Quantum probabilities as degrees of belief

I outline an argument for a subjective Bayesian interpretation of quantum probabilities as degrees of belief distributed subject to consistency constraints on a quantum rather than a classical event space. I show that the projection postulate of quantum mechanics can be understood as a noncommutative generalization of the classical Bayesian rule for updating an initial probability distribution on new information, and I contrast the Bayesian interpretation of quantum probabilities sketched here with an alternative approach defended by Chris Fuchs.     

Subjective probability and quantum certainty

In the Bayesian approach to quantum mechanics, probabilities—and thus quantum states—represent an agent's degrees of belief, rather than corresponding to objective properties of physical systems. In this paper we investigate the concept of certainty in quantum mechanics. Particularly, we show how the probability-1 predictions derived from pure quantum states highlight a fundamental difference between our Bayesian approach, on the one hand, and Copenhagen and similar interpretations on the other. We first review the main arguments for the general claim that probabilities always represent degrees of belief. We then argue that a quantum state prepared by some physical device always depends on an agent's prior beliefs, implying that the probability-1 predictions derived from that state also depend on the agent's prior beliefs. Quantum certainty is therefore always some agent's certainty. Conversely, if facts about an experimental setup could imply agent-independent certainty for a measurement outcome, as in many Copenhagen-like interpretations, that outcome would effectively correspond to a preexisting system property. The idea that measurement outcomes occurring with certainty correspond to preexisting system properties is, however, in conflict with locality. We emphasize this by giving a version of an argument of Stairs [(1983). Quantum logic, realism, and value-definiteness. Philosophy of Science, 50, 578], which applies the Kochen–Specker theorem to an entangled bipartite system.     

Is there a stability problem for Bayesian noncommutative probabilities?

A stability condition for Bayesian statistical inference, which Redei [(1992). When can non-commutative statistical inference be Bayesian? International Studies in the Philosophy of Science, 6, 129–132; (1998). Quantum logic in algebraic approach. Dordrecht: Kluwer Academic Publishers] formulated as a rationality constraint holding in classical probability theory, is shown to fail in quantum mechanics. That allegedly challenges a Bayesian interpretation of quantum probabilities. In this paper we demonstrate that Redei's argument does not apply to quantum mechanics. Moreover, we provide a solution to the problem of Bayesian noncommutative statistical inference arising from the violation of stability condition in general probability spaces.     

Uncomfortable bedfellows: Objective quantum Bayesianism and the von Neumann–Lüders projection postulate

In this paper I critically evaluate the justification of the von Neumann–Lüders projection postulate for state changes in projective measurement contexts from the objective quantum Bayesian perspective. I point out that the justification provided so far for the von Neumann–Lüders projection postulate is insufficient. I argue that the best way to correct this problem is to make an assumption, Benign Realism, which is contradictory to the objective quantum Bayesian interpretation of quantum states.     

A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one

Carlton Caves, Fuchs, and Schack (2002) have recently appealed to an argument of mine (Stairs, 1983) to address a problem for their subjective Bayesian account of quantum probability. The difficulty is that on the face of it, quantum mechanical probabilities of one appear to be objective, but in that case, the Born Rule would yield a continuum of probabilities between zero and one. If so, we end up with objective probabilities strictly between zero and one. The authors claim that objective probabilities of one leads to a dilemma: give up locality or fall into contradiction. I argue that this conclusion depends on an overly strong interpretation of objectivism about quantum probabilities.     

Multiplicity in Everett׳s interpretation of quantum mechanics

Everett׳s interpretation of quantum mechanics was proposed to avoid problems inherent in the prevailing interpretational frame. It assumes that quantum mechanics can be applied to any system and that the state vector always evolves unitarily. It then claims that whenever an observable is measured, all possible results of the measurement exist. This notion of multiplicity has been understood in different ways by proponents of Everett׳s theory. In fact the spectrum of opinions on various ontological questions raised by Everett׳s approach is rather large, as we attempt to document in this critical review. We conclude that much remains to be done to clarify and specify Everett׳s approach.     

An argument for ψ-ontology in terms of protective measurements

The ontological model framework provides a rigorous approach to address the question of whether the quantum state is ontic or epistemic. When considering only conventional projective measurements, auxiliary assumptions are always needed to prove the reality of the quantum state in the framework. For example, the Pusey–Barrett–Rudolph theorem is based on an additional preparation independence assumption. In this paper, we give a new proof of ψ-ontology in terms of protective measurements in the ontological model framework. The proof does not rely on auxiliary assumptions, and it also applies to deterministic theories such as the de Broglie–Bohm theory. In addition, we give a simpler argument for ψ-ontology beyond the framework, which is based on protective measurements and a weaker criterion of reality. The argument may be also appealing for those people who favor an anti-realist view of quantum mechanics.     

Typical worlds

Hugh Everett III presented pure wave mechanics, sometimes referred to as the many-worlds interpretation, as a solution to the quantum measurement problem. While pure wave mechanics is an objectively deterministic physical theory with no probabilities, Everett sought to show how the theory might be understood as making the standard quantum statistical predictions as appearances to observers who were themselves described by the theory. We will consider his argument and how it depends on a particular notion of branch typicality. We will also consider responses to Everett and the relationship between typicality and probability. The suggestion will be that pure wave mechanics requires a number of significant auxiliary assumptions in order to make anything like the standard quantum predictions.     

Decision theory and information propagation in quantum physics

In recent papers, Zurek [(2005). Probabilities from entanglement, Born's rule p_k=|ψ_k|² from entanglement. Physical Review A, 71, 052105] has objected to the decision-theoretic approach of Deutsch [(1999) Quantum theory of probability and decisions. Proceedings of the Royal Society of London A, 455, 3129–3137] and Wallace [(2003). Everettian rationality: defending Deutsch's approach to probability in the Everett interpretation. Studies in History and Philosophy of Modern Physics, 34, 415–438] to deriving the Born rule for quantum probabilities on the grounds that it courts circularity. Deutsch and Wallace assume that the many worlds theory is true and that decoherence gives rise to a preferred basis. However, decoherence arguments use the reduced density matrix, which relies upon the partial trace and hence upon the Born rule for its validity. Using the Heisenberg picture and quantum Darwinism—the notion that classical information is quantum information that can proliferate in the environment pioneered in Ollivier et al. [(2004). Objective properties from subjective quantum states: Environment as a witness. Physical Review Letters, 93, 220401 and (2005). Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe. Physical Review A, 72, 042113]—I show that measurement interactions between two systems only create correlations between a specific set of commuting observables of system 1 and a specific set of commuting observables of system 2. This argument picks out a unique basis in which information flows in the correlations between those sets of commuting observables. I then derive the Born rule for both pure and mixed states and answer some other criticisms of the decision theoretic approach to quantum probability.     

Quantum decoherence and the approach to equilibrium (II)

In a previous paper [Hemmo, M & Shenker, O (2003). Quantum decoherence and the approach to equilibrium I. Philosophy of Science, 70, 330–358] we discussed a recent proposal by Albert [(2000). Time and chance. Cambridge, MA: Harvard University Press. Chapter 7] to recover thermodynamics on a purely dynamical basis, using the quantum theory of the collapse of the quantum state of [Ghirardi, G, Rimini, A and Weber, T., (1986). Unified dynamics for microscopic and macroscopic systems. Physical Review, D 34, 470–479]. We proposed an alternative way to explain thermodynamics within no collapse interpretations of quantum mechanics. In this paper some difficulties faced by both approaches are discussed and solved: the spin echo experiments, and the problem of extremely light gases. In these contexts, we point out several ways in which the above quantum mechanical approaches as well as some other classical approaches to the foundations of statistical mechanics may be distinguished experimentally.     

Situated observation in Bohmian mechanics

Here we investigate what it might mean for a formulation of quantum mechanics to be empirically adequate. We begin by considering the measurement problem as an empirical problem and distinguishing between stronger and weaker varieties of empirical adequacy. A strongly adequate theory is one that explains the experiences of a physically situated observer. A formulation of quantum mechanics that provides such situated empirical adequacy also provides a particularly compelling response to the measurement problem. As a concrete example we consider how Bohmian mechanics explains the experience of a physically situated observer.     

The operational framework for quantum theories is both epistemologically and ontologically neutral

Operational frameworks are very useful to study the foundations of quantum mechanics, and are sometimes used to promote antirealist attitudes towards the theory. The aim of this paper is to review three arguments aiming at defending an antirealist reading of quantum physics based on various developments of standard quantum mechanics appealing to notions such as quantum information, non-causal correlations and indefinite causal orders. Those arguments will be discussed in order to show that they are not convincing. Instead, it is argued that there is conceptually no argument that could favour realist or antirealist attitudes towards quantum mechanics based solely on some features of some formalism. In particular, both realist and antirealist views are well accomodable within operational formulations of the theory. The reason for this is that the realist/antirealist debate is located at a purely epistemic level, which is not engaged by formal aspects of theories. As such, operational formulations of quantum mechanics are epistmologically and ontologically neutral. This discussion aims at clarifying the limits of the historical and methodological affinities between scientific antirealism and operational physics while engaging with recent discoveries in quantum foundations. It also aims at presenting various realist strategies to account for those developments.     

Experimental : The double standard in the quantum-information approach to the foundations of quantum theory

Among the alternatives of non-relativistic quantum mechanics (NRQM) there are those that give different predictions than quantum mechanics in yet-untested circumstances, while remaining compatible with current empirical findings. In order to test these predictions, one must isolate one's system from environmental induced decoherence, which, on the standard view of NRQM, is the dynamical mechanism that is responsible for the ‘apparent’ collapse in open quantum systems. But while recent advances in condensed-matter physics may lead in the near future to experimental setups that will allow one to test the two hypotheses, namely genuine collapse vs. decoherence, hence make progress toward a solution to the quantum measurement problem, those philosophers and physicists who are advocating an information-theoretic approach to the foundations of quantum mechanics are still unwilling to acknowledge the empirical character of the issue at stake. Here I argue that in doing so they are displaying an unwarranted double standard.     

Quantum reality: A pragmaticized neo-Kantian approach

Despite remarkable efforts, it remains notoriously difficult to equip quantum theory with a coherent ontology. Hence, Healey (2017, 12) has recently suggested that “quantum theory has no physical ontology and states no facts about physical objects or events”, and Fuchs et al. (2014, 752) similarly hold that “quantum mechanics itself does not deal directly with the objective world”. While intriguing, these positions either raise the question of how talk of ‘physical reality’ can even remain meaningful, or they must ultimately embrace a hidden variables-view, in tension with their original project. I here offer a neo-Kantian alternative. In particular, I will show how constitutive elements in the sense of Reichenbach (1920) and Friedman (1999, 2001) can be identified within quantum theory, through considerations of symmetries that allow the constitution of a ‘quantum reality’, without invoking any notion of a radically mind-independent reality. The resulting conception will inherit elements from pragmatist and ‘QBist’ approaches, but also differ from them in crucial respects. Furthermore, going beyond the Friedmanian program, I will show how non-fundamental and approximate symmetries can be relevant for identifying constitutive principles.     

Holism, physical theories and quantum mechanics

Motivated by the question what it is that makes quantum mechanics a holistic theory (if so), I try to define for general physical theories what we mean by `holism'. For this purpose I propose an epistemological criterion to decide whether or not a physical theory is holistic, namely: a physical theory is holistic if and only if it is impossible in principle to infer the global properties, as assigned in the theory, by local resources available to an agent. I propose that these resources include at least all local operations and classical communication. This approach is contrasted with the well-known approaches to holism in terms of supervenience. The criterion for holism proposed here involves a shift in emphasis from ontology to epistemology. I apply this epistemological criterion to classical physics and Bohmian mechanics as represented on a phase and configuration space respectively, and for quantum mechanics (in the orthodox interpretation) using the formalism of general quantum operations as completely positive trace non-increasing maps. Furthermore, I provide an interesting example from which one can conclude that quantum mechanics is holistic in the above mentioned sense, although, perhaps surprisingly, no entanglement is needed.     

Uncertainty and probability for branching selves

Everettian accounts of quantum mechanics entail that people branch; every possible result of a measurement actually occurs, and I have one successor for each result. Is there room for probability in such an account? The prima facie answer is no; there are no ontic chances here, and no ignorance about what will happen. But since any adequate quantum mechanical theory must make probabilistic predictions, much recent philosophical labor has gone into trying to construct an account of probability for branching selves. One popular strategy involves arguing that branching selves introduce a new kind of subjective uncertainty. I argue here that the variants of this strategy in the literature all fail, either because the uncertainty is spurious, or because it is in the wrong place to yield probabilistic predictions. I conclude that uncertainty cannot be the ground for probability in Everettian quantum mechanics.