Notation

Below is a brief list of the notation used in this book. In general, the Dirac notation is used for abstract vectors and operators that are expressed in a coordinate free manner, and traditional matrix algebra notation is used when a vector or an operator is expressed in terms of coordinates of a specific basis.

N is the dimension of a Hilbert space

a, b, c, x, y, z are scalars which can be complex numbers

X, Y, P, Q are matrices

diag[X] is a diagonal matrix formed from the N × 1 column matrix X

α, β, γ are often used to represent N × 1 column matrices of amplitudes

αi is one coordinate value of α; that is, a single amplitude

X† is the Hermitian transpose of X

X−1 is the inverse of the full rank matrix X

If α is an N × 1 column matrix, then α† is an 1 × N row matrix of conjugate values

(α† · β) is the inner product of two N × 1 column matrices (α, β)

ψ · φ† is the outer product matrix of two N × 1 column matrices (ψ, φ)

Tr[X] is the trace of the square matrix X

X ⊗ Y is the Kronecker product of two matrices

V = {|Vi⟩, i = 1, N} orthonormal basis, or W = {|Wi⟩, i = 1, N} for another one

|X⟩ is an abstract vector; it can be represented by a N × 1 matrix α once you choose a basis

How can quantum theory be used to model human performance on complex information processing tasks? Quantum computing and quantum information theory is relatively new, but it is already a highly developed field (Nielsen and Chuang, 2000). The concept of a quantum computer was introduced by Feynman in 1982 (Feynman, 1982). Soon afterwards, a universal quantum computer was formulated by David Deutsch in 1989 using quantum gates, which he demonstrated could perform computations not possible with classic Turing machines, including generating genuine random numbers and performing parallel calculations within a single register. Subsequently, new quantum algorithms were discovered by Peter Schor in 1984 and Lov Grover in 1997 that could solve important computational problems, such as factoring and searching, faster than any known Turing machine. However, all of these accomplishments were designed for actual quantum computers, and only very small versions have been realized so far. Moreover, if we are not working under the assumption that the brain is a quantum computer, then what has all of this to do with information processing by humans?

The answer is that quantum information processing theory provides new and powerful principles for modelling human performance. Currently, there are three general approaches to modelling information processing with humans. One is based on production rule systems such as used in Act-R (Anderson, 1993), EPIC (Meyer & Kieres, 1997), and Soar (Laird et al., 1987); a second is neural network (Grossberg, 1982) and connectionist network (Rumelhart & McClelland, 1986) models; and a third is Bayesian models (Griffiths et al., 2008).

Why should you be interested in quantum theory applied to Cognition and decision? Perhaps you are a physicist who is curious whether or not quantum principles can be applied outside of physics. In fact, that is one purpose of this book. Perhaps you are a cognitive scientist who is interested in representing concepts by vectors in a multidimensional feature space. This is essentially the way quantum theory works too. Perhaps you are a decision scientist who is trying to understand how people make decisions under uncertainty. Quantum theory could provide some interesting new answers. Generally speaking, quantum theory is a new theory for constructing probabilistic and dynamic systems, and in this book we apply this new theory to topics in cognition and decision. Later in this chapter we will give some specific examples, but let us step back at this point and try to understand the more general principles that support a quantum approach to cognition and decision.

Six reasons for a quantum approach to cognition and decision

Quantum physics is arguably the most successful scientific theoretical achievement that humans have ever created. It was created to explain puzzling findings that were impossible to understand using the older classical physical theory, and it achieved this by introducing an entirely new set of revolutionary principles. The older classical physical theory is now seen as a special case of the more general quantum theory. In the process of creating quantum mechanics, physicists also created a new theory of probabilistic and dynamic systems that is more general than the previous classic theory (Pitowski, 1989).

What about a process theory? How does quantum theory explain changes in confidence across time or how does quantum theory predict the time that it takes to make a decision? So far we have only made use of the structural part of quantum theory introduced in Chapter 2. This chapter introduces some of the basic principles for quantum dynamics.

It is useful to compare quantum dynamics with Markov dynamics (Howard, 1971). Markov theory is a general mathematical framework for describing probabilistic-dynamic systems, which is commonly used in all areas of cognitive and decision sciences. For example, it is the mathematical basis that underlies random walk/diffusion models of decision making (Busemeyer & Diederich, 2009), or stochastic models of information processing (Townsend & Ashby, 1983), or multinomial processing tree models of memory retrieval (Batchelder & Reiffer, 1999), or the even more general field of stochastic processes (Bhattacharya & Waymire, 1990).

Quantum theory provides an alternative general mathematical framework for describing probabilistic-dynamic systems (Gudder, 1979). However, quantum theory is similar in many ways to Markov theory, and so if you already know Markov theory, then it will be easy to learn about quantum dynamics too. This chapter introduces the Kolmogorov forward equation used to describe time evolution in Markov models, as well as the quantum analogue – the Schrödinger equation – which describes time evolution in quantum models. This chapter examines the similarities and differences between these two types of evolution, and why they are both useful for cognitive and decision modelling.

Can one really predict something new and interesting using quantum theory in the social or behavioral sciences? Consider the law of reciprocity described in Chapter 2: according to this law, the probability of transiting from one state to another is equal to the probability of making this transition in the opposite direction. Now this symmetrical property is a pretty bold prediction to make in the social and behavioral sciences. Does this really work?

To answer this question, in this chapter we empirically test this law by applying the quantum principles to an important empirical problem concerning the effects of question order on attitude judgments. For example, suppose you are asked “How happy are you with life in general?” This question could be preceded or followed by the question “How happy are you with your marriage?” If the marriage question comes first, then the happiness rating for the general life question tends to be substantially depressed, producing a large order effect (Tourangeau et al., 1991). Quantum physics was originally developed to understand how measurement affects the system under investigation, which led to Heisenberg's famous uncertainty principle. The potential reactivity to measurement by the person being measured has an even longer history in psychology (Tourangeau et al., 2000).

The effects of question order have been an important issue to survey researchers interested in studying beliefs and attitudes (Schuman & Presser, 1981). Measurement of a belief or attitude directs attention to a subset of a person's knowledge, which is then used in guiding subsequent judgments.

Can quantum theory provide new answers to puzzling results from decision research that have resisted formal explanations so far? This chapter presents several new contributions that quantum theory provides to decision research. However, in order to locate where quantum theory can make an important new contribution, we first need to provide some necessary background on decision theory and what has already been accomplished.

Decision science is a highly developed field that has produced a variety of sophisticated mathematical theories (Gilboa, 2009; Wakker, 2010). Some of these theories are rational theories, which prescribe optimal ways to make decisions. Others are psychological theories that describe the way people actually make decisions.

One of the most important rational decision theories is von Neumannn and Morgenstern's expected utility (EU) theory (von Neumann & Morgenstern, 1944). Expected utility theory is designed for decisions under risk; that is, a choice among actions defined as an objective probability distribution over possible payoffs. An important extension of (EU) theory is Savage's (1954) subjective expected utility (SEU) theory. This extends utility theory to decisions under uncertainty; that is, a choice among actions defined as functions that map un-certain events into payoffs, and the events may not have objective probabilities. A more general theory designed for decisions under both uncertainty and risk was presented by Anscombe and Aumann (1963). All of these theories are rational, in the sense that they are based on a small set of compelling axioms and the prescribed action logically follows from the axioms.

Learning is a criticai aspect of any intelligent cognitive system. How can this be done within a QIP approach? This is a relatively new field, but some progress has already been achieved (Ivancevic & Ivancevic, 2010). There are at least three ways to accomplish learning using quantum principles. One way is to update the agent's belief state based on experience (Schack et al., 2001), as done in Bayesian learning models (Griffiths et al., 2008). A second way is to update the weights in a unitary matrix using gradient descent of an error function (Zak & Williams, 1998), as done in connectionist learning models (Rumelhart & McClelland, 1986). A third way is to update the amplitudes assigned to control U gate actions based on rewards and punishments (Dong et al., 2010), as done with reinforcement learning algorithms (Sutton & Barto, 1998). This chapter reviews all three approaches.

Quantum state updating based on experience

For the first type of quantum learning model, consider how to update an agent's belief state based on experience. In Chapter 4 we presented a quantum model for probability judgments, and in that chapter the initial belief state denoted |ψ⟩ was given or assumed to be already known in advance – when new facts were presented, inferences were made from the known state |ψ⟩ using Luder's rule. However, where does this initial state |ψ⟩ come from? Now we examine how this initial state |ψ⟩ can be learned or estimated from experience. Principles borrowed from quantum state tomography can be used to model the estimation of a quantum state (Schack et al., 2001).

How can we use quantum theory to model other phenomena of interest to researchers in cognition and decision making? Quantum theory is not easy for researchers in cognition and decision making to accept. In fact, quantum mechanics was not easy for physicists to accept either, but it was forced on them by several paradoxical findings that could not be explained using classical physics. We have a similar problem in cognition and decision making – there are numerous paradoxical findings that just seem irrational according to classic probability theory. For example, under some conditions, people judge the probability of event A and B to be greater than the probability of event B, which is called the conjunction fallacy (Tversky & Kahneman, 1983). Also, under the same conditions, they judge the probability of A or B to be less than the probability of event A (Carlson & Yates, 1989), which is called the disjunction fallacy. In this chapter we examine how quantum probability theory explains these and other puzzling results from human probability judgment research. This chapter has two main parts. In the first part, we use a quantum model to derive qualitative predictions for conjunction errors, disjunction errors, and other closely related findings. The first section provides a general set of predictions that do not depend on specific assumptions about the features used to represent events, and the predictions are parameter free. In the second part, we examine the quantitative predictions of the quantum model for a Bayesian inference task, which we use to explain order effects on inference.

Rationale

The purpose of this book is to introduce the application of quantum theory to cognitive and decision scientists. At first sight it may seem bizarre, even ridiculous, to draw a connection between cognition and decision making – research lying within the realm of day-to-day human behavior – on the one hand and quantum mechanics – a highly successful theory for modelling subatomic phenomena – on the other hand. Yet there are good scientific reasons for doing so, which is leading a growing number of researchers to examine quantum theory as a way to understand perplexing findings and stubborn problems within their own fields. Hence this book. Given the nascent state of this field, some words of justification are warranted. The research just mentioned is not concerned with modelling the brain using quantum mechanics, nor is it directly concerned with the idea of the brain as a quantum computer. Instead it turns to quantum theory as a fresh conceptual framework for explaining empirical puzzles, as well as a rich new source of alternative formal tools. To convey the idea that researchers in this area are not doing quantum mechanics, various modifiers have been proposed to describe this work, such as quantum-like models of cognition, cognitive models based on quantum structure, or generalized quantum models.

There are two aspects of quantum theory which open the door to addressing problems facing cognition and decision in a totally new light.

A connection between quantum theory and human memory came out of cybernetics research in the early 1950s (von Foerster, 1950). It was not until modelling the activation of a word in human memory did the connection begin to appear in mainstream psychological literature. Target word activation was speculated as being similar to quantum entanglement because the target word and its associative network were being activated “in synchrony,” the core intuition behind the “spooky-action-at-a-distance” formula (Nelson et al., 2003). This chapter begins by developing a quantum-like model of the human mental lexicon and compares it against other memory models. Specific attention is paid to how these models describe the activation of a word in memory in the context of the extra- and intra-list cuing tasks. The chapter then covers how quantum theory can be applied to model human semantic space.

The human mental lexicon

A mental lexicon refers to the words that comprise a language, and its structure is defined here by the associative links that bind this vocabulary together. Such links are acquired through experience and the vast and semi-random nature of this experience ensures that words within this vocabulary are highly inter-connected, both directly and indirectly through other words. For example, the word planet becomes associated with earth, space, moon, and so on, and within this set, moon can become linked to earth and star. Words are so associatively interconnected with each other they meet the qualifications of a “small world” network wherein it takes only a few associative steps to move from any one word to any other in the lexicon (Steyvers & Tenenbaum, 2005).

Quantum theory has inspired many new ideas for understanding Cognition and decision making. Given the nascent state of this field, many new challenges still remain. What are the strengths and unique contributions of these ideas? What are likely next steps in its development? How can it be applied to practical problems in cognitive science? Is this a rational or an irrational system of reasoning? What, if any, are its connections to neuroscience? Should the hard problem of consciousness be considered? These are questions that we try to address in this chapter, albeit at times in a limited way. We end with a debate with a fictitious skeptic (actually previous reviewers) of these ideas. Let us start by reviewing what we have covered so far.

Contributions to cognition and decision

We begin by summarizing the conclusions reached in previous chapters regarding the usefulness of quantum theory for modelling phenomena in cognition and decision. Chapter 1 provided some motivation and Chapter 2 provided a theoretical introduction to quantum theory. Chapter 3 presented a quantum model to account for question order effects on survey research. An exact and parameter-free test was derived from the model, which produced surprisingly accurate predictions for results obtained from three different large Gallup poll survey studies. Chapter 4 described a quantum model of probability judgments to account for findings that produce violations of classic probability theory, including conjunction and disjunction fallacies and order effects on inference.

What can you learn about quantum theory in one chapter without knowing any physics? A complete presentation of quantum theory requires an entire textbook, but our goal for this chapter is to provide only the essential elements of the theory that we feel are relevant for modelling behavioral phenomena. Just like classical probability theory, quantum theory is based on a small set of axioms used to assign probabilities to events. This chapter is limited to what is called the structural principles of quantum theory, whereas Chapter 8 will set out the dynamic principles. For simplicity, only finite state systems will be described, although quantum theory can also be applied to continuous state systems. Even though this book is limited to finite dimensions, the number of dimensions can be arbitrarily large.

For finite state systems, the structural part of quantum theory is expressed in the formalism of linear algebra (for continuous state systems, it is expressed in the formalism of functional analysis). Consequently, a brief tutorial of linear algebra is presented along with our elementary introduction to quantum theory. Various notations are used to describe linear algebra, depending on the application field. Physicists like to use Dirac notation, invented by one of the founders of quantum theory, Paul Dirac (Dirac, 1958). Although the Dirac notation will be unfamiliar to many cognitive scientists, we will still use it as it is a succinct notation for expressing linear algebra. It helps the reader to see relations that are more difficult to identify using other formalisms.

Consider the concept combination “pet human.” In word association experiments, human subjects often produce the associate “slave” in relation to this combination. The striking aspect of this associate is that it is not produced as an associate of “pet” or “human” in isolation. In other words, the associate “slave” cannot be recovered from the constituent concepts. Such examples have been used in both cognitive science and philosophy to argue that concept combinations have a non-compositional semantics. This chapter will feature how various non-compositional accounts of concept combinations can be provided from quantum theory. Quantum theory is a theory which caters for the modelling of non-compositionality because the state of a quantum entangled system cannot be constructed from the states of its individual subsystems. Utilizing probabilistic methods developed for analyzing composite systems in quantum theory, we show that it is possible to classify concept combinations as having “classically compositional,” “pseudo-classically non-compositional,” or “non-classically non-compositional” semantics by determining whether the joint probability distribution modelling the combination is factorizable or not.

Concept combinations and cognition

The principle of semantic compositionality states the meaning of a (syntactically complex) whole is a function only of the meanings of its (syntactic) parts together with the manner in which these parts were combined (Pelletier, 1994). Whether the semantics of concepts are compositional, or not, has been somewhat of a battleground. On the one hand, there are authors like Fodor (1994) who are adamant that concepts are, and must be, compositional, but this position stands in contrast with the computational linguist Zadrozny (1994), who produced a theorem suggesting that: “the standard definition of compositionality is formally vacuous.”

One of the goals of quantum Cognition and decision is to explain findings that seem paradoxical from a classic probability point of view. This is important to justify the introduction of this new theory. So far we have used the theory to explain paradoxical findings from research on human probability judgments and conceptual combinations. These are both considered higher level cognitive functions that involve more complex reasoning. What about more primitive cognitive functions such as memory recognition? Do paradoxical findings with respect to classic probability theory occur in this basic area of cognition? If so, how can a quantum model help to explain these findings over and above current explanations? This chapter presents a new application of quantum theory to a puzzling phenomenon observed in human memory recognition called the episodic overdistribution (EOD) effect (Brainerd & Reyna, 2008). First we describe the phenomena and explain why it is a puzzle, and then we present a quantum solution to the puzzle and compare it with previous memory recognition models.

Episodic overdistribution effect

In the conjoint–recognition paradigm, participants are rehearsed on a single set T of memory targets (e.g., each member is a short description of an event). After a delay, a recognition test phase occurs, during which they are presented with a series of test probes that consist of trained targets from T, related non-targets from a different set R of distracting events (e.g., each member is a new event that has some meaningful relation to a target event), and an unrelated set U of non-target items (e.g., each member is completely unrelated to the targets).

Although the basic ideas of quantum computation are fairly straightforward, the detailed manipulations required to extract a useful result from an entangled superposition of answers can be rather complicated. There are, however, a small number of algorithms which are simple enough to explain using only elementary methods. The best example is Deutsch's algorithm: as this only requires two qubits its properties are amenable to brute-force matrix calculations, and these calculations can be simplified using a variety of short cuts, which also give some insight into how and why the algorithm works. We will also explore the simplest case of Grover's quantum search algorithm, which is once again simple enough to be tacked by brute force. Finally we will look briefly at two methods used to stabilize quantum computers against errors.

Deutsch's algorithm

The invention of Deutsch's algorithm can be taken as defining the start of modern quantum computation, and it remains a key example, exhibiting many of the key properties of quantum algorithms in a particularly simple form. (Note, however, that the version of Deutsch's algorithm described here is not in fact the original but a later modification which is both more powerful and easier to understand.)

Consider a binary function f from one bit to one bit, that is a function which takes in either 0 or 1 as its input and returns either 0 or 1 as its output. There are four such functions which may be conveniently labeled by their outputs as shown in Table 8.1.

We now have all the basic tools we need to describe systems of one and two qubits. In this final chapter of Part I we look at some of the consequences of the peculiar properties of qubits when they are used to encode information. Many of these results can be traced to the properties of quantum measurement.

Measuring a single qubit

A key result in quantum information theory is that it is impossible to accurately characterize a single qubit. In other words, there is no experiment, or sequence of experiments, which allows us to find out the state of a single quantum bit.

The reason for this problem is twofold. Firstly, we have to make some sort of decision about the basis we will use for the measurement. For example, when measuring a single qubit the most popular choice is to make a measurement in the computational basis. This is equivalent to asking the qubit whether it is in state |0⧽ or state |1⧽. If the qubit is indeed in one of the basis states the measurement process is simple, and we will get the obvious answer of 0 if it is in |0⧽ and 1 if it is in |1⧽. If, however, the qubit is in a superposition, such as α|0⧽+β|1⧽, then the situation is more difficult. Characterizing the state now means determining the values of the two complex numbers α and β, but the measurement can only return the answer 0 or 1, and for a superposition state one of these two answers will be returned at random, with probabilities |α|2 and |β|2, respectively.