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Quantum consciousness (sometimes called quantum mind) is the idea that consciousness requires quantum processes, as opposed the view of mainstream neurobiology in which the function of the brain is wholly classical, and quantum processes play no computational role.
While many attempts at a theory of quantum consciousness are pseudoscientific by naively claiming the strangeness of quantum mechanics is a parallel to the strangeness of consciousness, more sophisticated quantum consciousness theories are an attempt at a solution of the "combination problem"; the problem explaining how a system of classical neurons can combine to form a single subject of experience (also referred to as the "binding problem"). However, there is currently little experimental evidence of computationally relevant quantum processes in the human brain, in part due to the technical difficulty of probing the brain at sufficient spatial and temporal granularity.
Whether or not quantum effects influence thought is a valid topic for scientific investigation, but simply stating "quantum effects cause consciousness" explains nothing unless scientists can come up with some suggestion about how quantum effects could possibly cause consciousness. The argument goes:
- I don't understand consciousness.
- I don't understand quantum physics.
- Therefore, consciousness must be a function of quantum physics!
Please note: This should be distinguished from research into "quantum cognition," which applies quantum-mechanical mathematical models to human behavior in areas where classical probability theory fails to match observed human behavior. "Quantum cognition" does not assume that the underlying human consciousness is quantum-mechanical; it's simply that a few psychologists noted that the same concepts and equations used in quantum mechanics are for reasons unknown good analogies for actual human behavior where traditional probability theory suggests that actual behavior is irrational.
Also note that at the atomic level, quantum events (radioactive decay of atoms, probablistic collisions of molecules) obviously take place in the brain and affect neurons to some extent. However, such events are considered trivial and there is no evidence that they play any computationally relevant role.
Quanta and consciousness
Not all quantum mechanical interpretations assume quantum collapse happens, but if it does, one of many competing theories about how it happens is that consciousness causes collapse. Since these consciousness-based theories were developed, people who want to believe that consciousness is in some way special have been attracted to that type of quantum physics. Some adherents to mystical consciousness-based quantum physics have been woo-meisters and pseudoscientists proposing often expensive solutions to problems. Despite this respected scientists like Eugene Wigner were also attracted to quantum consciousness, a view Wigner later repudiated. At the present level of knowledge it is difficult for lay people to see how far quantum consciousness is a reasonable theory and how much it is wishful thinking. It should be noted however that unless substance dualism is true, which most scientists doubt, conscious minds should be able to collapse wave-functions just as much as unconscious photodetectors can. If this were not so, it would mean that conscious minds are made of some non-physical substance that unconscious photodetectors are not made of.
According to Chopra, "consciousness is nonlocal" and consciousness is "a field, a superposition of possibilities." It is a bit hard to see what Chopra means by this, it seems that a whole lot of different, perhaps incompatible possibilities (like the possibility that Schrödinger's cat is alive and the simultaneous possibility that the cat is dead) form a field and by some unexplained mechanism consciousness derives from this field of possibilities.
Mario Beauregard's brand of quantum theory is explained in-depth at the non-materialist neuroscience page. The crux of the quantum-related part of his argument is that the ion channels in neurons are small enough to be subject to quantum effects. This is similar to the materialist version of quantum consciousness. Other scientists reject many of Beauregard's postulates.
The most famous proponent of this theory is Roger Penrose, a renowned mathematician who has collaborated frequently with Stephen Hawking — that is, he differs from other quantum consciousness proponents in actually understanding something about quantum physics. You might recognize his name from the Penrose triangle or the Penrose tiling.
Penrose's argument starts off based on Gödel's incompleteness theorem, stating that the existence of the theorem demonstrated that the mind had the capability of thinking outside of an algorithmic fashion, i.e. that consciousness is non-computable. Quantum physics then gives him the out to argue that neurons, and thus the brain as a whole, operate in a probabilistic fashion. Somehow probabilistic fashions lead to consciousness. Max Tegmark, a man who received donations from Elon Musk to investigate existential risk from advanced artificial intelligence, claims the brain is simply too hot for quantum states to be influential. 
Penrose and Hameroff
Penrose then teamed up with Stuart Hameroff, who developed a similarly unscientific theory about quantum independently, to further this idea. They developed something called the Orchestrated Objective Reduction (Orch-OR) model. Most of it is dependent on Hameroff's assertion that the micro-tubules in neurons could have quantum effects on neuronal behavior, thus allowing the brain to behave as a quantum computer. Max Tegmark performed some mathematics and he saw any quantum effects within micro-tubules as subject to decoherence and thus not affecting brain activity. Further falsifications of the Orch-OR model have been performed. Penrose is an atheist and his arguments are usually used to support free will without invoking spirits, making this something like materialist woo.
Research has recently shown that anaesthesia's action differs from the model in Penrose and Hameroff's hypothesis, casting further doubt on the idea.
It should be noted that quantum computers exist. They are specialized devices that use the quantum properties of atoms and molecules for computing power. A quantum computer harnesses the quantum superposition of atoms and can hold a single bit (known as "qubit") in a state of 1, 0 and both at the same time. Effectively this means that a single atom used to perform a calculation is on, off, and on and off at the same time. The quantum computer can maintain these states — potentially all possible states — simultaneously before coalescing around a single calculation almost instantaneously. In principle, this allows quantum computers to perform some operations exponentially faster than classical computers, though there is currently no sound evidence to suggest that the human brain (or any other natural phenomenon) is or acts like a quantum computer.
Philosopher David Pearce published an essay entitled "Physicalistic Idealism: Does physicalism entail monistic idealism? An experimentally testable conjecture about the nature of the physical world." In it, he proposes a scientifically falsifiable hypothesis for testing the computational role of quantum processes in the brain as a solution to the phenomenal binding problem that arises from a standard conception of the brain as a wholly classical system. The experimental apparatus required to test and possibly falsify the conjecture is currently more advanced than what we have available.
All these theories
In short, the various exercises in quantum "flapdoodle" seem to demonstrate that many are uncomfortable with the facts that neurons operate on the all-or-nothing principle, i.e., a) they are either on or off, making them in effect similar to a computer's binary code; b) that, as far as we can tell according to modern science, they are subject to physical law and classical mechanics;  and c) we still don't have a full solution for the binding problem or an all-encompassing explanation for just why brains work the way they do. Of course not all of it is flapdoodle-this doesn't rule out the possibility of quantum effects, but no coherent mechanism able to be replicated by experiment has been proposed thus far.
As of late, it seems that Penrose and his followers have allowed their brand of quantum consciousness to bleed into Chopra's and that of other woo-meisters. This is rather unfortunate due to the fact that while many predictions made by the Orch-OR model have been wrong, it might have eventually shown promise in a protoscientific sense, while Chopra's problematic and tangled nonsense is mostly unfalsifiable with scientific instruments in their current stage of evolution or to initial inspection not even wrong.
The question of "quantum free will"
Even if quantum forces were discovered to exert influence over neuronal activity, this still doesn't necessarily prove free will, in great part because of the slippery definition of "free will." The probabilistic nature of quantum mechanics does not rule out determinism — rather than a strict "A then B" view of causality, it simply creates a more complex picture of causality in which future possibilities and probabilities are naturally constrained and "calculated." This might be thought of as rolling dice to determine your future action. It may be probabilistic, but it isn't freely chosen in the normal sense of the word.
- Quantum Consciousness, Hameroff's webpage
- Cold Numbers Unmake the Quantum Mind
- Is the Brain a Quantum Device? by Victor Stenger
- Conscious Machines by Marvin Minsky, a critique of some of Penrose's ideas
- If you don't talk to your kids about quantum computing, someone else will SMBC
- Note that this is the case even if, as with Penrose, you do know your quantum physics.
- "Wigner’s View of Physical Reality," Section 2.5 - Wigner’s later change of position
- What Quantum Physics Tells Us About Consciousness, Deepak Chopra
- This is Penrose's book The Emperor's New Mind and some of his later work condensed to a few sentences.
- Max Tegmark, "The Importance of Quantum Decoherence in Brain Processes," Phys. Rev. E 61 (2000) 4194-4206. Copy at arXiv.
- Max Tegmark, "The Importance of Quantum Decoherence in Brain Processes," Phys. Rev. E 61 (2000) 4194-4206. Copy at arXiv. Further explanation and commentary can be found here
- Reimers et al. Weak, strong, and coherent regimes of Fröhlich condensation and their applications to terahertz medicine and quantum consciousness. Proceedings of the National Academy of Sciences Online, Feb. 2009.
- Georgiev, Danko (2006). Falsifications of Hameroff-Penrose Orch OR Model of Consciousness and Novel Avenues for Development of Quantum Mind Theory. PhilSci Archive
- Sandra Handy via The Science Times: "Scientists Discovered Where Anesthesia Works On the Brain"
- See the Wikipedia article on Quantum computer.
- Physicalistic Idealism: An experimentally testable conjecture about the nature of the physical world.
- To borrow a phrase from Murray Gell-Mann.
- Similar, but not identical. See Isaac White's GOFAI's Biological Assumption and Where It Leads for differences between boolean logic and neural activity.
- More precisely, that classical mechanics provides very accurate approximations. Of course, neurons are subject to laws of quantum mechanics just like any object in the universe. However, quantum corrections are extremely small in magnitude.
- Hameroff has become a particularly bad offender, having gone on to be featured in What the Bleep Do We Know? and palling around with Chopra himself.
- This argument has been put forth by a number of physicists and philosophers, perhaps most recently in Stephen Hawking and Leonard Mlodinow's The Grand Design. pp. 72-73
- Although Luke Rhinehart would disagree.