A remark on Fuchs’ Bayesian interpretation of quantum mechanics

Quantum mechanics is a theory whose foundations spark controversy to this day. Although many attempts to explain the underpinnings of the theory have been made, none has been unanimously accepted as satisfactory. Fuchs has recently claimed that the foundational issues can be resolved by interpreting quantum mechanics in the light of quantum information. The view proposed is that quantum mechanics should be interpreted along the lines of the subjective Bayesian approach to probability theory. The quantum state is not the physical state of a microscopic object. It is an epistemic state of an observer; it represents subjective degrees of belief about outcomes of measurements. The interpretation gives an elegant solution to the infamous measurement problem: measurement is nothing but Bayesian belief updating in a analogy to belief updating in a classical setting. In this paper, we analyze an argument that Fuchs gives in support of this latter claim. We suggest that the argument is not convincing since it rests on an ad hoc construction. We close with some remarks on the options left for Fuchs’ quantum Bayesian project.     

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.     

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.     

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.     

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.     

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.     

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.     

Non-contextuality,finite precision measurement and the Kochen–Specker theorem

Meyer originally raised the question of whether non-contextual hidden variable models can, despite the Kochen–Specker theorem, simulate the predictions of quantum mechanics to within any fixed finite experimental precision (Phys. Rev. Lett. 83 (1999) 3751). Meyer's result was extended by Kent (Phys. Rev. Lett. 83 (1999) 3755). Clifton and Kent later presented constructions of non-contextual hidden variable theories which, they argued, indeed simulate quantum mechanics in this way (Proc. Roy. Soc. Lond. A 456 (2000) 2101).These arguments have evoked some controversy. Among other things, it has been suggested that the Clifton–Kent models do not in fact reproduce correctly the predictions of quantum mechanics, even when finite precision is taken into account. It has also been suggested that careful analysis of the notion of contextuality in the context of finite precision measurement motivates definitions which imply that the Clifton–Kent models are in fact contextual. Several critics have also argued that the issue can be definitively resolved by experimental tests of the Kochen–Specker theorem or experimental demonstrations of the contextuality of Nature.One aim of this paper is to respond to and rebut criticisms of the Meyer–Clifton–Kent papers. We thus elaborate in a little more detail how the Clifton–Kent models can reproduce the predictions of quantum mechanics to arbitrary precision. We analyse in more detail the relationship between classicality, finite precision measurement and contextuality, and defend the claims that the Clifton–Kent models are both essentially classical and non-contextual. We also examine in more detail the senses in which a theory can be said to be contextual or non-contextual, and in which an experiment can be said to provide evidence on the point. In particular, we criticise the suggestion that a decisive experimental verification of contextuality is possible, arguing that the idea rests on a conceptual confusion.     

An axiomatic formulation of the Montevideo interpretation of quantum mechanics

We make a first attempt to axiomatically formulate the Montevideo interpretation of quantum mechanics. In this interpretation environmental decoherence is supplemented with loss of coherence due to the use of realistic clocks to measure time to solve the measurement problem. The resulting formulation is framed entirely in terms of quantum objects. Unlike in ordinary quantum mechanics, classical time only plays the role of an unobservable parameter. The formulation eliminates any privileged role of the measurement process giving an objective definition of when an event occurs in a system.     

Intrinsically mixed states: an appreciation

An “intrinsically mixed” state is a mixed state of a system that is (in a sense to be elaborated) ‘orthogonal’ to every pure state of that system. Although the presence of such states in the quantum theories of infinite systems is well known to those who work with such theories, intrinsically mixed states are virtually unheralded in the philosophical literature. Rob Clifton was thoroughly familiar with intrinsically mixed states. I aim here to introduce them to a wider audience—and to encourage that audience to cultivate their acquaintance by suggesting that intrinsically mixed states undermine assumptions framing standard discussions of the quantum measurement problem.     

On the verge of <Emphasis Type="Italic">Umdeutung</Emphasis> in Minnesota: Van Vleck and the correspondence principle. Part two

This is the second installment of a two-part paper on developments in quantum dispersion theory leading up to Heisenberg’s Umdeutung paper. In telling this story, we have taken a 1924 paper by John H. Van Vleck in The Physical Review as our main guide. In this second part we present the detailed derivations on which our narrative in the first part rests. The central result that we derive is the Kramers dispersion formula, which played a key role in the thinking that led to Heisenberg’s Umdeutung paper. We derive classical formulae for the dispersion, emission, and absorption of radiation and use Bohr’s correspondence principle to construct their quantum counterparts both for the special case of a charged harmonic oscillator (Sect. 5) and for arbitrary non-degenerate multiply-periodic systems (Sect. 6). We then rederive these results in modern quantum mechanics (Sect. 7). This paper was written as part of a joint project in the history of quantum physics of the Max Planck Institut für Wissenschaftsgeschichte and the Fritz-Haber-Institut in Berlin. The authors gratefully acknowledge support from the Max Planck Institute for History of Science. The research of Anthony Duncan is supported in part by the National Science Foundation under grant PHY-0554660.     

Explaining Leibniz equivalence as difference of non-inertial appearances: Dis-solution of the Hole Argument and physical individuation of point-events

“The last remnant of physical objectivity of space–time” is disclosed in the case of a continuous family of spatially non-compact models of general relativity (GR). The physical individuation of point-events is furnished by the autonomous degrees of freedom of the gravitational field (viz., the Dirac observables) which represent—as it were—the ontic part of the metric field. The physical role of the epistemic part (viz. the gauge variables) is likewise clarified as embodying the unavoidable non-inertial aspects of GR. At the end the philosophical import of the Hole Argument is substantially weakened and in fact the Argument itself dissolved, while a specific four-dimensional holistic and structuralist view of space–time (called point-structuralism) emerges, including elements common to the tradition of both substantivalism and relationism. The observables of our models undergo real temporal change: this gives new evidence to the fact that statements like the frozen-time character of evolution, as other ontological claims about GR, are model dependent.     

Statistical estimation of optimal portfolios for non‐Gaussian dependent returns of assets

This paper discusses the asymptotic efficiency of estimators for optimal portfolios when returns are vector‐valued non‐Gaussian stationary processes. We give the asymptotic distribution of portfolio estimators ? for non‐Gaussian dependent return processes. Next we address the problem of asymptotic efficiency for the class of estimators ?. First, it is shown that there are some cases when the asymptotic variance of ? under non‐Gaussianity can be smaller than that under Gaussianity. The result shows that non‐Gaussianity of the returns does not always affect the efficiency badly. Second, we give a necessary and sufficient condition for ? to be asymptotically efficient when the return process is Gaussian, which shows that ? is not asymptotically efficient generally. From this point of view we propose to use maximum likelihood type estimators for g, which are asymptotically efficient. Furthermore, we investigate the problem of predicting the one‐step‐ahead optimal portfolio return by the estimated portfolio based on ? and examine the mean squares prediction error. Copyright © 2008 John Wiley & Sons, Ltd.     

Observability of States in Non-Linear Systems

In this paper we define two new properties—linear observability and L-observability—to aid the study of overspecification and unidentifiability in quasi-linear and non-linear state space time series. Various equivalence results are proved and illustrated and some general properties of observable and unobservable processes are derived. The results are often proved by using graphical methods (influence diagrams) for manipulating conditional independence statements embedded in the new definitions. © 1997 John Wiley & Sons, Ltd.     

The puzzle of half-integral quanta in the application of the adiabatic hypothesis to rotational motion

We present and discuss an interesting and puzzling problem Ehrenfest found in his first application of the adiabatic hypothesis, in 1913. It arose when trying to extend Planck׳s quantization of the energy of harmonic oscillators to a rotating dipole within the frame of the old quantum theory. Such an extension seemed to lead unavoidably to half-integral values for the rotational angular momentum of a system (in units of ℏ). We present the problem in its original form along with the (few) responses we have found to Ehrenfest׳s treatment. After giving a brief account of the classical and quantum adiabatic theorem, we also describe how Quantum Mechanics provides an explanation for this difficulty.     

On information-theoretic characterizations of physical theories

Clifton et al. (Found. Phys. 33 (2003) 1561) have recently argued that quantum theory is characterized by its satisfaction of three information-theoretic axioms. However, it is not difficult to construct apparent counterexamples to the CBH characterization theorem. In this paper, we discuss the limits of the characterization theorem, and we provide some technical tools for checking whether a theory (specified in terms of the convex structure of its state space) falls within these limits.     

The origin of the Everettian heresy

In 1956, Hugh Everett, then a PhD student at Princeton, proposed his “relative state” formulation of quantum mechanics. John Wheeler, who was Everett's advisor, recognized the originality and importance of such a proposal, but he denied that its non-conventional approach to measurement questioned the orthodox view. Indeed, Wheeler made serious efforts to obtain the blessing of Niels Bohr for Everett's ideas. These efforts gave rise to a lively debate with the Copenhagen group, the existence and content of which have been only recently disclosed by the discovery of unpublished documents. The analysis of such documents opens a window on the conceptual background of Everett's proposal, and illuminates at the same time some crucial aspects of the Copenhagen view of the measurement problem. Also, it provides an original insight into the interplay between philosophical and social factors which underlay the postwar controversies on the interpretation of quantum mechanics.