Speaker
Description
Abstract: Ontological models of quantum theory seek to realize quantum theory as a statistical theory over underlying degrees of freedom, whose measurement statistics are described by classical probability theory. Harrigan and Spekkens (2010) proposed a distinction between ontological models in which pure quantum states represent states of reality ($\psi$-ontic) or knowledge about reality ($\psi$-epistemic) depending on whether their associated probability distributions overlap, subsequently formalized into measure-theoretic form by Leifer (2014). In this talk, we examine whether this $\psi$-ontic/epistemic distinction truly captures the intended difference between ontic and epistemic conceptions of pure states, using measure-theoretic probability theory. We consider some mathematical alternatives relating to different notions of non-overlap between probability measures and compare them. We discuss the implications of these results for quantum ontology theorems -- theorems that rule out $\psi$-epistemic readings of pure states under certain assumptions about the ontological model -- such as the Pusey-Barrett-Rudolph Theorem (2012).
Extended Abstract:
Since the inception of quantum theory, there has been much debate over how to understand the theory. In particular, there is no uncontroversial agreement on how to understand the very notion of a quantum state. The idea of a state of a system is not so puzzling in classical physics, from which we inherit two notions. One is that of the state as physical (or ontic), where each state represents a distinct way that a system can be, like the states in classical mechanics. The other is of the state as statistical (or epistemic), where the state is compatible with different ways a system can be, with a probability assigned to each possibility, as the states in classical statistical mechanics. Pure states of quantum theory, however, do not seem to fit either mold straightforwardly. There is no doubt that quantum theory, understood as a framework under which one obtains correct predictions about outcomes of measurements and their probabilities, is empirically successful. But without an agreed-upon understanding of what quantum states represent, one lacks an explanation for this empirical success.
One observes formal similarities between probability theory and quantum theory; quantum theory can be viewed as a non-commutative generalization of classical probability theory. For want of an explanation, some have seeked to understand quantum states analogously to the way we understand classical probabilities. This motivates a statistical conception of quantum states that extends beyond an operational understanding: to take quantum states not just as a tool for calculating probabilities of measurement outcomes, but more fundamentally as representing probabilities of what the system is actually like.
Motivated by these ideas, the framework of ontological models of quantum theory seeks to describe theories that realize quantum theory as a statistical theory over some underlying degrees of freedom, over which the statistics of measurement outcomes are described by classical probability theory. Under this framework, quantum theory is taken to provide operational predictions for observations. Meanwhile, an ontological model posits a description of the same system using the aforementioned underlying degrees of freedom—the ontic state of the system—often taken to describe some underlying reality, which need not be directly empirically accessible to us. The familiar objects of quantum theory, such as the quantum state of the system, then correspond to some probability distribution over the ontic state space of the system. In order for a description of a physical system from an ontological model to be consistent with predictions of quantum theory, the probabilistic predictions of the ontological model should agree with those of quantum theory.
For pure states to be ontic, distinct pure states ought to correspond to different ways the system can be. Insofar as ontic states faithfully describe different ways the system can be, one expects each ontic state be compatible with at most one pure state in a given ontological model, In this case, the probabilities involved in the ontological model arise simply due to lack of knowledge of these underlying degrees of freedom, so quantum pure states need not be understood probabilistically relative to the given ontological model. Harrigan and Spekkens (2010) introduced the notion of an ontological model being $\psi$-ontic to capture this very idea. According to their definition, an ontological model is $\psi$-ontic if no pairs of probability distributions over ontic states corresponding to different pure states overlap, and is $\psi$-epistemic otherwise. This definition subsequently has been sharpened by Leifer (2014) into measure-theoretic form, which involves the vanishing total variation distance between a pair of probability measures each corresponding to distinct pure states.
We consider the question of whether the received $\psi$-ontic/epistemic distinction based on the total variation distance succeeds in capturing the original idea of each ontic state corresponding to at most one pure state, from the perspective of measure theory. Based on our analysis, we conclude that it does not. The key observation is that the standard definition of $\psi$-ontology only requires each pair of probability measures for distinct pure states to be non-overlapping, a notion captured by mutual singularity of probability measures. But it takes a notion of non-overlap for the entire collection of probability measures, called complete singularity, to capture the idea that each ontic state describes a system associated with at most one pure quantum state in an ontological model. As pairwise non-overlap is insufficient to establish this in general, we conclude that $\psi$-ontic models do not adequately capture the notion of reality of pure quantum states.
In light of this observation, we argue that one should be cautious about the purported conclusions of quantum ontology theorems such as the PBR Theorem. With the debate on the ontological status of pure states, previous analyses have focused on the assumptions these theorems impose on the ontological models, such as the Preparation Independence Postulate and its physical relevance. Our work instead questions the foundation of the conclusion of these theorems—whether the criterion of $\psi$-onticity is the correct mathematical characterization of ontological models where pure states are ontic to begin with. If the standard criterion for a $\psi$-ontic model based on the total variation distance is too weak to establish ontic pure quantum states, then these theorems have not established ontic pure states in the given ontological model, even taking for granted the assumptions they impose on the ontological model.