Let me take the time to briefly recapitulate the results of part II before moving on. Two complementary programs have been running alongside each other. One concerns the possibility of understanding quantum probability from a classical per- spective by means of a classical representation of a quantum probability space. The other concerns the possibility of understanding quantum mechanics from a hidden variable perspective. The main difficulty encountered is that neither is possible in such a way that all relevant aspects of the quantum are captured in a faithful way. The Kochen-Specker theorem provides a compact description of the problem: classical representations or hidden variables are impossible when one holds on to both NC and IP. Giving up either of these two assumptions was shown to be suf- ficient to turn the impossibility into a possibility. A striking aspect of these possi- bilities is that for the constructed measurable space the probability functions that obey the Born rule form a special subset of all admissible probability functions. So while one of the important messages of section 4.3 was that classical probability is a special case of quantum probability, here the opposite message prevails: quantum probability is a special case of classical probability. These are not two contradictory messages, nor does their simultaneous truth establish that classical and quantum probability are one and the same thing. The fact that every classical probability space can be represented by a quantum probability space and vice versa rather may be seen to be analogous to the claim that every circle can be fitted within a square and vice versa. This claim is obviously true, but it is also clear that its truth does not imply that circles are squares.10
From the hidden variable perspective the situation is not entirely surprising. One of the reasons for introducing hidden variables in the first place was to correct for the incompleteness of the quantum formalism. The hidden variables describe putative states of affairs that aren’t accounted for by the quantum state. Necessarily then, probability functions that are peaked around these states cannot be described
10
It deserves to be noted that a similar situation is well-known within the literature on Bell inequalities (Cirel’son, 1980; Popescu and Rohrlich, 1994; Brunner et al., 2014). Although under certain conditions these inequalities cannot be violated within a classical representation, relieving these conditions allows for stronger violations than that can be obtained with quantum probabil- ities. The tension there is often framed by a contrast between signaling interactions and general non-local interaction. Such a contrast however cannot explain the general mismatch between quantum probability spaces and their classical representatives. After all, as seen in this part, this mismatch already occurs for single particle systems.
by a quantum probability function. However, one still has to explain why in practice these non-quantum probability functions play no role. This is best understood by way of an example. Consider again Bohmian mechanics. As noted earlier, this theory does not neatly conform to the formalism of a contextual hidden variable theory as illustrated in section 6.3. Instead, the hidden variables have a physical story to tell, and this story is helpful in explaining the Born rule: it is used to show that this rule arises as an equilibrium distribution. Importantly, then, the Born rule is not fundamental in Bohmian mechanics. Other probability functions are not prohibited, but are just ‘atypical’, and this even allows, in principle, for empirical discrepancies with quantum mechanics (Valentini, 2010).
When it comes to understanding the formalism of quantum probability on its own, the hidden variable perspective doesn’t suffice. Putting it somewhat boldly, it ‘explains’ the formalism of quantum probability by stating that it is essentially incorrect, or at least incomplete. What I am hinting at is an important distinc- tion between the quantum representation of classical probability and the classical representation of quantum probability. In the first case there is a criterion that sin- gles out the classical from the quantum: classical probability corresponds, roughly speaking, to commutative probability. But conversely it isn’t easy to understand from a classical perspective, either formally or conceptually, what the special role of the quantum probability functions is. Without such an understanding it isn’t entirely satisfactory to speak of a faithful representation. This is not to say that faithful representations are an impossibility. But a more direct investigation of quantum probability may be more worth our time, and this is what I do in part III of this dissertation.
Worries about the possibility of deriving the Born rule within hidden variable theories seldom play a role in discussions on the pros and cons of hidden variables. As noted above, there are good reasons for this: the Born rule may just be a con- tingent rule. More common are the objections that “the introduction of hidden variables does not lead to new experimentally testable predictions”, or that “hid- den variable theories cannot be generalized to incorporate quantum field theories”. But the most traditional objection is of course that “hidden variable theories are necessarily non-local”. Correspondingly, there are also generic replies to these ob- jections. The first objection misses the point: the aim of hidden variables is to solve the measurement problem, or, more modestly, to simply give an interpreta- tion of quantum mechanics, not to necessarily lead to new physics. The second is an undecided issue, and in the third one may be applying a double standard: it is not evident that a rejection of hidden variables automatically gives you a local interpretation of quantum mechanics.
It is not my aim here to take sides by scrutinizing these objections and replies. The main point is rather that there are more stringent objections against hidden variables than those given by the Kochen-Specker theorem. However, when it comes
Taking stock
7.6
to the more formal questions of classical representations of quantum probability, the Kochen-Specker theorem already provides enough restrictions to be pessimistic. The general aspect of all ways to circumvent the theorem, is that the formalism of quantum probability is no longer fundamental. Now while this is of course a logical possibility, its denial is a logical possibility as well. In that case, letting go of classical probability, and instead focusing directly on quantum probability, is the road towards a physical understanding of quantum probability. This then is the project in the next part.
Part III
Quantum logic and quantum
probability
8
Orthodox quantum logic
8.1
Introduction
The central theme of this dissertation is the investigation of quantum probability in order to obtain a better understanding of its mathematical formalism. Thus far, the investigation focused on a comparison with classical probability. In chapter 4 a first purely mathematical comparison of the two formalisms was made. In part II the comparison focused on the (im)possibility of embedding quantum probability within the classical framework. Both these investigations led to useful insights. In chapter 4 the view emerged that classical probability can be viewed as a special case of quantum probability. In part II, on the other hand, the opposite view emerged. For any classical representation of quantum probability it was found that the quantum probability functions form a strict subset of the classical probability functions.1
These two opposing views are complementary rather than contradictory: they highlight different aspects of quantum probability. On the first view classical prob- ability identifies with the commutative side of quantum probability. The richer structure of non-commutative probability is an essential ingredient of the advan- tages of quantum computation over classical computation (Nielsen and Chuang, 2010). On the second view the quantum probability functions can be understood as incomplete specifications of the actual (hidden variable) state. The total set of probability functions then also contains complete specifications of the states.
In this part of the dissertation I undertake a more direct investigation of quan- tum probability. The aim is to obtain a reformulation of the formalism in which the mathematical symbols have a clear physical interpretation. In the remainder of this section I elaborate on what this aim is, but two remarks are already in order. First, the aim presupposes that in the current formulation of quantum probability a clear physical interpretation is absent. This is a claim I defend in this chapter. Second, in the investigation I adopt an empiricist stance. The aim is to obtain a
1
formulation of quantum probability that connects smoothly with the way quantum probability is used to formulate empirical statements. Little or no assumptions are made concerning the correct metaphysics for quantum probability, and in particular no attempt is made to provide a solution to the measurement problem.
The program in this part of the dissertation is somewhat unorthodox, and it is therefore useful to separate it from a program that looks much like it. The summary above sketches a picture of quantum probability spaces lying in between two classical probability spaces. A commutative subspace of the quantum probability space on the inside, and a hidden variable space enveloping the quantum space on the outside. A similar picture is well-known in the foundations of quantum mechanics. Although quantum mechanics violates Bell inequalities, and thus contains non- local correlations, these correlations are found to be non-maximal: they satisfy the so-called Tsirelson bound (Cirel’son, 1980), which may be violated by non-local hidden variable theories. Quantum mechanics thus occupies a special place within the space of all possible theories. The quest to understand this particular space was triggered by Wheeler’s famous question: “How come the quantum?” (1990). In this program, the aim is to reconstruct the formalism of quantum mechanics from physical principles.
In this dissertation I aim for something more modest and specialized. Part of the investigation of quantum probability is to try to reconstruct its formalism. But in this reconstruction I do not shy from picking (non-probabilistic) ingredients from quantum mechanics. Thus while in the reconstruction of quantum mechanics one may seek an underpinning for the use of Hilbert space theory, I take this mathe- matical structure for granted. Then, given this structure, the aim is to provide a conceptual reading/reformulation of this structure on which one can introduce a natural formalism of probability.
To understand the strategy, it is useful to make a final comparison with classical probability. Consider again the definitions of classical and quantum probability spaces alongside each other.
Definition 8.1.
A classical probability space is a triplet (Ω,F,P) such that
A quantum probability space is a triplet (H, L(H),P) such that
1. Ω is a set, 1. His a Hilbert space, 2. F is aσ-algebra of subsets of Ω con-
taining Ω,
2. L(H) is the collection of all projec- tion operators onH,
3. P is a function from F to the inter- val [0,1],
3. Pis a function fromL(H) to the in- terval [0,1],
4. P(Ω) = 1, 4. P(1) = 1,
5. P(S
n∆n) = PnP(∆n) for any
countable sequence of mutually dis- joint sets.
5. P(P
nPn) = PnP(Pn) for any
countable sequence of mutually or- thogonal projections.
Introduction
8.1
The most striking difference is in the first two points, which mark the replace- ment of the familiar set theory with the Hilbert space formalism. This suggests that once one can determine why it makes sense to adopt a Hilbert space struc- ture for the appropriate domain of probability functions,2 one is already close to an understanding of the whole of quantum probability. That is, given a conceptual underpinning of 1 and 2, it is reasonable to assume that 3, 4 and 5 can be motivated by appealing to an analogy with classical probability. Thus explaining the role of L(H) really can be considered the central task.
To get a feeling for what one may be looking for in an explanation of 1 and 2, it is useful to make a comparison with the classical case. Here, the adoption of 1 and 2 is so ‘trivial’ that it becomes hard to find a direct motivation for it in modern literature. The use of set theory is deeply entrenched in our mathematical thinking, and so one has to search for answers to the question of why a set-theoretical structure is the appropriate domain of probability functions. In most of the math textbooks on probability, the sets in aσ-algebra are identified with ‘events’, usually taken to be a primitive term. This is not surprising since, pragmatically, it is easy to work with this notion. For example, considering the roll of a die, the set of possible outcomes is {1,2,3,4,5,6}. The set {2,4,6} can then be identified with the event of rolling an even number. Thus it is plausible to model the probability of events with the values assigned to the corresponding sets by a function P.
When accepting the notion of ‘event’ as a primitive concept, a similar under- standing can be given for the use of projection operators. This link is given by the Born postulate. The event that the measurement of an observable A yields an outcome in the set ∆ is identified with the projection operator µA(∆) (with
A the operator associated with A). This primitive use of the notion of ‘event’ is quite common and, pragmatically, it is of course as natural as it is in the classical case. But more care is required when this notion starts playing a role in philo- sophical discussions of quantum probability. A good example of such use is in the information-theoretical interpretation of Bub and Pitowsky:
On this interpretation, the structure of Hilbert space, i.e., the non- Boolean algebra of Hilbert space subspaces, defines the structure of a quantum event space, just as, classically, a Boolean algebra, the subsets of a set (phase space), defines the structure of a classical event space. Gleason’s theorem then determines all possible probability measures on this structure as given by quantum states (pure and mixed) according to the trace rule, where the probabilities are interpreted as degrees of
2
To contrast with the program of reconstructing quantum mechanics: It is not required here that the use of Hilbert space theory comes out as the necessarily correct formalism. It suffices if it can be shown to be a possibly correct formalism.
belief or measures of uncertainty about events in the Bayesian sense. (Bub, 2007, p. 239)
Here, the quantum event structure is taken for granted and the interpretation of quantum probability is a story about this formalism, rather than something that provides an explanation of the formalism. This is problematic because it is not clear whether the notion of ‘event’ can be unpacked in such a way that it is compatible with the given interpretation. In fact, there is even cause for serious doubt, as there are good reasons to believe that events should be modeled by sets rather than projection operators. An interpretation of quantum probability should thus come equipped with tools to counter these reasons.
Note that the problem is not specific to the adoption of a Bayesian interpreta- tion. The same worry also applies for example to the consistent histories approach of Griffiths (2013), in which the quantum event structure plays a similar primitive role, but where probabilities are understood as objective chances. But for the sake of definiteness it is useful to unpack this problem further with an eye on Bayesian interpretations. According to the Bayesian, probabilities are degrees of belief of a rational agent. The usual elaboration goes as follows. The formula P(A) = x is taken to express the idea that a rational agent believes with degree x that the proposition (represented by) A is true. It is relatively straightforward to relate events to propositions. The probability of the event that the roll of a die yields an even number can be understood as the degree of belief that the proposition “the roll of the die yields an even number” is true. Thus any motivation for a particu- lar mathematical modeling of propositions also provides a motivation for using the same model for events.
In philosophy and classical logic, propositions are regularly associated with sets. This idea at least dates back to the works of Boole and Venn in the 19th cen- tury. The standard way to make the identification with sets is to construct the Lindenbaum-Tarski algebra from some formal languageL. IfLobeys classical logic this algebra is Boolean. Stone’s representation theorem then shows that this algebra can be identified with an algebra of sets. In short, if one accepts that ultimately probabilities are the kind of things to be assigned to propositions, and that classical logic is correct, then one ends up with a Boolean algebra of sets as the domain of probability functions.3 Admittedly, this does not immediately give one the classical probability framework. The algebras obtained need not be a σ-algebras, and the sets come equipped with a special topology (i.e., the Stone topology). However, these are just details in comparison to the mismatch with the quantum framework.
3
It seems then that Popper (1959, p. 320) was somewhat reckless when criticizing Kolmogorov’s framework as being “less ‘formal”’ than his own because “He [Kolmogorov] interprets the arguments of the probability functor assets” (emphasis in original) while priding himself for being completely silent about what kind of mathematical objects are in the domain of his own probability functions.
Introduction
8.1
One therefore cannot just replace the classical event structure with a non- Boolean structure and keep the classical interpretations as if nothing has changed. This of course does not imply thatL(H) cannot play the role of an event-set. But it makes any approach that takes this role as a separate fundamental assumption sus- pect. The aim then of this part of this dissertation is to gain a better understanding of L(H). In other words, the focus is on quantum logic.4
I start in this chapter and the next one with a direct approach. That is, I take the setL(H) as given and investigate the possibility to view this set as an algebraic structure of propositions. The obvious starting point is a discussion of the orthodox quantum logic of Birkhoff and von Neumann (1936), which is given in section 8.2. For reasons discussed in section 8.3, this logic is found to be unsatisfactory for providing a direct conceptual foundation for quantum probability. Consequently,