Quantum Foundations

The upheaval in quantum theory in the mid-1920s is often presented as a Kuhnian paradigm shift. The new quantum mechanics, to use a building metaphor, was erected on the ruins of the old quantum theory, brought down by an accumulation of anomalies. In our book Constructing Quantum Mechanics (2019/2023), Tony Duncan and I use a different building metaphor. As suggested by the subtitles of the two volumes of our book, The Scaffold: 1900–1923 and The Arch: 1923–1927, we see the architects of the new quantum mechanics using parts of the old quantum theory as scaffolds to build the arch of the new one. In this talk, after sketching the underlying alternative to Kuhn's account of scientific revolutions in general, I will give an overview, as non-technical as possible, of the genesis of quantum mechanics in the period 1900–1927.

I claim that the idea of "reconstructing" quantum mechanics along the lines of quantum probability was there from the beginning of the theory as we know it, in von Neumann's paper "Probabilistic construction of quantum mechanics". In this paper, von Neumann derives (and often introduces for the first time) a slew of now-familiar concepts in quantum mechanics, such as both pure and mixed states (respectively as wavefunctions and—generally unnormalized—density operators), the general form of the Born rule for expectation values, a rather general form of transition probabilities, and a recognisable version of the collapse postulate for the general case of maximal measurements and in a special case of non-maximal measurements. He does so axiomatically, starting from two probabilistic axioms that are neutral between classical and quantum probability and two axioms characterising quantum mechanical quantities in terms of a non-commutative algebra (that of self-adjoint operators in Hilbert space). The presentation of this material in von Neumann's 1932 book is usually understood as a "no-hidden-variables" theorem. However, the sense in which a "completion" of the theory is ruled out is merely that a theory with dispersion-free states would require a different algebraic structure. Grete Hermann pointed this out already in the 1930s (she later retracted the additional claim of "circularity"), and it is arguably also how von Neumann himself saw his result. The more fundamental disagreement between von Neumann and Hermann (and arguably between von Neumann and Bohr) turns out to be about the nature of the quantum state and the status of collapse.

We show that classical mechanics can be recovered as the high-entropy limit of quantum mechanics. The mathematical limit $\hbar \to 0$ can be recovered by decreasing entropy of pure states to minus infinity, in the same way that non-relativistic mechanics can be recovered mathematically by increasing the speed of light c to plus infinty. Physically, these limits are more appropriately understood as a high entropy limit and low speed limit respectively, representing approximations that are independent of underlying mechanism. With this approach, the classical limit is both formally and conceptually similar to the non-relativistic limit, and is independent of interpretation. It also gives an intuitive understanding to the Dirac correspondence principle: it is looking for a theory with lower entropy bound that, at high entropy, recovers classical mechanics. Given that the Moyal bracket is the unique one-parameter Lie-algebraic deformation of the Poisson bracket, quantum mechanics is the only theory that can provide such a lower bound on the entropy.

The Heisenberg microscope provides a powerful mental image of the measurement process of quantum mechanics (QM), attempting to explain the uncertainty relation through an uncontrollable back-action from the measurement device. However, Heisenberg's proposed back-action uses features that are not present in the QM description of the world, and according to Bohr not present in the world. Therefore, Bohr argues, the mental image proposed by Heisenberg should be avoided. Later developments by Bell and Kochen-Specker shows that a model that contains the features used for the Heisenberg microscope is in principle possible but must necessarily be nonlocal and contextual. In this paper we will re-examine the measurement process within a restriction of QM known as Stabilizer QM, that still exhibits for example Greenberger-Horne-Zeilinger nonlocality and Peres-Mermin contextuality. The re-examination will use a recent extension of stabilizer QM, the Contextual Ontological Model (COM), where the system state gives a complete description of future measurement outcomes reproducing the quantum predictions, including the mentioned phenomena. We will see that the resulting contextual Heisenberg microscope back-action can be completely described within COM, and that the associated randomness originates in the initial state of the pointer system, exactly as in the original description of the Heisenberg microscope. The presence of contextuality, usually seen as prohibiting ontological models, suggests that the contextual Heisenberg microscope picture could perhaps be enabled in general QM.

On March 27th, 1927, Heisenberg published the paper in which he introduced the uncertainty relations through his well-known $\gamma$-ray microscope thought experiment. Since then, the debates and commentaries over the origin of the uncertainty relations have continued for a century (see, for instance, Bacciagaluppi and Valentini 2009; Beller 1999; Brown and Redhead 1981; Busch 1990; Hilgevoord and Uffink 1985,2024; Jammer 1974; Uffink 1990). Going back to 1927 through a selection of letters -- including an unpublished letter from Jordan to Rosenfeld -- this talk aims to add a crucial piece to this century-old puzzle by shedding new light on the differences between Heisenberg and Bohr's conceptual basis of the uncertainty relations. According to the received view, Bohr derives the uncertainty relations from the \textit{wave-particle duality} of light. By contrast, it is commonly stated that Heisenberg based them on the \textit{discontinuity} in the interaction between the electron and the light quantum, i.e., the uncontrollable change resulting from the Compton effect (cf. Camilleri 2009). I will challenge the received view by arguing that: (i) Bohr’s derivation of the uncertainty relations fundamentally relies on the use of \textit{classical concepts}, consistent with their central role in his broader interpretation of quantum mechanics. (ii) The supposed contrast rooted in the wave-particle duality of light may stem from Heisenberg’s 1927 misrepresentation (or misinterpretation) of Bohr's ideas on the $\gamma$-ray microscope. This misrepresentation may have influenced the reception of Bohr’s interpretation, particularly regarding his ideas on the conceptual justification for the uncertainty relations, and contributed to obscuring the deeper significance of Bohr's complementarity principle - fully comprehensible only in light of (i).

We articulate a collection of desiderata for an account of the dynamical quantities of a physical theory, and we present a theory that meets these desiderata in the case of quantum mechanics. Our theory retains a distinction between the values of dynamical quantities and the truth values of sentences asserting that a system has a particular value of a particular quantity. This allows our theory to incorporate the phenomenon of quantum indeterminacy as a pattern in the properties instantiated by a system in a particular quantum state, while also retaining the semantics of classical logic. We contrast our theory with quantum logic, which flattens the distinction between quantity values and truth values. We discuss the historical origin of the assumptions leading to this feature of quantum logic, and we address a series of objections that have been raised to previous attempts to reconcile quantum indeterminacy with classical logic.

Since its inception, the quantum theory has been fraught with interpretive challenges. It has often been complained that the physical content of the theory is obscured by the mathematical resources used to articulate it. In recent decades, many have sought to reorganize the conceptual resources of quantum theory in terms of operational principles with clear physical meaning. These quantum reconstructions have been so successful that some people take them as evidence that the rightful subject matter of quantum theory—what the theory is about—is not the dynamical evolution of material systems at all, but rather the operational capacities of agents. Here, I argue against this agentive turn. Specifically, I show that such an account is not recommended by the success of quantum reconstruction and is, in fact, incompatible with it. I show how the operational language of quantum reconstruction principles can be given a clear, non-agentive interpretation. I further explain the practical and heuristic value of employing this operational language in facilitating the aims of the quantum reconstruction program.

We ask why finite-dimensional quantum states can be represented as mixtures of pure states on complex projective space, with reversible transitions given by unitary transformations up to phase. We derive this from two principles. First, determinism of probabilities: the evolution on the state space is smooth and reversible, and each state specifies observable probabilities together with auxiliary variables that drive the deterministic trajectory. Second, invariance: every reversible transformation preserves a physical quantity, so trajectories are tangent to level sets of a real function. From these assumptions we show that the state space is isomorphic to complex projective space and the reversible transformations form a subgroup of the projective unitary group. This yields a minimal reconstruction of quantum theory and a route to more transparent physical interpretations.

Wernher von Braun is supposed to have said, "Research is what I'm doing when I don't know what I'm doing." The path to a new idea is seldom straight, and travelers seldom know what their actual destination will be. I intend to illustrate all this by describing the origin of the word "qubit" and the concept of quantum data compression in the early days of quantum information theory.

In this talk, I trace one particular, personal thread in the history of quantum foundations and quantum information. It starts with a course taught by John Wheeler in the academic year 1977-1978 and ends with the 1992 discovery of the theoretical possibility of quantum teleportation. The story depends crucially on two insightful questions, posed by Asher Peres and Charles Bennett.

At the 1981 Physics of Computation Conference, held at MIT’s Endicott House, **John A. Wheeler** presented a paper entitled **“The Computer and the Universe”** where he put forward the idea of one day understanding ‘physics as information’ (Wheeler 1982). By the early 1980s, it was already well established that information is inseparably tied to physical degrees of freedom and therefore subject to physical law. Yet, several conference presentations advanced the counterpart idea: that physical laws themselves can be viewed as algorithms for information processing, and that their ultimate form must respect the limits of physically executable computation (see Landauer 1982; Fredkin 1982; Kantor 1982; Zuse 1982). After a distinguished career at Princeton, Wheeler joined the University of Texas at Austin in 1976, where he founded the Center for Theoretical Physics. His time there offers a perspective on how the informational turn, highlighted at the 1981 conference, influenced the trajectory of quantum mechanics research in the latter half of the twentieth century. During his time at Texas, Wheeler became intensely focused on the question, **“How come the quantum?”** He sought a deeper principle from which the quantum formalism could be derived. In this search, quantum theory became increasingly intertwined with computing and information theory. As director, he assembled a diverse group of graduate students, postdoctoral fellows, and visiting researchers, creating a dynamic research environment to help tackle this question. The group engaged deeply with emerging developments in computer science—reading widely on parallel computing, Turing machines, cellular automata, and cybernetics—and often held informal seminars where these topics converged with foundational issues in physics, particularly debates on the many-worlds interpretation (Everett, 1957). For Wheeler, the informational turn culminated in his proposal of **“it from bit,”** the idea that every physical entity—every “it”—derives its meaning from fundamental units of information, or “bits” (Wheeler, 1989). For many of the researchers who passed through Austin, this perspective translated into concrete efforts that laid the conceptual and technical groundwork for what would later be recognized as quantum information theory, quantum computation, and novel approaches to the foundations of quantum mechanics. This influence is particularly evident in the work of **David Deutsch** (quantum Turing machines), **William Wootters** (quantum distinguishability and teleportation), **Wojciech Zurek** (decoherence), and **Benjamin Schumacher** (quantum coding), with Wootters and Schumacher together introducing the concept of the qubit. The decade Wheeler spent at Austin provides a fresh perspective on the expanding role of computation and information theory in shaping modern physics in the late twentieth century. Computers were not merely pragmatic tools; they became conceptual models and heuristic devices that that helped steer the direction of research in quantum physics.