{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T08:32:58Z","timestamp":1762072378308,"version":"build-2065373602"},"reference-count":139,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T00:00:00Z","timestamp":1666742400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Tiny Blue Dot Foundation"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A hypothesis is presented that non-separability of degrees of freedom is the fundamental property underlying consciousness in physical systems. The amount of consciousness in a system is determined by the extent of non-separability and the number of degrees of freedom involved. Non-interacting and feedforward systems have zero consciousness, whereas most systems of interacting particles appear to have low non-separability and consciousness. By contrast, brain circuits exhibit high complexity and weak but tightly coordinated interactions, which appear to support high non-separability and therefore high amount of consciousness. The hypothesis applies to both classical and quantum cases, and we highlight the formalism employing the Wigner function (which in the classical limit becomes the Liouville density function) as a potentially fruitful framework for characterizing non-separability and, thus, the amount of consciousness in a system. The hypothesis appears to be consistent with both the Integrated Information Theory and the Orchestrated Objective Reduction Theory and may help reconcile the two. It offers a natural explanation for the physical properties underlying the amount of consciousness and points to methods of estimating the amount of non-separability as promising ways of characterizing the amount of consciousness.<\/jats:p>","DOI":"10.3390\/e24111539","type":"journal-article","created":{"date-parts":[[2022,10,26]],"date-time":"2022-10-26T09:59:37Z","timestamp":1666778377000},"page":"1539","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Non-Separability of Physical Systems as a Foundation of Consciousness"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1106-8310","authenticated-orcid":false,"given":"Anton","family":"Arkhipov","sequence":"first","affiliation":[{"name":"MindScope Program, Allen Institute, Seattle, WA 98109, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,26]]},"reference":[{"key":"ref_1","unstructured":"Koch, C. (2004). The Quest for Consciousness: A Neurobiological Approach, Roberts & Company."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1038\/nrn.2016.22","article-title":"Neural correlates of consciousness: Progress and problems","volume":"17","author":"Koch","year":"2016","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_3","first-page":"200","article-title":"Facing up to the problem of consciousness","volume":"2","author":"Chalmers","year":"1995","journal-title":"J. Conscious. Stud."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Tononi, G. (2004). An information integration theory of consciousness. BMC Neurosci., 5.","DOI":"10.1186\/1471-2202-5-42"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1038\/nrn.2016.44","article-title":"Integrated information theory: From consciousness to its physical substrate","volume":"17","author":"Tononi","year":"2016","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Oizumi, M., Albantakis, L., and Tononi, G. (2014). From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003588"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Penrose, R. (1989). The Emperor\u2019s New Mind: Concerning Computers, Minds, and the Laws of Physics. The Emperor\u2019s New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford University Press.","DOI":"10.1093\/oso\/9780198519737.001.0001"},{"key":"ref_8","unstructured":"Penrose, R. (1994). Shadows of the Mind: A Search for the Missing Science of Consciousness, Oxford University."},{"key":"ref_9","first-page":"36","article-title":"Conscious events as orchestrated space-time selections","volume":"3","author":"Hameroff","year":"1996","journal-title":"J. Conscious. Stud."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/0378-4754(96)80476-9","article-title":"Orchestrated reduction of quantum coherence in brain microtubules: A model for consciousness","volume":"40","author":"Hameroff","year":"1996","journal-title":"Math. Comput. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.plrev.2013.08.002","article-title":"Consciousness in the universe: A review of the \u2018Orch OR\u2019 theory","volume":"11","author":"Hameroff","year":"2014","journal-title":"Phys. Life Rev."},{"key":"ref_12","unstructured":"Baars, B.J. (1988). A Cognitive Theory of Consciousness, Cambridge University Press."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"749868","DOI":"10.3389\/fpsyg.2021.749868","article-title":"Global Workspace Theory (GWT) and Prefrontal Cortex: Recent Developments","volume":"12","author":"Baars","year":"2021","journal-title":"Front. Psychol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"14529","DOI":"10.1073\/pnas.95.24.14529","article-title":"A neuronal model of a global workspace in effortful cognitive tasks","volume":"95","author":"Dehaene","year":"1998","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"8520","DOI":"10.1073\/pnas.1332574100","article-title":"A neuronal network model linking subjective reports and objective physiological data during conscious perception","volume":"100","author":"Dehaene","year":"2003","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"776","DOI":"10.1016\/j.neuron.2020.01.026","article-title":"Conscious Processing and the Global Neuronal Workspace Hypothesis","volume":"105","author":"Mashour","year":"2020","journal-title":"Neuron"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1016\/j.neubiorev.2017.07.013","article-title":"How do the brain\u2019s time and space mediate consciousness and its different dimensions? Temporo-spatial theory of consciousness (TTC)","volume":"80","author":"Northoff","year":"2017","journal-title":"Neurosci. Biobehav. Rev."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Rosenthal, D. (2005). Consciousness and Mind, Oxford University Press.","DOI":"10.1093\/oso\/9780198236979.001.0001"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/s11229-006-9106-0","article-title":"Same Old, Same Old: The Same-Order Representational Theory of Consciousness and the Division of Phenomenal Labor","volume":"160","author":"Weisberg","year":"2008","journal-title":"Synthese"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1016\/j.tics.2011.05.009","article-title":"Empirical support for higher-order theories of conscious awareness","volume":"15","author":"Lau","year":"2011","journal-title":"Trends Cogn Sci."},{"key":"ref_21","unstructured":"Banerjee, R., and Chakrabarti, B. (2007). A higher order Bayesian decision theory of consciousness. Models of Brain and Mind, Elsevier."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1080\/17588928.2011.565121","article-title":"Human consciousness and its relationship to social neuroscience: A novel hypothesis","volume":"2","author":"Graziano","year":"2011","journal-title":"Cogn Neurosci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"500","DOI":"10.3389\/fpsyg.2015.00500","article-title":"The attention schema theory: A mechanistic account of subjective awareness","volume":"6","author":"Graziano","year":"2015","journal-title":"Front. Psychol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.jtbi.2017.05.032","article-title":"A mathematical model of embodied consciousness","volume":"428","author":"Rudrauf","year":"2017","journal-title":"J. Theor. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2571","DOI":"10.3389\/fpsyg.2018.02571","article-title":"The Projective Consciousness Model and Phenomenal Selfhood","volume":"9","author":"Williford","year":"2018","journal-title":"Front. Psychol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1504","DOI":"10.3389\/fpsyg.2020.01504","article-title":"Information Closure Theory of Consciousness","volume":"11","author":"Chang","year":"2020","journal-title":"Front Psychol."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1016\/j.neubiorev.2020.07.019","article-title":"Neural signs and mechanisms of consciousness: Is there a potential convergence of theories of consciousness in sight?","volume":"118","author":"Northoff","year":"2020","journal-title":"Neurosci. Biobehav. Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1038\/s41583-022-00587-4","article-title":"Theories of consciousness","volume":"23","author":"Seth","year":"2022","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1016\/j.neubiorev.2011.12.003","article-title":"Distilling the neural correlates of consciousness","volume":"36","author":"Aru","year":"2012","journal-title":"Neurosci. Biobehav. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1016\/j.tics.2020.07.006","article-title":"Cellular Mechanisms of Conscious Processing","volume":"24","author":"Aru","year":"2020","journal-title":"Trends Cogn. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1080\/17588921003731586","article-title":"How neuroscience will change our view on consciousness","volume":"1","author":"Lamme","year":"2010","journal-title":"Cogn. Neurosci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.chaos.2015.03.014","article-title":"Consciousness as a state of matter","volume":"76","author":"Tegmark","year":"2015","journal-title":"Chaos Solitons Fractals"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Northoff, G. (2013). Unlocking the Brain: Volume 2: Consciousness, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199826995.001.0001"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1161\/STROKEAHA.114.007969","article-title":"Why Is the Distinction Between Neural Predispositions, Prerequisites, and Correlates of the Level of Consciousness Clinically Relevant?","volume":"46","author":"Northoff","year":"2015","journal-title":"Stroke"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"113788","DOI":"10.1016\/j.bbr.2022.113788","article-title":"Temporo-spatial Theory of Consciousness (TTC)\u2013Bridging the gap of neuronal activity and phenomenal states","volume":"424","author":"Northoff","year":"2022","journal-title":"Behav. Brain Res."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Del Cul, A., Baillet, S., and Dehaene, S. (2007). Brain Dynamics Underlying the Nonlinear Threshold for Access to Consciousness. PLoS Biol., 5.","DOI":"10.1371\/journal.pbio.0050260"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Herzog, M.H., Kammer, T., and Scharnowski, F. (2016). Time Slices: What Is the Duration of a Percept?. PLoS Biol., 14.","DOI":"10.1371\/journal.pbio.1002433"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"826","DOI":"10.1016\/j.tics.2020.07.001","article-title":"All in Good Time: Long-Lasting Postdictive Effects Reveal Discrete Perception","volume":"24","author":"Herzog","year":"2020","journal-title":"Trends Cogn. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"niy007","DOI":"10.1093\/nc\/niy007","article-title":"On the axiomatic foundations of the integrated information theory of consciousness","volume":"2018","author":"Bayne","year":"2018","journal-title":"Neurosci. Conscious."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1103\/PhysRev.40.749","article-title":"On the Quantum Correction For Thermodynamic Equilibrium","volume":"40","author":"Wigner","year":"1932","journal-title":"Phys. Rev."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1119\/1.2957889","article-title":"Wigner functions and Weyl transforms for pedestrians","volume":"76","author":"Case","year":"2008","journal-title":"Am. J. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/BF00733376","article-title":"A tomographic approach to Wigner\u2019s function","volume":"17","author":"Bertrand","year":"1987","journal-title":"Found. Phys."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2847","DOI":"10.1103\/PhysRevA.40.2847","article-title":"Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase","volume":"40","author":"Vogel","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0375-9601(96)00107-7","article-title":"Symplectic tomography as classical approach to quantum systems","volume":"213","author":"Mancini","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_45","first-page":"419","article-title":"Center of mass tomography for reconstructing quantum states of multipartite systems","volume":"328","author":"Arkhipov","year":"2004","journal-title":"Phys. Lett Sect. A Gen. At. Solid State Phys."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"012101","DOI":"10.1103\/PhysRevA.71.012101","article-title":"Quantum transitions in the center-of-mass tomographic probability representation","volume":"71","author":"Arkhipov","year":"2005","journal-title":"Phys. Rev. A At. Mol. Opt. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Landau, L.D., and Lifshitz, E.M. (1958). Quantum Mechanics: Non-Relativistic Theory. V. 3 of Course of Theoretical Physics, Pergamon Press.","DOI":"10.1063\/1.3062347"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2009.02.004","article-title":"Entanglement detection","volume":"474","year":"2009","journal-title":"Phys. Rep."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Tegmark, M. (2016). Improved Measures of Integrated Information. PLoS Comput. Biol., 12.","DOI":"10.1371\/journal.pcbi.1005123"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Barrett, A.B., and Seth, A.K. (2011). Practical Measures of Integrated Information for Time-Series Data. PLoS Comput. Biol., 7.","DOI":"10.1371\/journal.pcbi.1001052"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Mediano, P.A.M., Seth, A.K., and Barrett, A.B. (2019). Measuring Integrated Information: Comparison of Candidate Measures in Theory and Simulation. Entropy, 21.","DOI":"10.3390\/e21010017"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Kitazono, J., Kanai, R., and Oizumi, M. (2018). Efficient Algorithms for Searching the Minimum Information Partition in Integrated Information Theory. Entropy, 20.","DOI":"10.3390\/e20030173"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"18803","DOI":"10.1038\/s41598-020-75943-4","article-title":"A measure for intrinsic information","volume":"10","author":"Barbosa","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Mayner, W.G.P., Marshall, W., Albantakis, L., Findlay, G., Marchman, R., and Tononi, G. (2018). PyPhi: A toolbox for integrated information theory. PLoS Comput. Biol., 14.","DOI":"10.1371\/journal.pcbi.1006343"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"nix017","DOI":"10.1093\/nc\/nix017","article-title":"Computing integrated information","volume":"2017","author":"Krohn","year":"2017","journal-title":"Neurosci Conscious."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1103\/RevModPhys.76.981","article-title":"Nobel Lecture: On superconductivity and superfluidity (what I have and have not managed to do) as well as on the \u201cphysical minimum\u2019\u2019 at the beginning of the XXI century","volume":"76","author":"Ginzburg","year":"2004","journal-title":"Rev. Mod. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1103\/RevModPhys.74.99","article-title":"The world of the complex Ginzburg-Landau equation","volume":"74","author":"Aranson","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1038\/416211a","article-title":"Bose\u2013Einstein condensation of atomic gases","volume":"416","author":"Anglin","year":"2002","journal-title":"Nature"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1103\/RevModPhys.71.463","article-title":"Theory of Bose-Einstein condensation in trapped gases","volume":"71","author":"Dalfovo","year":"1999","journal-title":"Rev. Mod. Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1103\/RevModPhys.73.307","article-title":"Bose-Einstein condensation in the alkali gases: Some fundamental concepts","volume":"73","author":"Leggett","year":"2001","journal-title":"Rev. Mod. Phys."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"3169","DOI":"10.1142\/S0217732310034626","article-title":"BCS as foundation and inspiration: The transmutation of symmetry","volume":"25","author":"Wilczek","year":"2010","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"424","DOI":"10.1113\/jphysiol.1952.sp004716","article-title":"Measurement of current-voltage relations in the membrane of the giant axon of Loligo","volume":"116","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1113\/jphysiol.1952.sp004764","article-title":"A quantitative description of membrane current and its application to conduction and excitation in nerve","volume":"117","author":"Hodgkin","year":"1952","journal-title":"J. Physiol."},{"key":"ref_64","unstructured":"Koch, C. (2004). Biophysics of Computation: Information Processing in Single Neurons, Oxford University Press."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1177\/107385849700300110","article-title":"The Thalamic Intralaminar Nuclei: A Role in Visual Awareness","volume":"3","author":"Purpura","year":"1997","journal-title":"Neuroscientist"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1846","DOI":"10.1126\/science.282.5395.1846","article-title":"Consciousness and Complexity","volume":"282","author":"Giulio","year":"1998","journal-title":"Science"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"9603","DOI":"10.1523\/JNEUROSCI.3218-16.2017","article-title":"Are the Neural Correlates of Consciousness in the Front or in the Back of the Cerebral Cortex? Clinical and Neuroimaging Evidence","volume":"37","author":"Boly","year":"2017","journal-title":"J. Neurosci."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1038\/nrn3687","article-title":"The log-dynamic brain: How skewed distributions affect network operations","volume":"15","author":"Mizuseki","year":"2014","journal-title":"Nat. Rev. Neurosci."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"19790","DOI":"10.1073\/pnas.1314922110","article-title":"Quantifying causal emergence shows that macro can beat micro","volume":"110","author":"Hoel","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1038\/s41593-021-00911-8","article-title":"Causal reductionism and causal structures","volume":"24","author":"Grasso","year":"2021","journal-title":"Nat. Neurosci."},{"key":"ref_71","doi-asserted-by":"crossref","unstructured":"Northoff, G., and Zilio, F. (2022). From Shorter to Longer Timescales: Converging Integrated Information Theory (IIT) with the Temporo-Spatial Theory of Consciousness (TTC). Entropy, 24.","DOI":"10.3390\/e24020270"},{"key":"ref_72","first-page":"31","article-title":"The human brain in numbers: A linearly scaled-up primate brain","volume":"3","year":"2009","journal-title":"Front. Hum. Neurosci."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1159\/000437413","article-title":"Mammalian Brains Are Made of These: A Dataset of the Numbers and Densities of Neuronal and Nonneuronal Cells in the Brain of Glires, Primates, Scandentia, Eulipotyphlans, Afrotherians and Artiodactyls, and Their Relationship with Body Mass","volume":"86","author":"Catania","year":"2015","journal-title":"Brain Behav. Evol."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1038\/nature18933","article-title":"A multi-modal parcellation of human cerebral cortex","volume":"536","author":"Glasser","year":"2016","journal-title":"Nature"},{"key":"ref_75","unstructured":"Simon, C. (2018). Can quantum physics help solve the hard problem of consciousness?. A hypothesis based on entangled spins and photons. arXiv, arXiv180903490."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.brainresrev.2005.11.008","article-title":"The centrifugal visual system of vertebrates: A comparative analysis of its functional anatomical organization","volume":"52","author":"Ward","year":"2006","journal-title":"Brain Res. Rev."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.neunet.2014.09.003","article-title":"Deep learning in neural networks: An overview","volume":"61","author":"Schmidhuber","year":"2015","journal-title":"Neural Netw."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"593","DOI":"10.1016\/j.aop.2015.08.020","article-title":"Quantum cognition: The possibility of processing with nuclear spins in the brain","volume":"362","author":"Fisher","year":"2015","journal-title":"Ann. Phys. (N. Y.)"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1524\/zpch.1976.101.1-6.371","article-title":"Magnetic Field Dependence of the Geminate Recombination of Radical Ion Pairs in Polar Solvents","volume":"101","author":"Schulten","year":"1976","journal-title":"Z. F\u00fcr Phys. Chem."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1524\/zpch.1978.111.1.001","article-title":"A Biomagnetic Sensory Mechanism Based on Magnetic Field Modulated Coherent Electron Spin Motion","volume":"111","author":"Schulten","year":"1978","journal-title":"Z. F\u00fcr Phys Chem."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/S0006-3495(00)76629-X","article-title":"A Model for Photoreceptor-Based Magnetoreception in Birds","volume":"78","author":"Ritz","year":"2000","journal-title":"Biophys. J."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1073\/pnas.0711968106","article-title":"Chemical magnetoreception in birds: The radical pair mechanism","volume":"106","author":"Rodgers","year":"2009","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1039\/C9FD00049F","article-title":"How quantum is radical pair magnetoreception?","volume":"221","author":"Fay","year":"2020","journal-title":"Faraday Discuss."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"1925","DOI":"10.1037\/xge0000585","article-title":"True contextuality beats direct influences in human decision making","volume":"148","author":"Basieva","year":"2019","journal-title":"J. Exp. Psychol. Gen."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"14006","DOI":"10.1088\/0031-8949\/2014\/T163\/014006","article-title":"Violation of contextual generalization of the Leggett\u2013Garg inequality for recognition of ambiguous figures","volume":"2014","author":"Asano","year":"2014","journal-title":"Phys. Scr."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"214","DOI":"10.14704\/nq.2008.6.3.178","article-title":"A Preliminary Experimental Verification On the Possibility of Bell Inequality Violation in Mental States","volume":"6","author":"Conte","year":"2008","journal-title":"Neuroquantology"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1126\/science.aad7174","article-title":"Classical entanglement?","volume":"350","author":"Ebrahim","year":"2015","journal-title":"Science"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"1762","DOI":"10.1007\/s10701-020-00319-7","article-title":"Quantum Versus Classical Entanglement: Eliminating the Issue of Quantum Nonlocality","volume":"50","author":"Khrennikov","year":"2020","journal-title":"Found. Phys."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1038\/s41377-021-00521-w","article-title":"Entanglement goes classically high-dimensional","volume":"10","author":"Zhan","year":"2021","journal-title":"Light. Sci. Appl."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1023\/A:1018703709245","article-title":"A Classical Analogy of Entanglement","volume":"28","author":"Spreeuw","year":"1998","journal-title":"Found. Phys."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"45020","DOI":"10.1063\/1.4982732","article-title":"Non-separable states in a bipartite elastic system","volume":"7","author":"Deymier","year":"2017","journal-title":"AIP Adv."},{"key":"ref_92","unstructured":"Girard, M.F. (2019). Hamilton-Jacobi Approach to the Quantization of Classically Non-Separable but Integrable Two-Dimensional Systems: The Role of the Classical Caustics. arXiv."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Marshall, W., Albantakis, L., and Tononi, G. (2018). Black-boxing and cause-effect power. PLoS Comput. Biol., 14.","DOI":"10.1371\/journal.pcbi.1006114"},{"key":"ref_94","first-page":"573","article-title":"Phase Transitions in Reservoir-Driven Open Systems with Applications to Lasers and Superconductors","volume":"46","author":"Hepp","year":"1973","journal-title":"Helv. Phys. Acta"},{"key":"ref_95","unstructured":"B\u00f3na, P. (2012). Extended Quantum Mechanics. arXiv."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"168643","DOI":"10.1016\/j.aop.2021.168643","article-title":"Entanglement of classical and quantum short-range dynamics in mean-field systems","volume":"434","author":"Bru","year":"2021","journal-title":"Ann. Phys. N. Y."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"1283","DOI":"10.1103\/PhysRev.184.1283","article-title":"Wigner Quantum Density Function in the Classical Limit. Development of a Three-Dimensional WKBJ-Type Solution","volume":"184","author":"Schipper","year":"1969","journal-title":"Phys. Rev."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"2726","DOI":"10.1103\/PhysRevLett.84.2726","article-title":"Peres-Horodecki Separability Criterion for Continuous Variable Systems","volume":"84","author":"Simon","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"2722","DOI":"10.1103\/PhysRevLett.84.2722","article-title":"Inseparability Criterion for Continuous Variable Systems","volume":"84","author":"Duan","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"62412","DOI":"10.1103\/PhysRevA.105.062412","article-title":"Classical analogs of the covariance matrix, purity, linear entropy, and von Neumann entropy","volume":"105","author":"Diaz","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"230502","DOI":"10.1103\/PhysRevLett.95.230502","article-title":"Inseparability Criteria for Continuous Bipartite Quantum States","volume":"95","author":"Shchukin","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"22111","DOI":"10.1103\/PhysRevA.84.022111","article-title":"Entanglement in the classical limit: Quantum correlations from classical probabilities","volume":"84","author":"Matzkin","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"4665","DOI":"10.1103\/PhysRevE.62.4665","article-title":"Entropy and Wigner functions","volume":"62","author":"Manfredi","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_104","unstructured":"Chakraborty, A., and Sensarma, R. (2018). Wigner Function and Entanglement Entropy for Bosons from Non-Equilibrium Field Theory. arXiv."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"184306","DOI":"10.1103\/PhysRevB.102.184306","article-title":"Entanglement entropy of fermions from Wigner functions: Excited states and open quantum systems","volume":"102","author":"Moitra","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"32314","DOI":"10.1103\/PhysRevA.65.032314","article-title":"Computable measure of entanglement","volume":"65","author":"Vidal","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"42327","DOI":"10.1103\/PhysRevA.66.042327","article-title":"Entanglement properties of the harmonic chain","volume":"66","author":"Audenaert","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_108","doi-asserted-by":"crossref","unstructured":"Boly, M., Sasai, S., Gosseries, O., Oizumi, M., Casali, A., Massimini, M., and Tononi, G. (2015). Stimulus Set Meaningfulness and Neurophysiological Differentiation: A Functional Magnetic Resonance Imaging Study. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0125337"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"1748","DOI":"10.3389\/fpsyg.2017.01748","article-title":"EEG Differentiation Analysis and Stimulus Set Meaningfulness","volume":"8","author":"Mensen","year":"2017","journal-title":"Front. Psychol."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1162\/jocn_a_01278","article-title":"Differentiation Analysis of Continuous Electroencephalographic Activity Triggered by Video Clip Contents","volume":"30","author":"Mensen","year":"2018","journal-title":"J. Cogn. Neurosci."},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"ENEURO.0280-21.2021","DOI":"10.1523\/ENEURO.0280-21.2021","article-title":"Measuring stimulus-evoked neurophysiological differentiation in distinct populations of neurons in mouse visual cortex","volume":"9","author":"Mayner","year":"2022","journal-title":"Eneuro"},{"key":"ref_112","doi-asserted-by":"crossref","unstructured":"Haun, A., and Tononi, G. (2019). Why Does Space Feel the Way it Does? Towards a Principled Account of Spatial Experience. Entropy, 21.","DOI":"10.3390\/e21121160"},{"key":"ref_113","doi-asserted-by":"crossref","unstructured":"Grasso, M., Haun, A.M., and Tononi, G. (2021). Of maps and grids. Neurosci. Conscious., 202.","DOI":"10.1093\/nc\/niab022"},{"key":"ref_114","unstructured":"Lafferty, J., Williams, C., Shawe-Taylor, J., Zemel, R., and Culotta, A. (2010). Inferring Stimulus Selectivity from the Spatial Structure of Neural Network Dynamics. Advances in Neural Information Processing Systems, Curran Associates, Inc.. Available online: https:\/\/proceedings.neurips.cc\/paper\/2010\/file\/19b650660b253761af189682e03501dd-Paper.pdf."},{"key":"ref_115","doi-asserted-by":"crossref","unstructured":"Gao, P., Trautmann, E., Yu, B., Santhanam, G., Ryu, S., Shenoy, K., and Ganguli, S. (2017). A theory of multineuronal dimensionality, dynamics and measurement. bioRxiv, 214262.","DOI":"10.1101\/214262"},{"key":"ref_116","doi-asserted-by":"crossref","unstructured":"Recanatesi, S., Ocker, G.K., Buice, M.A., and Shea-Brown, E. (2019). Dimensionality in recurrent spiking networks: Global trends in activity and local origins in connectivity. PLoS Comput. Biol., 15.","DOI":"10.1371\/journal.pcbi.1006446"},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1103\/RevModPhys.76.1267","article-title":"Decoherence, the measurement problem, and interpretations of quantum mechanics","volume":"76","author":"Schlosshauer","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_118","unstructured":"Hooft, G. (2016). The Cellular Automaton Interpretation of Quantum Mechanics, Springer."},{"key":"ref_119","doi-asserted-by":"crossref","unstructured":"Lalo\u00eb, F. (2019). Do We Really Understand Quantum Mechanics?, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/9781108569361"},{"key":"ref_120","first-page":"1728","article-title":"Loss of Differentiation and Complexity in the Sleeping Human Brain: A Multi-Scale Analysis","volume":"14","author":"Pigorini","year":"2021","journal-title":"Brain Stimul Basic Transl. Clin. Res. Neuromodulation"},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"198ra105","DOI":"10.1126\/scitranslmed.3006294","article-title":"A Theoretically Based Index of Consciousness Independent of Sensory Processing and Behavior","volume":"5","author":"Casali","year":"2013","journal-title":"Sci. Transl. Med."},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"1280","DOI":"10.1016\/j.brs.2019.05.013","article-title":"A fast and general method to empirically estimate the complexity of brain responses to transcranial and intracranial stimulations","volume":"12","author":"Comolatti","year":"2019","journal-title":"Brain Stimul."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"2681","DOI":"10.1073\/pnas.0913008107","article-title":"Breakdown in cortical effective connectivity during midazolam-induced loss of consciousness","volume":"107","author":"Ferrarelli","year":"2010","journal-title":"Proc. Natl Acad Sci. USA"},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"1308","DOI":"10.1093\/brain\/awr340","article-title":"Recovery of cortical effective connectivity and recovery of consciousness in vegetative patients","volume":"135","author":"Rosanova","year":"2012","journal-title":"Brain"},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1016\/j.tins.2020.05.008","article-title":"From Complexity to Consciousness","volume":"43","author":"Bayne","year":"2020","journal-title":"Trends Neurosci."},{"key":"ref_126","unstructured":"Nielsen, M.A., and Chuang, I.L. (2010). Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge University Press."},{"key":"ref_127","doi-asserted-by":"crossref","unstructured":"Jozsa, R. (1997). Entanglement and Quantum Computation. arXiv.","DOI":"10.1049\/ic:19970792"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/j.tcs.2004.03.041","article-title":"Quantum computing without entanglement","volume":"320","author":"Biham","year":"2004","journal-title":"Theor. Comput. Sci."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1103\/RevModPhys.84.1655","article-title":"The classical-quantum boundary for correlations: Discord and related measures","volume":"84","author":"Modi","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"80501","DOI":"10.1103\/PhysRevLett.104.080501","article-title":"Unified View of Quantum and Classical Correlations","volume":"104","author":"Modi","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"200501","DOI":"10.1103\/PhysRevLett.101.200501","article-title":"Experimental Quantum Computing without Entanglement","volume":"101","author":"Lanyon","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"13236","DOI":"10.1103\/PhysRevResearch.3.013236","article-title":"Measures of space-time nonseparability of electromagnetic pulses","volume":"3","author":"Shen","year":"2021","journal-title":"Phys. Rev. Res."},{"key":"ref_133","unstructured":"Bruza, P., Sofge, D., Lawless, W., Widdows, D., and van Rijsbergen, K. (2010). The Role of Non-Factorizability in Determining \u201cPseudo-Classical\u201d Non-Separability. Quantum Informatics for Cognitive, Social, and Semantic Processes: Papers from the AAAI Fall Symposium, AAAI Publications (Association for the Advancement of Artificial Intelligence). Available online: https:\/\/eprints.qut.edu.au\/39336\/."},{"key":"ref_134","first-page":"1","article-title":"Entangled Kernels\u2013Beyond Separability","volume":"22","author":"Huusari","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"ref_135","unstructured":"Miszczak, J.A. (2015). Separable and non-separable data representation for pattern discrimination. arXiv."},{"key":"ref_136","doi-asserted-by":"crossref","unstructured":"Liu, Y., Li, W.-J., Zhang, X., Lewenstein, M., Su, G., and Ran, S.-J. (2022, October 22). Entanglement-Based Feature Extraction by Tensor Network Machine Learning. Front. Appl. Math. Stat. Available online: https:\/\/www.frontiersin.org\/article\/10.3389\/fams.2021.716044.","DOI":"10.3389\/fams.2021.716044"},{"key":"ref_137","unstructured":"Singh, A. (2020, January 13\u201318). Non-separable Non-stationary random fields. Proceedings of the 37th International Conference on Machine Learning, PMLR, III HD, online. Available online: https:\/\/proceedings.mlr.press\/v119\/wang20g.html."},{"key":"ref_138","doi-asserted-by":"crossref","unstructured":"Albantakis, L., Hintze, A., Koch, C., Adami, C., and Tononi, G. (2014). Evolution of Integrated Causal Structures in Animats Exposed to Environments of Increasing Complexity. PLoS Comput. Biol., 10.","DOI":"10.1371\/journal.pcbi.1003966"},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"5472","DOI":"10.3390\/e17085472","article-title":"The Intrinsic Cause-Effect Power of Discrete Dynamical Systems\u2014From Elementary Cellular Automata to Adapting Animats","volume":"17","author":"Albantakis","year":"2015","journal-title":"Entropy"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1539\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:03:21Z","timestamp":1760144601000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/11\/1539"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,26]]},"references-count":139,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["e24111539"],"URL":"https:\/\/doi.org\/10.3390\/e24111539","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,10,26]]}}}