y.
Musser, George (16 October 2012). "A New Enlightenment" . Scientific American. 307 (5): 76–81. doi:10.1038/scientificamerican1112-76.
Allais, M. (1953). "Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'ecole Americaine". Econometrica. 21 (4): 503–546. doi:10.2307/1907921.
Ellsberg, D. (1961). "Risk, ambiguity, and the Savage axioms". Quarterly Journal of Economics. 75 (4): 643–669. doi:10.2307/1884324.
Machina, M. J. (2009). "Risk, Ambiguity, and the Rank-Dependence Axioms". American Economic Review. 99 (1): 385–392. doi:10.1257/aer.99.1.385.
Aerts, D.; Sozzo, S.; Tapia, J. (2012). "A quantum model for the Ellsberg and Machina paradoxes". In Busemeyer, J.; Dubois, F.; Lambert-Mogilansky, A. (eds.). Quantum Interaction 2012. LNCS. 7620. Berlin: Springer. pp. 48–59.
Aerts, D.; Sozzo, S.; Tapia, J. (2014). "Identifying quantum structures in the Ellsberg paradox". International Journal of Theoretical Physics. 53: 3666–3682.arXiv:1302.3850. doi:10.1007/s10773-014-2086-9.
La Mura, P. (2009). "Projective expected utility". Journal of Mathematical Psychology. 53 (5): 408–414.arXiv:0802.3300. doi:10.1016/j.jmp.2009.02.001.
Kak, S. (2017). Incomplete Information and Quantum Decision Trees. IEEE International Conference on Systems, Man, and Cybernetics. Banff, Canada, October. doi:10.1109/SMC.2017.8122615.
Tversky, A.; Kahneman, D. (1983). "Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment". Psychological Review. 90 (4): 293–315. doi:10.1037/0033-295X.90.4.293.
Bond, Rachael L.; He, Yang-Hui; Ormerod, Thomas C. (2018). "A quantum framework for likelihood ratios". International Journal of Quantum Information. 16 (1): 1850002.arXiv:1508.00936. Bibcode:2018IJQI...1650002B. doi:10.1142/s0219749918500028. ISSN 0219-7499.
Aerts, D. (2009). "Quantum structure in cognition". Journal of Mathematical Psychology. 53 (5): 314–348.arXiv:0805.3850. doi:10.1016/j.jmp.2009.04.005.
Busemeyer, J. R.; Pothos, E.; Franco, R.; Trueblood, J. S. (2011). "A quantum theoretical explanation for probability judgment 'errors'". Psychological Review. 118 (2): 193–218. doi:10.1037/a0022542.
Trueblood, J. S.; Busemeyer, J. R. (2011). "A quantum probability account of order effects in inference". Cognitive science. 35 (8): 1518–1552. doi:10.1111/j.1551-6709.2011.01197.x.
Aerts, D.; Broekaert, J.; Smets, S. (1999). "The liar paradox in a quantum mechanical perspective". Foundations of Science. 4: 115–132. doi:10.1023/A:1009610326206.
Aerts, D.; Aerts, S.; Broekaert, J.; Gabora, L. (2000). "The violation of Bell inequalities in the macroworld". Foundations of Physics. 30: 1387–1414. doi:10.1023/A:1026449716544.
Osherson, D. N.; Smith, E. E. (1981). "On the adequacy of prototype theory as a theory of concepts". Cognition. 9 (1): 35–58. doi:10.1016/0010-0277(81)90013-5.
Hampton, J. A. (1988). "Overextension of conjunctive concepts: Evidence for a unitary model for concept typicality and class inclusion". Journal of Experimental Psychology: Learning, Memory, and Cognition. 14 (1): 12–32. doi:10.1037/0278-7393.14.1.12.
Hampton, J. A. (1988). "Disjunction of natural concepts". Memory & Cognition. 16: 579–591. doi:10.3758/BF03197059.
Aerts, D.; Gabora, L. (2005). "A state-context-property model of concepts and their combinations I: The structure of the sets of contexts and properties". Kybernetes. 34 (1&2): 167–191.
Aerts, D.; Gabora, L. (2005). "A state-context-property model of concepts and their combinations II: A Hilbert space representation". Kybernetes. 34 (1&2): 192–221.
Gabora, L.; Aerts, D. (2002). "Contextualizing concepts using a mathematical generalization of the quantum formalism". Journal of Experimental and Theoretical Artificial Intelligence. 14 (4): 327–358.
Widdows, D.; Peters, S. (2003). Word Vectors and Quantum Logic: Experiments with negation and disjunction. Eighth Mathematics of Language Conference. pp. 141–154.
Bruza, P. D.; Cole, R. J. (2005). "Quantum logic of semantic space: An exploratory investigation of context effects in practical reasoning". In Artemov, S.; Barringer, H.; d'Avila Garcez, A. S.; Lamb, L. C.; Woods, J. (eds.). We Will Show Them: Essays in Honour of Dov Gabbay. College Publications. ISBN 1-904987-11-7.
Aerts, D. (2009). "Quantum particles as conceptual entities: A possible explanatory framework for quantum theory". Foundations of Science. 14: 361–411.arXiv:1004.2530. doi:10.1007/s10699-009-9166-y.
Aerts, D.; Broekaert, J.; Gabora, L.; Sozzo, S. (2013). "Quantum structure and human thought". Behavioral and Brain Sciences. 36 (3): 274–276. doi:10.1017/S0140525X12002841.
Aerts, Diederik; Gabora, Liane; Sozzo, Sandro (September 2013). "Concepts and Their Dynamics: A Quantum-Theoretic Modeling of Human Thought". Topics in Cognitive Science. 5 (4): 737–772.arXiv:1206.1069. doi:10.1111/tops.12042. PMID 24039114.
Aerts, D.; Sozzo, S. (2012). "Quantum structures in cognition: Why and how concepts are entangled". In Song, D.; Melucci, M.; Frommholz, I. (eds.). Quantum Interaction 2011. LNCS. 7052. Berlin: Springer. pp. 116–127. ISBN 978-3-642-24970-9.
Aerts, D.; Sozzo, S. (2014). "Quantum entanglement in concept combinations". International Journal of Theoretical Physics. 53: 3587–3603.arXiv:1302.3831. doi:10.1007/s10773-013-1946-z.
Kak, S. (1996). "The three languages of the brain: quantum, reorganizational, and associative". In Pribram, Karl; King, J. (eds.). Learning as Self-Organization. Mahwah, NJ: Lawrence Erlbaum Associates. pp. 185–219. ISBN 0-8058-2586-X.
Kak, S. (2013). "Biological memories and agents as quantum collectives". NeuroQuantology. 11: 391–398.[unreliable source?]
Kak, S. (2014). "Observability and computability in physics". Quantum Matter. 3: 172–176. doi:10.1166/qm.2014.1112.[unreliable source?]
Van Rijsbergen, K. (2004). The Geometry of Information Retrieval. Cambridge University Press. ISBN 0-521-83805-3.
Widdows, D. (2006). Geometry and meaning. CSLI Publications. ISBN 1-57586-448-7.
Aerts, D.; Czachor, M. (2004). "Quantum aspects of semantic analysis and symbolic artificial intelligence". Journal of Physics A. 37: L123–L132.arXiv:quant-ph/0309022.
Sorah, Michael. "Parserless Extraction; Using a Multidimensional Transient State Vector Machine" (PDF).
Atmanspacher, H., Filk, T., Romer, H. (2004). Quantum zeno features of bi-stable perception. Biological Cybernetics 90, 33–40.
Conte, Elio; Todarello, Orlando; Federici, Antonio; Vitiello, Francesco; Lopane, Michele; Khrennikov, Andrei; Zbilut, Joseph P. (March 2007). "Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics". Chaos, Solitons & Fractals. 31 (5): 1076–1088.arXiv:0710.5092. Bibcode:2007CSF....31.1076C. doi:10.1016/j.chaos.2005.09.061.
Conte, E., Khrennikov, A., Todarello, O., Federici, A., Zbilut, J. P. (2009). Mental states follow quantum mechanics during perception and cognition of ambiguous figures. Open Systems and Information Dynamics 16, 1–17.
Conte, E., Khrennikov A., Todarello, O., De Robertis, R., Federici, A., Zbilut, J. P. (2011). On the possibility that we think in a quantum mechanical manner: An experimental verification of existing quantum interference effects in cognitive anomaly of Conjunction Fallacy. Chaos and Complexity Letters 4, 123–136.
Conte, E., Santacroce, N., Laterza, V., Conte, S., Federici A., Todarello, O. (2012). The brain knows more than it admits: A quantum model and its experimental confirmation. Electronic Journal of Theoretical Physics 9, 72–110.
Anton Amann: The Gestalt Problem in Quantum Theory: Generation of Molecular Shape by the Environment, Synthese, vol. 97, no. 1 (1993), pp. 125–156, jstor 20117832
Haven E. and Khrennikov A. Quantum Social Science, Cambridge University Press, 2012.
Asano, M., Ohya, M., Tanaka, Y., Basieva, I., Khrennikov, A., 2011. Quantum-like model of brain's functioning: Decision making from decoherence. Journal of Theoretical Biologyvol. 281, no. 1, pp. 56–64.
Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Yamato, I. 2013. Non-Kolmogorovian Approach to the Context-Dependent Systems Breaking the Classical Probability Law Foundations of Physics, vol. 43, no 7, pp. 895–911.
Asano, M., Basieva, I., Khrennikov, A., Ohya, M., Tanaka, Y. Yamato, I. 2012. Quantum-like model for the adaptive dynamics of the genetic regulation of E. coli’s metabolism of glucose/lactose. System Synthetic Biology vol. 6(1–2) pp. 1–7.
B.J. Hiley: Particles, fields, and observers, Volume I The Origins of Life, Part 1 Origin and Evolution of Life, Section II The Physical and Chemical Basis of Life, pp. 87–106 (PDF)
Basil J. Hiley, Paavo Pylkkänen: Naturalizing the mind in a quantum framework. In Paavo Pylkkänen and Tere Vadén (eds.): Dimensions of conscious experience, Advances in Consciousness Research, Volume 37, John Benjamins B.V., 2001, ISBN 90-272-5157-6, pages 119–144

Paavo Pylkkänen. "Can quantum analogies help us to understand the process of thought?" (PDF). Mind & Matter. 12 (1): 61–91. p. 83–84.