Pothos & Busemeyer (P&B) present the Dirac formalism of quantum probability (DQP) as a potential direction for cognitive modeling. It is important to stress that the quantum probability (QP) approach is a modeling framework. It does not provide guidance in designing experiments or generating testable predictions. It is not a theory. P&B do not show how the framework could be used to build predictive theories: all the examples listed are post hoc descriptive models. Therefore, the target article should be judged on its merits as a descriptive framework.
A framework has to allow experimenters to average behavior over (or within) participants. Unfortunately, the pure states of P&B's Dirac formalism cannot be directly averaged. Experimental psychologists have addressed this methodological constraint of DQP by ad hoc fixes (Bruza et al. Reference Bruza, Kitto, Nelson and McEvoyc2009; Franco Reference Franco2009). To avoid such fixes or creating statistics outside the framework, one has to introduce mixed states (see sect. 2.4 of Nielsen & Chuang [Reference Nielsen and Chuang2000]), which physicists use to reason about open quantum systems.
A theoretical expressiveness constraint is that the time evolution in DQP is inherently periodic. After a recurrence time, the evolution will be arbitrary close to (experimentally indistinguishable from) the identity, as a consequence of the Poincaré recurrence theorem for closed quantum systems. In terms of decision making, this means that given enough deliberation time, a participant will always return to a mental state indistinguishable from the one before deliberation. As the QP framework has no natural time scales, this issue can be swept under the rug by hoping that the recurrence time is astronomical compared with typical deliberation. To overcome this limitation in a principled manner, we must look beyond the Dirac formalism and allow irreversible evolutions.
In quantum information theory, irreversible evolutions are introduced via quantum channels (see sect. 8.2 of Nielsen & Chuang [Reference Nielsen and Chuang2000]). Like the mixed states, and unlike P&B's pure states and unitary evolution, quantum channels can be averaged; a methodological asset. The combination of quantum channels and mixed states is often described as the “von Neumann formalism” (NQP; or “Lindbland form” if we think in terms of master equations) and is the standard approach for dealing with open quantum systems. In physics, it is possible to alternate between NQP and DQP by explicitly modeling the environment to close the system. However, this is an unreasonable constraint for modeling an open system such as the mind. When modeling the decision “Is Linda a bank teller or feminist?” psychologists do not close the system by explicitly considering all the other concurrent thoughts (e.g., “What will I have for dinner?”).
Modellers must choose between NQP and DQP. The preceding methodological and theoretical considerations compel us to select NQP. Because a Markov chain is a type of quantum channel, this choice makes the framework a generalization of classical probability (CP; see Kaznatcheev [submitted] for a complete treatment) and renders section 4.3 of the target article moot. We cannot use global properties of the transition operator (such as the law of total-probability versus double-stochasticity) to empirically decide between CP and NQP, because any transition matrix consistent with CP will also be consistent with NQP.
If we take NQP as a generalization of CP, then we are faced with the difficulty of justifying the increase in complexity. Physicists are able to justify the difficulty of quantum mechanics over classical local hidden variable theories through experimental violations of Bell's inequality. Physical tests of Bell's inequality rely on being able to separate entangled subsystems (local property of the transition operator) far enough and perform measurements fast enough that the subsystems cannot communicate without violations of special relativity. Although Aerts and Sozzo (Reference Aerts and Sozzo2011b) are able to produce joint probability functions that violate Bell's inequality in psychological settings, they cannot draw the strong conclusions of physicists because their experiments cannot create a space-like separation between the subsystems (in their case the systems are concepts within a single mind). In this psychological setting, local and global hidden variable theories cannot be distinguished, and a classical explanation cannot be ruled out. Without an operationalist test (such as Bell's) or reductionist grounding in the physics of quantum systems (which seems unlikely [Litt et al. Reference Litt, Eliasmith, Kroon, Weinstein and Thagard2006]), we are left with only practical considerations as criteria to decide between the CP and NQP frameworks.
Because QP is notorious for being unintuitive and difficult to understand, great care is required when trying to reconcile QP with classical intuition. For example, non-orthogonal states are impossible to perfectly distinguish (Chefles Reference Chefles2000). P&B make this error in the classical reasoning of their model in section 3.1:
[I]n situations such as this, the more probable possible outcome is evaluated first…Therefore, the conjunctive statement involves first projecting onto the feminist basis vector, and subsequently projecting on the [non-orthogonal] bank teller one. (emphasis ours)
Unfortunately, the subject cannot determine the more overlapped state by any process inside QP. P&B have to step outside their framework and provide an ad hoc fix of the sort they dismiss in CP.
From a psychological perspective, it is encouraging that the QP framework addresses issues in perception, categorization, memory, decision making, and similarity judgments. However, it lacks tools to cover learning and development. In contrast, Bayesian CP is capable of learning and inference (Griffiths et al. Reference Griffiths, Kemp, Tenenbaum and Sun2008), and has made inroads into modeling development (Shultz Reference Shultz2007). Comparable extensions of the QP framework would be welcomed. If QP cannot be extended to learning and development, then it is unlikely to replace the popular Bayesian CP approach.
In conclusion, we agree that a QP approach should be pursued in psychology, but we suggest viewing it from the open systems perspective of NQP. DQP is restricted to closed systems, but the mind is an open system, and, therefore, NQP yields a better modeling framework. However, it is unclear how to experimentally rule out ad hoc CP interpretations without physically separating psychological subsystems. P&B's survey of QP's significant progress in covering a range of psychological phenomena is promising, and we look forward to future coverage of learning and development.
Pothos & Busemeyer (P&B) present the Dirac formalism of quantum probability (DQP) as a potential direction for cognitive modeling. It is important to stress that the quantum probability (QP) approach is a modeling framework. It does not provide guidance in designing experiments or generating testable predictions. It is not a theory. P&B do not show how the framework could be used to build predictive theories: all the examples listed are post hoc descriptive models. Therefore, the target article should be judged on its merits as a descriptive framework.
A framework has to allow experimenters to average behavior over (or within) participants. Unfortunately, the pure states of P&B's Dirac formalism cannot be directly averaged. Experimental psychologists have addressed this methodological constraint of DQP by ad hoc fixes (Bruza et al. Reference Bruza, Kitto, Nelson and McEvoyc2009; Franco Reference Franco2009). To avoid such fixes or creating statistics outside the framework, one has to introduce mixed states (see sect. 2.4 of Nielsen & Chuang [Reference Nielsen and Chuang2000]), which physicists use to reason about open quantum systems.
A theoretical expressiveness constraint is that the time evolution in DQP is inherently periodic. After a recurrence time, the evolution will be arbitrary close to (experimentally indistinguishable from) the identity, as a consequence of the Poincaré recurrence theorem for closed quantum systems. In terms of decision making, this means that given enough deliberation time, a participant will always return to a mental state indistinguishable from the one before deliberation. As the QP framework has no natural time scales, this issue can be swept under the rug by hoping that the recurrence time is astronomical compared with typical deliberation. To overcome this limitation in a principled manner, we must look beyond the Dirac formalism and allow irreversible evolutions.
In quantum information theory, irreversible evolutions are introduced via quantum channels (see sect. 8.2 of Nielsen & Chuang [Reference Nielsen and Chuang2000]). Like the mixed states, and unlike P&B's pure states and unitary evolution, quantum channels can be averaged; a methodological asset. The combination of quantum channels and mixed states is often described as the “von Neumann formalism” (NQP; or “Lindbland form” if we think in terms of master equations) and is the standard approach for dealing with open quantum systems. In physics, it is possible to alternate between NQP and DQP by explicitly modeling the environment to close the system. However, this is an unreasonable constraint for modeling an open system such as the mind. When modeling the decision “Is Linda a bank teller or feminist?” psychologists do not close the system by explicitly considering all the other concurrent thoughts (e.g., “What will I have for dinner?”).
Modellers must choose between NQP and DQP. The preceding methodological and theoretical considerations compel us to select NQP. Because a Markov chain is a type of quantum channel, this choice makes the framework a generalization of classical probability (CP; see Kaznatcheev [submitted] for a complete treatment) and renders section 4.3 of the target article moot. We cannot use global properties of the transition operator (such as the law of total-probability versus double-stochasticity) to empirically decide between CP and NQP, because any transition matrix consistent with CP will also be consistent with NQP.
If we take NQP as a generalization of CP, then we are faced with the difficulty of justifying the increase in complexity. Physicists are able to justify the difficulty of quantum mechanics over classical local hidden variable theories through experimental violations of Bell's inequality. Physical tests of Bell's inequality rely on being able to separate entangled subsystems (local property of the transition operator) far enough and perform measurements fast enough that the subsystems cannot communicate without violations of special relativity. Although Aerts and Sozzo (Reference Aerts and Sozzo2011b) are able to produce joint probability functions that violate Bell's inequality in psychological settings, they cannot draw the strong conclusions of physicists because their experiments cannot create a space-like separation between the subsystems (in their case the systems are concepts within a single mind). In this psychological setting, local and global hidden variable theories cannot be distinguished, and a classical explanation cannot be ruled out. Without an operationalist test (such as Bell's) or reductionist grounding in the physics of quantum systems (which seems unlikely [Litt et al. Reference Litt, Eliasmith, Kroon, Weinstein and Thagard2006]), we are left with only practical considerations as criteria to decide between the CP and NQP frameworks.
Because QP is notorious for being unintuitive and difficult to understand, great care is required when trying to reconcile QP with classical intuition. For example, non-orthogonal states are impossible to perfectly distinguish (Chefles Reference Chefles2000). P&B make this error in the classical reasoning of their model in section 3.1:
[I]n situations such as this, the more probable possible outcome is evaluated first…Therefore, the conjunctive statement involves first projecting onto the feminist basis vector, and subsequently projecting on the [non-orthogonal] bank teller one. (emphasis ours)
Unfortunately, the subject cannot determine the more overlapped state by any process inside QP. P&B have to step outside their framework and provide an ad hoc fix of the sort they dismiss in CP.
From a psychological perspective, it is encouraging that the QP framework addresses issues in perception, categorization, memory, decision making, and similarity judgments. However, it lacks tools to cover learning and development. In contrast, Bayesian CP is capable of learning and inference (Griffiths et al. Reference Griffiths, Kemp, Tenenbaum and Sun2008), and has made inroads into modeling development (Shultz Reference Shultz2007). Comparable extensions of the QP framework would be welcomed. If QP cannot be extended to learning and development, then it is unlikely to replace the popular Bayesian CP approach.
In conclusion, we agree that a QP approach should be pursued in psychology, but we suggest viewing it from the open systems perspective of NQP. DQP is restricted to closed systems, but the mind is an open system, and, therefore, NQP yields a better modeling framework. However, it is unclear how to experimentally rule out ad hoc CP interpretations without physically separating psychological subsystems. P&B's survey of QP's significant progress in covering a range of psychological phenomena is promising, and we look forward to future coverage of learning and development.