Theory-independent realism

Scientific knowledge is assumed to go beyond mere appearances and to convey reliable information about that which our senses cannot directly apprehend. In Physics, a large fraction of our scientific knowledge stands upon quantum theory, which remains coherent with the data acquired for more than a century. However, unlike classical mechanics, whose mathematical formulation aligns with our everyday experiences, quantum theory is challenging to make sense of. To abstract a consistent story from its formalism, or to interpret it, we are required to give up at least one of the metaphysical assumptions founding our classical intuition. One of such assumptions is realism.

Broadly speaking, realism tells us about the definiteness of the physical properties of a system at any instant of time independently of observers. Its seminal definition was given by Einstein, Podolsky, and Rosen (EPR) in 1935 [1], without which the first hint of entanglement appearing in the literature would not have been possible. There, it refers to the capability of predicting with certainty the value assigned to a physical property without disturbing the system, such that, whenever this is the case, it is said that there is an element of reality assigned to that physical property. Conceiving an experiment where measurements are performed on two parts of an entangled system, with these measurements being events located outside of each other’s light cones, EPR argue that incompatible observables should be simultaneous elements of reality. The authors then concluded that this would constitute a proof of the incompleteness of quantum mechanics. Put under scrutiny by Bell in 1964, this claim was discredited in Ref. [2], where it was shown that the predictions made by quantum mechanics conflicted with any theory of local hidden variables. Quantum systems that violate Bell’s inequalities, now understood to violate the hypothesis of local causality [3], have been experimentally verified through loophole-free tests [4, 5, 6, 7, 8, 9].

Even though it is possible to envision fully realist interpretations of quantum mechanics at the expense of other metaphysical requirements for the theory, such as in Bohmian mechanics [10], which sacrifices locality, the concept of realism underpins the debate around entanglement or Bell nonlocality. Later developments of the concept of realism, such as that proposed by Fine in Ref. [11], refer to joint probability distributions assigned to the measurements of different observables. The probabilistic profile of the joint measurability of non-commuting observables contrasts with this definition, highlighting a clash between incompatibility and realism. Furthermore, the measurement problem, or what makes a measurement a measurement [12], is an issue that invites us to answer how realism, displayed in classical phenomena, emerges.

Newer proposals for realism criteria, such as the one devised by Bilobran and Angelo (BA) in Ref. [13], make the concept more tangible. BA, in particular, do this by providing a quantifiable and operational criterion that was later employed in the development of a concept of nonlocality [14, 15, 16, 17] and axiomatized [18]. Further developments applied this notion to quantum resources theory [19] and foundational elements in quantum mechanics [20, 21, 22, 23, 24, 25, 26, 27]. However, every proposal in the physics literature for a realism criterion pertains to the context of quantum mechanics. A definition for realism which is theory-independent is lacking, and this work aims at filling this gap.

Besides deepening our understanding of realism itself, a theory-independent approach allows us to better understand not only quantum mechanics per se but also the relation between quantum mechanics and other physical theories. More so, it equips us with a framework suitable to scrutinize physical theories that are still to be developed, once quantum mechanics may not be the fundamental theory of Nature. Work [28] gives us a clear example of a theory-independent proposal shedding light on the concept of macrorealism [29].

A concise and meaningful criterion for realism, fully encoded in a mathematical syllogism, that is independent of any particular theory but still pertains to the realm of physics, is introduced in this work through generalized probabilistic theories (GPT). The GPT framework is constructed as a mathematical structure that describes generic states, transformations, and measurements. In this framework, any physical theory can be viewed as a particular GPT defined by a set of rules that assign probabilities to outcomes for the measurements of physical properties. Our criterion relies on the probabilities themselves, rather than on the rules that determine them. Notwithstanding, we go further and, inspired by the works of BA, we propose two ways for quantifying, in a non-realist theory, the extent to which a state deviates from our criterion of realism. These sought-after measures are obtained using the concepts of robustness and the Kullback-Leibler divergence [30], and they constitute two theory-independent quantifiers of irrealism.

The article is structured as follows. In Section II we review preliminary concepts, summarizing BA’s criterion of realism and skimming through some building blocks of the GPT framework, presenting only the minimal amount necessary for the understanding of the following sections. Our criterion is then laid out in Section III, and in Section IV, the theory-independent quantifiers of irrealism are presented along with some numerical analysis. They are tested in quantum mechanics and contrasted with BA’s notion of irrealism. Section V closes the paper with our conclusions.

In a theory where a given observable is real (has a definite value regardless of observation), the role of a measurement is merely to reveal the already established element of reality. If the information about the outcome of a performed measurement is lost or somehow inaccessible, then the state of knowledge remains the same as before the measurement. In other words, unrevealed measurements are innocuous in realist theories. This will be the fundamental premise behind the criterion we now propose.

To implement this principle, consider a generic state $\epsilon$ of a state space $\mathscr{K}$ with dimension $d$, and physical properties $\mathcal{X}=\{x_{i}\}_{i=1}^{d}$ and $\mathcal{Y}=\{y_{j}\}_{j=1}^{d}$ defined by the set of their possible outcomes. The criterion of realism requires that the probabilities assigned to any outcome $x_{i}$ upon the measurement of any property $\mathcal{X}$ on a state $\epsilon$ remains unchanged if an unrevealed measurement of $\mathcal{Y}$ was performed beforehand. Symbolically, if

then $\mathcal{Y}$ is termed an element of reality given the state $\epsilon$. In the right-hand term, sub-index $\Phi_{\mathcal{Y}}(\epsilon)$ stands for a post unrevealed measurement of $\mathcal{Y}$ over state $\epsilon$. The term is expressed via conditional probabilities as

Note that the knowing of $y_{j}$ may alter the preparation $\epsilon$. The notation for conditional probabilities denotes that $\epsilon$ was the original state before the knowledge yielded by a precise measurement of $\mathcal{Y}$. Criterion (4) gives the core of this work. It does not demand every physical state $\epsilon$ to satisfy (4). When that is the case, that physical theory is considered realist. If only a set of states $\epsilon$ meets (4), those states are said to be $\mathcal{Y}$-realist.

Before moving on, it is opportune to note that our realism criterion can be validated through the assumption that joint probability distributions $p(x_{i},y_{j})$ exist for all $\mathcal{X}$, which makes contact with Fine’s approach to determinism [11]. If $p(x_{i},y_{j})$ exists without any concern a priori with the order with which $\mathcal{X}$ and $\mathcal{Y}$ are probed, and if the definition of conditional probability holds, then one can write $p(x_{i},y_{j})=p_{\epsilon}(x_{i}|y_{j})\,p_{\epsilon}(y_{j})=p_{\epsilon}(y_{j}|x_{i})\,p_{\epsilon}(x_{i})$ (Bayes’ rule). Since $\sum_{j}p_{\epsilon}(y_{j}|x_{i})=1$, substituting Bayes’ rule into the definition (5) immediately retrieves the realism criterion (4).

Let $\mathscr{C}$ be the state space where generic $\epsilon$ reside and $\mathscr{C}_{\mathcal{Y}}$ the state space for $\mathcal{Y}$-realist states. While $\mathscr{C}$ is convex by construction, the convexity of $\mathscr{C}_{\mathcal{Y}}$ remains to be proved. Considering the one-parameter state $\epsilon^{\prime}_{\lambda}=(1-\lambda)\epsilon^{\prime}_{1}+\lambda\epsilon^{\prime}_{2}$, with $\lambda\in[0,1]$ and ${\epsilon^{\prime}_{1},\epsilon^{\prime}_{2}}\in\mathscr{C}_{\mathcal{Y}}$, the proof is complete if $\epsilon^{\prime}_{\lambda}\in\mathscr{C}_{\mathcal{Y}}$. Suppose a binary random variable $\Lambda$ with assigned probabilities $(1-\lambda)$ and $\lambda$. A rule may be given such that, depending on the outcome of $\Lambda$, $\mathcal{X}$ will be measured in either $\epsilon^{\prime}_{1}$ or $\epsilon^{\prime}_{2}$. The probabilistic profile for this scenario is represented by $p_{\epsilon^{\prime}_{\lambda}}(x_{i})=p_{(1-\lambda)\epsilon^{\prime}_{1}+\lambda\epsilon^{\prime}_{2}}(x_{i})$. Due to the $\mathcal{Y}$-realism status of $\epsilon^{\prime}_{1,2}$,

However, the right-hand side of this equation can also represent a preparation and mixing of two ensembles of $\epsilon^{\prime}_{1}$ and $\epsilon^{\prime}_{2}$ with relative populations $(1-\lambda)$ and $\lambda$, respectively, with the measurement of $\mathcal{Y}$ first and $\mathcal{X}$ second from a randomly selected state from the mixed ensemble, $\epsilon^{\prime}_{\lambda}$. That is,

This last equality, which mirrors the fact that $\Phi$ is an affine map, can be connected to the previous equations to complete the proof.

Classical mechanics gives the simplest case for realist theories. The state space $\mathscr{C}$ becomes a $2n$-dimensional phase space, where $n$ represents the dimensions of position and momentum. A state $\epsilon$ in this space is one point that characterizes a configuration of a physical system. The realism of the framework is implied by the space of $\mathcal{Y}$-realist states being equal to the total phase space, $\mathscr{C}=\mathscr{C}_{\mathcal{Y}}$ for every $\mathcal{Y}$. In classical statistical mechanics—that is, classical mechanics supplemented with subjective uncertainties and Liouville’s equation—the diagnosis is no different. For every generalized coordinate $q_{i}$ and canonical momentum $\pi_{j}$, a joint probability distribution $\mathcal{P}(q_{i},\pi_{j})\,dq_{i}d\pi_{j}$ can be found, which, as discussed above, validates the realism criterion (4).

In the mathematical framework of quantum mechanics (QM), the left and right-hand sides of Eq. (4) become:

with $\Phi_{Y}$ defined as in Eq. (2). A state $\epsilon$ is represented by a density operator $\rho$ and a physical property $\mathcal{X}$ by an observable $X=\sum_{i}x_{i}X_{i}$, being $X_{i}$ projectors corresponding to measurement outcomes $x_{i}$. In this context, criterion (4) reads $\operatorname{Tr}[X_{i}\rho]=\operatorname{Tr}[X_{i}\Phi_{Y}(\rho)]$ and, because of the linearity and cyclic properties of the trace, this equation can be written as:

A sufficient condition for this equation to hold is for $\rho$ to be a $Y$-realist state according to criterion (1). The fact that this condition is also necessary is proven in Appendix A.

If two generic states in a physical theory fall outside $\mathscr{C}_{\mathcal{Y}}$, thus violating criterion (4) for a physical property $\mathcal{Y}$, it should be possible to quantify and compare the degrees of violation of these states. Thus, we propose two different methodologies for quantifying irrealism, the complement of realism.

Despite a rich body of literature on violations of hypotheses such as locality, causality, or even macrorealism [29] or local-friendliness [33], realism is a notion that sparks debates over its very definition. Additionally, quantifiable notions of realism transcending the scope of quantum theory are still lacking. This work fills this gap by bringing together different criteria and aiming to provide material for the emergence of a consensus on the subject.

By only requiring the probabilities assigned to the outcomes of measurements of physical properties in a physical system, we have crafted a theory-independent realism criterion, Eq. (4). It presents itself as a legitimate generalization of BA’s criterion and also Fine’s criterion, and it retrieves the results expected for both quantum and classical mechanics. Deviance measures for this criterion, the so-called irrealism measures, were developed in two independent ways, making use of the notion of robustness and the divergence of Kullback-Leibler. Although the robustness of irrealism [Eq. (11)] and the divergence of realism [Eq. (13)] are not conceptually equivalent, neither to each other nor to BA’s irreality measure, our case study highlighted a similar behavior for the three quantities. This provides evidence that both the robustness of irrealism and the divergence of realism are candidates for bona fide irreality quantifiers.

The numerical analysis of the robustness of irrealism in a qubit system, as presented in this work, is computationally feasible. However, calculations for higher-dimensional systems require the development of sophisticated computational strategies. We expect future research efforts in this direction to yield insights. Concurrently, our ongoing research delves into the interplay between realism and context incompatibility, particularly from a theory-independent perspective. Finally, this work paves the way for a deeper understanding of the emergence of classical realism, and we expect a revisit of Wigner’s friend scenarios under this light to bear fruit.