Pothos & Busemeyer (P&B) argue for the application of quantum probability (QP) theory to cognitive modeling in a function-first or top-down approach that begins with the postulation of vectors in a low-dimensional space (sect. 2.1), but consideration of the high-dimensional complex-valued wave function underlying the state vector will expand the value of QP in cognitive science. To this end, we should import two premises from quantum mechanics. The first is that the fundamental reality is the wave function. In cognitive science, this corresponds to postulating spatially distributed patterns of neural activity as the elements of the cognitive state space. Therefore, the basis vectors used in QP are basis functions for an infinite (or very high) dimensional Hilbert space. The second premise is that the wave function is complex valued and that wave functions combine with complex coefficients, which is the main reason for interference and other non-classical phenomena. The authors acknowledge this (sects. 2.3, 3.3, Appendix), but they do not make explicit use of complex numbers in the target article.
There are several possible analogs in neurophysiology of the complex-valued wave function, but perhaps the most obvious is the distribution of neural activity across a region of cortex; even a square millimeter of which can have hundreds of thousands of neurons. The dynamics are defined by a time-varying Hamiltonian, with each eigenstate being a spatial distribution of neurons firing at a particular rate. The most direct representations of the magnitude and phase (or argument) of a complex quantity are the rate and relative phase of neural impulses.
The target article specifies that a decision corresponds to measurement of a quantum state, which projects the cognitive state into a corresponding eigenspace, but it is informative to consider possible mechanisms. For example, the need to act definitely (such as coming to a conclusion to answer a question) can lead to mutually competitive mechanisms, such as among the minicolumns in a macrocolumn, which create dynamic attractors corresponding to measurement eigenspaces. Approach to the attractor amplifies certain patterns of activity at the expense of others. Orthogonal projectors filter the neural activity and win the competition with a probability proportional to the squared amplitude of their inner products with the wave function. (In the case in which impulse phases encode complex phases, matching occurs when the phases are delayed in such a way that the impulses reinforce.) The winner may positively reinforce its matched signal components while the loser negatively reinforces its matched components. Regardless of mechanism, during collapse, the energy of the observed eigenstate of the question (measurement) operator captures the energy of the orthogonal eigenstates (this is the effect of renormalization). The projection switches a jumble of frequencies and phases into a smaller, more coherent collection, corresponding to the outcome (observed) eigenspace. This competition also explains the prioritization of more likely outcomes (sect. 3.1).
The target article (sect. 2.1) suggests that a QP model of cognition begins by postulating basis vectors and qualitative angles between alternative question bases (significantly, only real rotations are discussed). As a consequence, a QP model is treated as a low-dimensional vector space. This is a reasonable, top-down strategy for defining a QP cognitive model, but it can be misleading. There is no reason to suppose that particular question bases are inherent in a cognitive Hilbert space. There may be a small number of “hard-wired” questions, such as fight-or-flight, but the vast majority is learned. Certainly this is the case for questions corresponding to lexical categories such as (un-)happy and (un-)employed.
Investigation of the dynamics of cognitive wave function collapse would illuminate the mechanisms of decision making, but also the processes by which observables are organized. This would allow modeling of changes in the question bases, either temporary through context effects, or longer lasting through learning. Furthermore, many question bases are ad hoc, as when we ask, “Do you admire Telemachus in the Odyssey?” How such ad hoc projectors are organized requires looking beneath a priori basis vectors to the underlying neural wave functions and the processes shaping them.
Certainly one of the most interesting consequences of applying to QP to cognition is the analysis of incompatible questions. The approach described in the target article (sect. 2.2) begins by postulating that incompatible questions correspond to alternative bases for a vector space. The qualitative angle between the question bases is estimated by a priori analysis of whether the questions interfere with each other.
In quantum mechanics, however, the uncertainty principle is a consequence of non-commuting measurement operators, and the degree of non-commutativity can be quantified. Two measurement operators P and Q commute if PQ = QP, that is, if the operator PQ−QP is identically 0. If they fail to commute, then PQ−QP measures the degree of non-commutativity, which is expressed in quantum mechanics by the commutator [P,Q] = PQ−QP. It is relatively easy to show that this implies an uncertainty relation:
$\Delta P \Delta Q \geq \left\vert \left\langle \left[P\comma \; Q \right]\right\rangle \right\vert$
. That is, the product of the uncertainties on a state is bounded below by the absolute mean value of the commutator on the state. Suppose H is a measurement that returns 1 for
$\left\vert {\rm happy} \right\rangle$
and 0 for
$\left\vert {\rm unhappy} \right\rangle$
, and E is a measurement that returns 1 for
$\left\vert {\rm employed} \right\rangle$
and 0 for
$\left\vert {\rm unemployed} \right\rangle$
. If
and the magnitude of the commutator applied to an arbitrary state |ψ〉 is ||[H,E]| ψ〉||=|ab|.
Might we design experiments to measure the commutators and so quantify incompatibility among questions? Certainly there are difficulties, such as making independent measurements of both PQ and QP for a single subject, or accounting for intersubject variability in question operators. But making such measurements would put more quantitative teeth into QP as a cognitive model.
Pothos & Busemeyer (P&B) argue for the application of quantum probability (QP) theory to cognitive modeling in a function-first or top-down approach that begins with the postulation of vectors in a low-dimensional space (sect. 2.1), but consideration of the high-dimensional complex-valued wave function underlying the state vector will expand the value of QP in cognitive science. To this end, we should import two premises from quantum mechanics. The first is that the fundamental reality is the wave function. In cognitive science, this corresponds to postulating spatially distributed patterns of neural activity as the elements of the cognitive state space. Therefore, the basis vectors used in QP are basis functions for an infinite (or very high) dimensional Hilbert space. The second premise is that the wave function is complex valued and that wave functions combine with complex coefficients, which is the main reason for interference and other non-classical phenomena. The authors acknowledge this (sects. 2.3, 3.3, Appendix), but they do not make explicit use of complex numbers in the target article.
There are several possible analogs in neurophysiology of the complex-valued wave function, but perhaps the most obvious is the distribution of neural activity across a region of cortex; even a square millimeter of which can have hundreds of thousands of neurons. The dynamics are defined by a time-varying Hamiltonian, with each eigenstate being a spatial distribution of neurons firing at a particular rate. The most direct representations of the magnitude and phase (or argument) of a complex quantity are the rate and relative phase of neural impulses.
The target article specifies that a decision corresponds to measurement of a quantum state, which projects the cognitive state into a corresponding eigenspace, but it is informative to consider possible mechanisms. For example, the need to act definitely (such as coming to a conclusion to answer a question) can lead to mutually competitive mechanisms, such as among the minicolumns in a macrocolumn, which create dynamic attractors corresponding to measurement eigenspaces. Approach to the attractor amplifies certain patterns of activity at the expense of others. Orthogonal projectors filter the neural activity and win the competition with a probability proportional to the squared amplitude of their inner products with the wave function. (In the case in which impulse phases encode complex phases, matching occurs when the phases are delayed in such a way that the impulses reinforce.) The winner may positively reinforce its matched signal components while the loser negatively reinforces its matched components. Regardless of mechanism, during collapse, the energy of the observed eigenstate of the question (measurement) operator captures the energy of the orthogonal eigenstates (this is the effect of renormalization). The projection switches a jumble of frequencies and phases into a smaller, more coherent collection, corresponding to the outcome (observed) eigenspace. This competition also explains the prioritization of more likely outcomes (sect. 3.1).
The target article (sect. 2.1) suggests that a QP model of cognition begins by postulating basis vectors and qualitative angles between alternative question bases (significantly, only real rotations are discussed). As a consequence, a QP model is treated as a low-dimensional vector space. This is a reasonable, top-down strategy for defining a QP cognitive model, but it can be misleading. There is no reason to suppose that particular question bases are inherent in a cognitive Hilbert space. There may be a small number of “hard-wired” questions, such as fight-or-flight, but the vast majority is learned. Certainly this is the case for questions corresponding to lexical categories such as (un-)happy and (un-)employed.
Investigation of the dynamics of cognitive wave function collapse would illuminate the mechanisms of decision making, but also the processes by which observables are organized. This would allow modeling of changes in the question bases, either temporary through context effects, or longer lasting through learning. Furthermore, many question bases are ad hoc, as when we ask, “Do you admire Telemachus in the Odyssey?” How such ad hoc projectors are organized requires looking beneath a priori basis vectors to the underlying neural wave functions and the processes shaping them.
Certainly one of the most interesting consequences of applying to QP to cognition is the analysis of incompatible questions. The approach described in the target article (sect. 2.2) begins by postulating that incompatible questions correspond to alternative bases for a vector space. The qualitative angle between the question bases is estimated by a priori analysis of whether the questions interfere with each other.
In quantum mechanics, however, the uncertainty principle is a consequence of non-commuting measurement operators, and the degree of non-commutativity can be quantified. Two measurement operators P and Q commute if PQ = QP, that is, if the operator PQ−QP is identically 0. If they fail to commute, then PQ−QP measures the degree of non-commutativity, which is expressed in quantum mechanics by the commutator [P,Q] = PQ−QP. It is relatively easy to show that this implies an uncertainty relation:
$\Delta P \Delta Q \geq \left\vert \left\langle \left[P\comma \; Q \right]\right\rangle \right\vert$
. That is, the product of the uncertainties on a state is bounded below by the absolute mean value of the commutator on the state. Suppose H is a measurement that returns 1 for
$\left\vert {\rm happy} \right\rangle$
and 0 for
$\left\vert {\rm unhappy} \right\rangle$
, and E is a measurement that returns 1 for
$\left\vert {\rm employed} \right\rangle$
and 0 for
$\left\vert {\rm unemployed} \right\rangle$
. If
and the magnitude of the commutator applied to an arbitrary state |ψ〉 is ||[H,E]| ψ〉||=|ab|.
Might we design experiments to measure the commutators and so quantify incompatibility among questions? Certainly there are difficulties, such as making independent measurements of both PQ and QP for a single subject, or accounting for intersubject variability in question operators. But making such measurements would put more quantitative teeth into QP as a cognitive model.