Typically, human decision making is emotionally “hot” and does not conform to “cold” classical probability (CP) theory. As quantum probability (QP) theory emphasises order, context, superimposition states, and nonlinear dynamic effects, one of its major strengths may be its power to unify formal modeling and realistic psychological theory (e.g., information uncertainty, anxiety, and indecision, as seen in the Prisoner's Dilemma).

There is another meeting place for quantum physics and psychology, both within and outside of cognitive modeling. In physics it is known as the issue of classical (probabilistic) determinism, and in psychology it is known as the issue of selective influences. The formalisms independently developed in the two areas for dealing with these issues turn out to be identical, opening ways for mutually beneficial interactions.

Quantum probability theory offers a viable alternative to classical probability, although there are some ambiguities inherent in transferring the quantum formalism to a less determined realm. A number of physicists are now looking at the applicability of quantum ideas to the assessment of physics learning, an area particularly suited to quantum probability ideas.

We are sympathetic to the general ideas presented in the article by Pothos & Busemeyer (P&B): Heisenberg's uncertainty principle seems naturally relevant in the social and behavioral sciences, in which measurements can affect the people being studied. We propose that the best approach for developing quantum probability models in the social and behavioral sciences is not by directly using the complex probability-amplitude formulation proposed in the article, but rather, more generally, to consider marginal probabilities that need not be averages over conditionals.

Quantum probability (QP) theory provides an alternative account of empirical phenomena in decision making that classical probability (CP) theory cannot explain. Cognitive architectures combine probabilistic mechanisms with symbolic knowledge-based representations (e.g., heuristics) to address effects that motivate QP. They provide simple and natural explanations of these phenomena based on general cognitive processes such as memory retrieval, similarity-based partial matching, and associative learning.

Evolution would favor organisms that can make recurrent decisions in accordance with classical probability (CP) theory, because such choices would be optimal in the long run. This is illustrated by the base-rate fallacy and probability matching, where nonhumans choose optimally but humans do not. Quantum probability (QP) theory may be able to account for these species differences in terms of orthogonal versus nonorthogonal representations.

The “Orch OR” theory suggests that quantum computations in brain neuronal dendritic-somatic microtubules regulate axonal firings to control conscious behavior. Within microtubule subunit proteins, collective dipoles in arrays of contiguous amino acid electron clouds enable “quantum channels” suitable for topological dipole “qubits” able to physically represent cognitive values, for example, those portrayed by Pothos & Busemeyer (P&B) as projections in abstract Hilbert space.

I consider the general problem of category conjunctions in the light of Pothos & Busemeyer (P&B)'s quantum probability (QP) account of the conjunction fallacy. I argue that their account as presented cannot capture the “guppy effect” – the case in which a class is a better member of a conjunction A^B than it is of either A or B alone.

We concentrate on two aspects of the article by Pothos & Busemeyer (P&B): the relationship between classical and quantum probability and quantum probability as a basis for rational decisions. We argue that the mathematical relationship between classical and quantum probability is not quite what the authors claim. Furthermore, it might be premature to regard quantum probability as the best practical rational scheme for decision making.

We highlight methodological and theoretical limitations of the authors' Dirac formalism and suggest the von Neumann open systems approach as a resolution. The open systems framework is a generalization of classical probability and we hope it will allow cognitive scientists to extend quantum probability from perception, categorization, memory, decision making, and similarity judgments to phenomena in learning and development.

Although the quantum probability (QP) can be useful to model the context effect, it is not relevant to the order effect, conjunction fallacy, and other related biases. Although the issue of potentiality, which is the intuition behind QP, is involved in the context effect, it is not involved in the other biases.

According to the foundations of economic theory, agents have stable and coherent “global” preferences that guide their choices among alternatives. However, people are constrained by information-processing and memory limitations and hence have a propensity to avoid cognitive load. We propose that this in turn will encourage them to respond to “local” preferences and goals influenced by context and memory representations.

Faced with probabilistic relationships between causes and effects, quantum theory assumes that deterministic causes do not exist, and that only incomplete probabilistic expressions of knowledge are possible. As in its application to physics, this fundamental epistemological stance severely limits the ability of quantum theory to provide insight and understanding in human cognition.

Pothos & Busemeyer (P&B) provide a compelling case that quantum probability (QP) theory is a better match to human judgment than is classical probability (CP) theory. However, any theory (QP, CP, or other) phrased solely at the computational level runs the risk of being underconstrained. One suggestion is to ground QP accounts in mechanism, to leverage a wide range of process-level data.

Use of quantum probability as a top-down model of cognition will be enhanced by consideration of the underlying complex-valued wave function, which allows a better account of interference effects and of the structure of learned and ad hoc question operators. Furthermore, the treatment of incompatible questions can be made more quantitative by analyzing them as non-commutative operators.

A lot of research in cognition and decision making suffers from a lack of formalism. The quantum probability program could help to improve this situation, but we wonder whether it would provide even more added value if its presumed focus on outcome models were complemented by process models that are, ideally, informed by ecological analyses and integrated into cognitive architectures.

Pothos & Busemeyer (P&B) argue convincingly that quantum probability offers an improvement over classical Bayesian probability in modeling the empirical data of cognitive science. However, a weakness related to restrictions on the dimensionality of incompatible physical observables flows from the authors' “agnosticism” regarding quantum processes in neural substrates underlying cognition. Addressing this problem will require either future research findings validating quantum neurophysics or theoretical expansion of the uncertainty principle as a new, neurocognitively contextualized, “local” symmetry.

Pothos & Busemeyer (P&B) argue that quantum probability (QP) provides a descriptive model of behavior and can also provide a rational analysis of a task. We discuss QP models using Marr's levels of analysis, arguing that they make most sense as algorithmic level theories. We also highlight the importance of having clear interpretations for basic mechanisms such as interference.

We focus on two issues: (1) an unusual, counterintuitive prediction that quantum probability (QP) theory appears to make regarding multiple sequential judgments, and (2) the extent to which QP is an appropriate and comprehensive benchmark for assessing judgment. These issues highlight how QP theory can fall prey to the same problems of arbitrariness that Pothos & Busemeyer (P&B) discuss as plaguing other models.

Quantum theory is a powerful framework for probabilistic modeling of cognition. Strong empirical evidence suggests the context- and order-dependent representation of human judgment and decision-making processes, which falls beyond the scope of classical Bayesian probability theories. However, considering behavior as the output of underlying neurobiological processes, a fundamental question remains unanswered: Is cognition a probabilistic process at all?