The notion that the theory of measurement and the logic of quantum probability can be applied to mental states is almost as old as quantum theory itself, and has drawn the attention of such outstanding physicists and mathematicians as Bohr (who adapted James's notion of complementarity to physics [Pais Reference Pais1991]) and Penrose (Reference Penrose1989). Pothos & Busemeyer's (P&B's) analysis provides a cogent statement of the case for using quantum probability. The prospect that quantum theory can be useful outside the realm of atomic and subatomic physics is, or ought to be, exciting to the community of physicists. We feel that we, as physicists with a strong interest in cognitive aspects of physics learning, can contribute to the discussion.

In its simple form, quantum theory asserts that the maximally determined states of a quantum system constitute a Hilbert space, with certain rules for the evolution of the system in time and for calculating the probabilities of various measurement outcomes. Hilbert spaces are generalizations of the ordinary two and three-dimensional vector spaces one might have encountered in high school or college. A Hilbert space is a collection of vectors with a defined inner product that allows one to determine the extent to which the vectors point in the same direction. If |ψ> and |ϕ> are vectors in the Hilbert space, then we denote the scalar product as < ${\bf\varphi}$ | ψ> using the notation introduced by Dirac (Reference Dirac1958) and adopted by P&B. For atomic systems, |ψ> is a complex quantity, vector, or function having real or imaginary parts whose scalar product with itself equals one, <ψ|ψ> = 1. The probability that a measurement of a property made on the system in state |ψ> yields a measured value consistent with the state | ${\rm\varphi}$ > is given by the absolute square |< ${\rm\varphi}$ |ψ>|². In strict quantum theory, once the measurement consistent with |φ>is made, the system is fully in the state | ${\rm\varphi}$ >, with < ${\rm\varphi}$ | ${\rm\varphi}$ > = 1. This re-normalization process is not apparent in the analysis of the target article. It is, however, not needed if one considers only relative probabilities.

Measurable quantities in quantum theory are represented by operators that transform the vectors of the Hilbert space. Measurable quantities A and B are compatible when the operators A and B commute, that is, for all possible states | $\rm \varphi$ >, AB| $\rm\varphi$ >BA| $\rm\varphi$ >. The non-commutation of any A and B is ultimately determined by the basic lack of commutation of position and momentum of a particle, which gives rise to the uncertainty principle for the operators for position and momentum, which are conjugate operators in a technical sense (Bohr's complementarity applied to the most basic mechanical measurements). When quantum probability is applied to mental states or likelihood judgments, as is done by P&B, one must make a number of assumptions about the predicates “is a feminist” or “is a bank teller,” which fall outside the realm of physics in which complementary attributes are easier to define.

For systems not maximally determined, that is, systems that have not passed through a sufficient number of filters to determine the values of a full set of commuting observables, a single state vector in Hilbert space cannot provide a complete description. Such systems are more generally said to be in mixed states, which are incoherent superpositions of the pure quantum states, and are described by a density operator on the Hilbert space of possible maximally specified states. In terms of probable outcomes of measurements, the mixed state allows interpolation between the realm of Bayseian probability in which P(AB|A) is P(A)P(B|A) and quantum probability, in which the Baysian assumptions absolutely do not hold for sequential measurements of incompatible measurements.

Treatment of mixed states is possible using the density matrix formalism introduced by Von Neumann in 1927, which is concisely summarized by Fano (Reference Fano1957). The trace of the density matrix (sum of its diagonal elements) is unity, as the system must be in some state. Multiply the density matrix by itself, however, and the trace is unity only if the system is in a pure, that is, maximally characterized, state. For systems in a mixed state, it is possible to define an entropy function that characterizes our lack of knowledge about the state. Making measurements on a system in a mixed state produces a mixed state of lower entropy or a pure state.

An important illustration of the possibilities suggested by the density matrix is provided by the work of Bao et al. (Reference Bao, Hogg and Zollman2002). In studying the mental models used by classes of undergraduates in answering a group of related physics questions, they assign a vector of unit length to each student, depending on the models used by the student, and then from a density matrix to represent the entire class. As the class evolves with instruction, the density matrix changes and approaches that for a pure state. As the students progress in their learning, the density matrix changes, as testing is a form of measurement. Ultimately, one ends up with a pure state for the well-instructed class.

Our comments are not intended to disparage the importation of ideas from quantum probability to applications in psychology, but rather to draw attention to the richness of the formalism and some possible pitfalls. We hope that there will be continuing interaction among quantum theorists, physics teachers, and psychologists, to their mutual benefit.