States vs. Changes of States: A Reformulation of the Ontic vs. Epistemic Distinction in Quantum Mechanics

Abstract

In this paper, I challenge the distinction between “epistemic” and “ontic” states propounded by Harrigan and Spekkens (Found Phys 40:125–157, 2010) by pointing out that because knowledge is factive, any state that represents someone’s knowledge about a physical system thereby also represents something about the physical system itself, so there is no such thing as “mere knowledge”. This criticism leads to the reformulation of the main question of the debate: instead of asking whether a given state is ontic or epistemic, we should instead ask whether a given change of a state is ontic or epistemic. In particular, in the context of quantum mechanics, one can ask whether the collapse of the quantum state could be understood as an epistemically successful change of the observer’s beliefs about the complete state of the system that is not associated with any change in the physical reality. I argue that the answer to this question should be in the negative because it is possible that, in a series of measurements, the collapse rule tells us to update a certain state to a different one and then back to the same state; if both of these updates are merely changes of our beliefs, then they could not both be epistemically successful.

1 Introduction

Quantum mechanics is famously difficult to interpret, mostly because, in its standard version, it involves two incompatible rules governing the change of quantum states: the Schrödinger evolution and quantum state collapse (which is used only in special circumstances, namely just after the “measurement” has been made). This has led some thinkers to the idea that at least some aspects of the formalism of quantum mechanics should be interpreted epistemically.

A contribution to this discussion has been made by Harrigan and Spekkens (2010; henceforth HS), who distinguish between an “ontic” and an “epistemic” understanding of quantum states in terms of non-overlapping vs. overlapping supports of associated probability distributions. The fact that their definitions are formulated in purely mathematical terms enabled the proof of theorems concerning this distinction. For example, Pusey et al. (2012; henceforth PBR) have shown that (given some additional assumptions) quantum states cannot be “epistemic” in the sense of HS. If the additional assumptions PBR made are tenable and the definitions of “ontic” and “epistemic” are adequate, then this result is a strong argument in favour of interpreting quantum states as representing something physically real.

However, the adequacy of HS’s framework has been questioned (e.g., Oldofredi and López 2020; Hance et al. 2022). In the current paper, I offer a critical assessment of the conceptual side of this debate, giving novel arguments for the philosophical inappropriateness of HS’s terminology. I also propose a reformulation of the central question of the debate and provide a new argument in favour of one of the answers to the reformulated question. Before I explain this in more detail, I introduce some terminology.

By the “physical state of the system”, I mean the state that the system is objectively in (at some given time). If a set of states is considered in an abstract way, without any reference to what it represents, its elements will be called “theoretical states”. Quantum states, which are the subject of the present paper, are an example of theoretical states. A crucial definition is as follows:

Definition 1

(Ontic and epistemic states) If a given theoretical state can represent the (possible) physical state of a system (perhaps in an incomplete way), then it is called an ontic state. If a given theoretical state can represent the (possible) beliefs of some observer about the physical state of a system, it is called an epistemic state.¹

Notice that these definitions do not exclude the idea that the same theoretical state can be both ontic and epistemic: it is conceivable that the same theoretical state can represent a physical state of the system and the observer’s knowledge about this state (cf. Sect. 3.3 and footnote 18).

The following is the plan for this paper. After reviewing HS’s definitions of “ontic” and “epistemic”² states (Sect. 2), I spend some time introducing the epistemological concepts that are crucial for the debate (Sects. 3.1–3.2). Then, I argue (Sect. 3.3) that if a theoretical state represents someone’s knowledge about the system (and, as such, it is epistemic), it is thereby also ontic—that is, it also represents something about physical reality. This is because knowledge is standardly understood as factive, that is, as entailing that what is known actually holds. This argumentation will be further supported by analysing a simple and more intuitive example with mass (Sects. 3.4–3.5). Therefore, I will conclude that the distinction made by HS is not conceptually adequate. Some other possible readings of this distinction will also be considered (Sect. 3.6–3.7), with the conclusion that they are either inconsistent with some of HS’s assumptions or also inadequate. I propose that the question of whether a given state is ontic or epistemic should be replaced by a different one: whether a given change of a state is ontic or epistemic, that is, whether it is a change in the physical reality or merely in our beliefs about it (Sect. 3.8). Then, I present a novel argument that a particular kind of change of quantum states, namely the measurement-induced collapse, cannot be interpreted as an epistemically successful change of beliefs about the complete state of the system (Sect. 4). In Sect. 5, I consider the options that remain for the defenders of the epistemic view on this kind of quantum collapse. In Sect. 6, I extend my analysis to some variants of quantum mechanics that do without the concept of measurement-induced collapse. Finally, Sect. 7 provides a short summary.

2 The Standard Definition of “Ontic” and “Epistemic” States

The distinction between “ontic” and “epistemic” states was introduced by HS (2010) within the framework of ontological models.³ They start with an operational formulation of quantum mechanics, the primitive terms of which are preparations (denoted by P) and measurement procedures (denoted by M). Quantum states (denoted by $\psi$ or $\phi$, possibly with labels) are assumed to be in one-to-one correspondence with preparation procedures. The operational formulation of a theory gives us the probabilities of the outcomes of different measurements given different preparations (i.e., probabilities that the measurement M will give the outcome k for the preparation P, denoted by Pr(k|M, P)). Such operational formulation of quantum mechanics can be associated⁴ with its ontological model, which postulates the set of complete⁵ states the system might be in (this set is denoted by $\Lambda$ and its elements by $\lambda$). By definition, complete states capture all the information about the system (all its properties at a given time). With each preparation procedure (and, therefore, with each quantum state) a probability distribution $Pr (\lambda | P)$ over $\Lambda$ is associated, which determines what the probability is that the system created by means of the preparation P is in the state $\lambda$. The ontological model needs to agree with the predictions of quantum mechanics in the sense that $\int d \lambda \ Pr (k | M, \lambda ) \ Pr(\lambda | P)$ should recover the values given by the Born rule.

Given this framework, HS claim that quantum states, which they understand as probability distributions over the set of complete states $\Lambda$ (i.e., $Pr (\lambda | P)$), might either represent the physical reality or represent “an observer’s knowledge of reality rather than reality itself” (2010, p. 126). The former are characterised by having non-overlapping supports, whereas the supports of the latter may overlap.⁶ Their full classification of the types of ontological models is as follows (2010, pp. 129–134):

• An ontological model is $\psi$-complete if quantum states are complete states (see HS 2010, p. 131 for technical details); otherwise, it is $\psi$-incomplete.

• An ontological model is $\psi$-ontic if for any pair of preparation procedures, $P_{\psi }$ and $P_{\phi }$, associated with distinct quantum states $\psi$ and $\phi$, we have $Pr (\lambda | P_{\psi }) \ Pr (\lambda | P_{\phi }) = 0$ for all $\lambda$; otherwise it is $\psi$-epistemic.

An important feature of HS’s proposal is that a state does not need to be complete for it to be “ontic”. Concerning terminology, HS prefer to attribute the property of being “ontic” or “epistemic” to models, whereas I attribute this property to states, but these two ways of speaking are equivalent and easily inter-translatable. In $\psi$-complete models, quantum states are complete states;⁷ in $\psi$-ontic models, quantum states are “ontic” states; and in $\psi$-epistemic models, quantum states are “epistemic” states. Quantum states considered in abstraction from what they represent are, in my nomenclature, called theoretical states. Therefore, I will use the following definition for expressing HS’s distinction:

Definition 2

(“Ontic” and “epistemic” states) Consider a set of theoretical states that are probability distributions over a certain state space (whose elements are interpreted as possible complete states of some physical systems). If a given theoretical state is such that its support does not overlap with the support of any other theoretical state in this set, it is called an “ontic” state; if its support does overlap with the support of some other theoretical state in this set, then it is called an “epistemic” state.

Our question is whether Def. 2 captures the same distinction as Def. 1.

3 Criticism of the Standard Definitions of “Ontic” and “Epistemic” States

In this section, I argue that the definitions of $\psi$-ontic and $\psi$-epistemic models coined by HS are conceptually inadequate.⁸ I begin by carefully introducing the epistemological concepts involved in the debate (Sects. 3.1–3.2). Building on this, I argue (Sects. 3.3–3.7) that “epistemic” and “ontic” states in the sense of HS (see Def. 2) can be both epistemic and ontic in the sense of representing both someone’s beliefs and physical reality (see Def. 1) at the same time. Finally, I propose the reformulation of the debate in terms of changes of states rather than states themselves (Sect. 3.8).

3.1 A Short Primer on Some Epistemological Concepts

The aim of this subsection is to introduce certain epistemological concepts that will be used in the argumentation. The presentation will be much more detailed than in any paper I am aware of that is devoted to the quantum ontic vs. epistemic debate, but at the same time, it will be not very detailed compared with what is available in the epistemological literature. I will not be assuming any particular epistemological theory and instead rely solely on theses that most contemporary epistemologists would agree with. The following common notation will be used: S will denote an epistemic subject (who may be also called “agent” or “observer”), and p will denote a proposition towards which S can have various attitudes (called “propositional attitudes”).

First, epistemologists distinguish between two families of propositional attitudes: full beliefs and partial beliefs; the latter are also called credences or degrees of belief (see, e.g., Jackson 2020, Genin and Huber 2021). Full beliefs are an all-or-nothing matter. For any p, S can believe p, disbelieve p or suspend judgement with respect to p; the fact that beliefs can come in various strengths is not taken into account here. In contrast, partial beliefs are graded and are often modelled by real numbers between 0 and 1. These numbers capture the strength of the subject’s beliefs. If S believes that p to a degree $d_S$ and $S'$ believes that p to a degree $d_{S'}$ such that $d_{S'} > d_S$, then $S'$ believes p stronger than S does.

Second, beliefs can be true or false. What this means precisely is a matter of debate, but here, we do not need to go into the details. We will only assume that whether a belief is true or false depends on what the world is like (which is in accordance with the classical conception of truth). This is surely a controversial assumption in philosophy, but it seems to be implicit in the discussion the current paper contributes to, so it is not problematic in this context.⁹S’s belief that p is true if it is the case that p and false otherwise. In particular, S’s belief that a system Q is in a state $\lambda _0$ is true if the system Q is indeed in the state $\lambda _0$ and false otherwise.

Third, partial beliefs, as they are usually conceived, are closely related to probabilities. There are arguments that the partial beliefs of a rational agent should satisfy the axioms of Kolmogorovian probability (see Genin and Huber 2021, Sect. 3.1.3 and references therein). This connection can be used in at least two ways. On the one hand, if I want my partial beliefs to be rational, I should better ensure that they satisfy the axioms of probability—so this connection gives me a constraint that my partial beliefs should satisfy. On the other hand, given some probabilities whose nature is unknown, this connection opens the possibility of interpreting such probabilities as representing the partial beliefs of some subject (instead of representing, e.g., relative frequencies or propensities). It is the latter way of exploiting the connection between partial beliefs and probabilities that will be of interest here. If quantum states are probability distributions over $\Lambda$, then they could be understood as representing one’s partial beliefs about the system being in one of the states belonging to $\Lambda$.

Fourth, various analyses of knowledge have been proposed in epistemology (see, e.g., Ichikawa and Steup 2018), but fortunately, the point I am going to make depends only on the assumptions that most of these analyses share. Knowledge is usually regarded as bearing a close relation to beliefs, on the one hand, and truth, on the other hand. “S knows p” means that S believes that p, p is true, and some further conditions are satisfied. Contemporary epistemologists have hotly debated what these further conditions should be (e.g., justification, causal connection to the fact that makes p true, safety, etc.), but they usually accept the first two conditions. There are exceptions to this rule, but they do not seem to be relevant in our context, so I will not discuss them here for the sake of brevity.

This way of thinking about knowledge has the following consequences. If we claim that S knows that the system Q is in the state $\lambda _0$, then this presupposes that S believes that the system Q is in the state $\lambda _0$, as well as that the system Q is indeed in the state $\lambda _0$ (we have substituted “the system Q is in the state $\lambda _0$” for p in the general analysis of the previous paragraph). The latter feature of the concept of knowledge is called “factivity”. If some p is known, then p must be true; that is, it must be the fact that p (this is where the name “factivity” comes from). In other words, saying that “S knows that p but p is false” or “S knows that p but p is not really the case” would be a contradiction.

Fifth, the relation between full and partial beliefs is a complicated issue, and epistemologists do not agree how exactly it should be approached. One idea is to set a certain threshold such that if S has a partial belief that p with a value equal to or greater than this threshold, then S also believes¹⁰p in the sense of having a full belief that p. However, it is not clear, for example, what the value of this threshold should be and whether it should be the same in all contexts.

3.2 Full Beliefs, Partial Beliefs and Quantum States

The importance of the problem of the relation between full beliefs and partial beliefs for our considerations comes from the fact that we want to analyse what it means to say that quantum states represent knowledge; yet, knowledge is usually defined as a full belief satisfying certain constraints, whereas quantum states, being probability distributions, can only represent partial beliefs, not full beliefs. Therefore, it is not clear how exactly the claim that a quantum state represents knowledge should be understood (and, more generally, what it could mean to attribute the status of knowledge to partial beliefs). One minimal reading of such claims, which I think captures the intentions of HS and their followers, is that a quantum state represents S’s knowledge iff it represents S’s partial beliefs about the state of some physical system and assigns a non-zero probability to the complete state the system is actually in (and some further conditions necessary for knowledge are satisfied).

This can be related to the standard way of understanding knowledge in the following way. Assume that $\Lambda$ is the set of all states the system Q might be in and that S knows that this is the case, so S can ascribe to the system Q only states belonging to $\Lambda$. Then, the following principle (which may be called the Full Belief–Partial Belief Link) holds:

If $\Lambda _0 \subseteq \Lambda$ is the largest subset of $\Lambda$ such that for any $\lambda \in \Lambda _0$, S’s degree of belief that the system Q is actually in the state $\lambda$ is non-zero,¹¹ then S believes (in the sense of having a full belief) that Q is in one of the states belonging to $\Lambda _0$ and is not in any state belonging to $\Lambda \setminus \Lambda _0$.

S’s partial beliefs are knowledge (in the minimal sense we are interested in here) iff the corresponding full belief given by the above Full Belief–Partial Belief Link is knowledge in the standard sense.

3.3 Quantum States and the Factivity of Knowledge

After this epistemological introduction, let us return to our main topic: the interpretation of quantum states. Could they represent the mere observer’s knowledge? In light of our observation that knowledge is standardly regarded as factive, it is difficult to make sense of such a claim. If a state represents an observer as knowing that p, then it thereby implicitly represents the reality as being such that p is the case. There is a complication here arising from the fact that quantum states, if interpreted epistemically, do not represent full beliefs but rather partial beliefs, and knowledge is usually understood as a full belief satisfying certain additional conditions. This complication does not change the essence of my objection, but it makes it technically more challenging to express.

Consider a quantum system Q, a quantum state $\psi$ (associated with a probability distribution over $\Lambda$), and an observer S whose partial beliefs about which state Q is in are represented by $\psi$. The last assumption means that for any $\lambda \in \Lambda$, $\psi$ assigns to $\lambda$ probability $Pr (\lambda ) = p_0$ iff S’s degree of belief that Q is in the state $\lambda$ is $p_0$.¹² Denote by $\Lambda _0$ the subset of $\Lambda$ that contains all and only elements of $\Lambda$ to which $\psi$ assigns a non-zero probability. By our Full Belief–Partial Belief Link, in such a case S believes (in the sense of having a full belief) that Q is in one of the states belonging to $\Lambda _0$ and is not in any state belonging to $\Lambda \setminus \Lambda _0$. If this belief is false, then we cannot attribute knowledge to S. If this belief is true, then we can say that S knows that Q is in one of the states belonging to $\Lambda _0$ and is not in any state belonging to $\Lambda \setminus \Lambda _0$ (provided that some additional conditions necessary for knowledge are satisfied; I set this issue aside as irrelevant for our discussion). Our question is the following: In this assertion about S’s knowledge, do we use $\psi$ to merely represent S’s knowledge or also (some part of) the physical reality? The answer is, as already observed, that we would not be allowed to assert that S knows that Q is in one of the states belonging to $\Lambda _0$ and is not in any state belonging to $\Lambda \setminus \Lambda _0$ if it was not the case that Q is in one of the states belonging to $\Lambda _0$ and is not in any state belonging to $\Lambda \setminus \Lambda _0$.¹³ Therefore, in the assertion about S’s knowledge, we use $\psi$ in two ways: first, explicitly, in our statement that S knows that Q is in one of the states that are in the support of the probability distribution associated with $\psi$ and, second, implicitly, in our presupposition that Q is indeed in one of the states that are in the support of the probability distribution associated with $\psi$. In the statement, we interpret $\psi$ epistemically (i.e., as representing beliefs of the observer), but in the presupposition, we interpret $\psi$ ontically (i.e., as representing something about the physical reality). We cannot make this statement without making this presupposition (unless we reject the factivity of knowledge), which means that we cannot use $\psi$ in an epistemic way without using it also in an ontic way.

Therefore, if a quantum state is epistemic in the sense of Def. 1, then it is also ontic in the sense of this definition. A similar (but weaker) connection holds in the other direction. Assume that $\psi$ represents the state of the system Q (i.e., it is ontic) and that, in principle, it is possible for some observer S to know that this is the case. Then, to represent S’s (possible) knowledge about the state of the system, we need to use the state $\psi$ again. Therefore, if a quantum state is ontic in the sense of Def. 1 and, in principle, is knowable, then it is epistemic in the sense of this definition (which, to recall, is weak in the sense that it identifies epistemic states as those that can represent possible beliefs). The connection here is weaker than the one established in the previous paragraph because it is not fully universal—it does not hold for ontic states that cannot be known even in principle; but quantum states are assumed to be knowable, so this restriction is irrelevant for us. Therefore, any quantum state is both ontic and epistemic in the sense of Def. 1—that is, it can represent the physical state of the system and the observer’s knowledge about this state. This implies that Defs. 1 and 2 cannot coincide because, in the latter, “ontic” and “epistemic” states are two disjoint classes of states.

Is there any way for the proponents of HS’s definition of “ontic” and “epistemic” states to avoid this argument? An obvious move is to talk about mere beliefs, without presupposing anything about their truth values. However, HS’s definition of “epistemic” states seems to presuppose that we have to be dealing with true beliefs, not just any beliefs. If we allowed treating quantum states as representing both partial beliefs that assign non-zero probability to the actual complete state of the system and partial beliefs that assign zero probability to the actual state of the system, this would undermine HS’s rationale behind defining “epistemic” states as having overlapping supports because, then, two quantum states with disjoint supports could represent partial beliefs about the actual state of the same system of two different agents as we do not require the actual complete state to be in the supports of both of them.

3.4 The Mass Example

To illustrate the irrelevance of the distinction between states with overlapping and non-overlapping supports for the issue of their being ontic or epistemic, let us consider the following simple example. Suppose we are investigating the masses of objects. Consider the following three sets of possible mass states (where the subscript “M” stands for mass):

• ${\Lambda }_{M} = {\mathbb {R}}^{+}$;

• $O_M = \{(0 \text { kg}, 4 \text { kg}], (4 \text { kg}, 8 \text { kg}], (8 \text { kg}, 12 \text { kg}], \ldots \}$;

• $E_M = \{(0 \text { kg}, 4 \text { kg}], (1 \text { kg}, 5 \text { kg}], (2 \text { kg}, 6 \text { kg}], \ldots \}$.

Using HS’s terminology, one should say that the states belonging to $\Lambda _M$ are complete (these are the exact masses of objects, which are expressed by real numbers), whereas the states belonging to $O_M$ and $E_M$ are incomplete. Furthermore, the states belonging to $O_M$ do not overlap, whereas some of the states belonging to $E_M$ do overlap, which means that the former should be regarded as “ontic”, whereas the latter should be regarded as “epistemic” in HS’s sense.¹⁴

Now, the question becomes the following: Are we willing to say that there is a fundamental metaphysical difference between the states belonging to $O_M$ and the states belonging to $E_M$? This seems a very implausible view for the following reasons.

First, any state belonging to either $O_M$ or $E_M$ can be used to represent both a mass of a physical object and an observer’s knowledge about that mass. If the mass of an object happens to belong to a given interval in $O_M$ or $E_M$, then this interval represents the actual mass of that object (albeit in an imprecise way). If some observer knows about this, then the same state represents that observer’s knowledge.

Second, there in no difference in objectivity between $O_M$ and $E_M$: it is not the case that states belonging to $E_M$ are subjective in some sense in which the elements of $O_M$ are not. If the mass of the object is 3.5 kg, then it is objectively true that its mass is between 0 and 4 kg (so it can be described by the first state in $O_M$), but it is no less objectively true that it is between 1 kg and 5 kg, as well as between 2 kg and 6 kg, and so on (so it can also be described by any of the first four states in $E_M$). Given the exact mass 3.5 kg, it follows by pure mathematics to which intervals this mass value belongs.

Third, irrespective of whether we use states belonging to $O_M$ or $E_M$, we will represent the mass of the object as belonging to a certain interval of length 4. Therefore, changing from $E_M$ to $O_M$ does not increase our precision. If our intuition was that only fully precise properties deserve to be called ontic (cf. Sect. 3.7), then this would not enable us to attribute different ontological statuses to $O_M$ and $E_M$: the elements of both are equally imprecise.

Fourth, another obvious obstacle for treating the two sets as having metaphysically different statuses is the fact that $O_M$ is a subset of $E_M$. Do the states belonging to $O_M$ cease to represent reality whenever this set is extended to $E_M$? This sounds rather absurd.

3.5 Some Objections to the Mass Example

In this section, I discuss four objections to my mass example presented in Sect. 3.4 that attempt to undermine either its internal adequacy or relevance for the assessment of the quantum case.

The first objection is that I have used arbitrarily defined sets of states to make my point, but the sets we actually use (e.g., in quantum mechanics) are not chosen at will, and their choice is a result of a conglomerate of experimental and theoretical considerations. If such considerations (the objection might go) lead to a set of non-overlapping states, this could be only because we are “cutting nature at the joints” and are revealing some objective distinctions in physical reality itself.

To see why this is not true, let us return to our mass example. Suppose we have a weighing scale with weights of 4 kg each. The scale has two arms. We put the object whose mass we want to measure on the left arm and one weight on the right arm. If the left arm is above the right arm, we know that the object has a mass between 0 kg and 4 kg. If the left arm is below the right arm, we put another weight on the right arm. If now the left arm is below the right arm, we know that the object has a mass between 4 kg and 8 kg. If not, then we put yet another weight on the right arm and so forth. We repeat the procedure until the left arm is above the right arm. The states we can detect in this way are precisely the elements of $O_M$. However, this does not mean that we have revealed some deep division in the nature of mass, namely that masses come in chunks of 4 kg each. Instead, this is a result of our accidental epistemic constraints—the measurement device that is available to us can detect only these states. This illustrates that the states that we actually use can be non-overlapping for rather epistemic reasons.

This does not mean (of course) that the elements of $O_M$ should be regarded as merely epistemic states. They can reveal something about the mass of the object we are investigating—namely, its actual value with a given precision. The elements of $O_M$ can be used to represent our knowledge that a physical object has a mass between 0 kg and 4 kg or between 4 kg and 8 kg (and so on), but also to represent the object itself having such a value of mass. And the same, mutatis mutandis, is true for $E_M$.

The second objection is that the mass example is inadequate because it involves only one property and HS’s idea is that quantum states capture some properties of the system, whereas other properties (if there are any) are captured by the hidden variables, so considering multiple properties is essential. In response, let us observe that HS assume only that quantum states are probability distributions over $\Lambda$, whereas properties do not enter their formal framework. Therefore, all our analyses for the mass case carry over straightforwardly to the case with multiple properties; the only difference is that now the elements of $\Lambda _M$ are n-tuples of real numbers, and the elements of sets $O_M$ and $E_M$ are sets of such n-tuples (but cf. Sect. 3.7 for more on the topic of multiple properties).

The third objection (suggested to me by an anonymous reviewer) is that, in the case of inexact value attributions (such as the mass example in Sect. 3.4 and the keys example in Sect. 3.6), we should not assert knowledge claims because the factivity property would then require us to accept disjunctive facts. To illustrate this thesis, the following example is invoked: Suppose that I know that my sister is either in Indonesia or Cambodia, but I do not know in which of these countries she is located; then, in fact, I do not have knowledge of where my sister is; claiming otherwise would commit us to accepting a disjunctive fact (her being either in Indonesia or Cambodia), which is absurd because she is not “smeared out”—she is located at exactly one place.

In response, let me first observe that the proposed view on knowledge would be unusually restrictive. In contemporary epistemology, the operators such as “S knows that...” or “I know that...” are typically regarded as applicable to any proposition, which means that for any proposition p, one can sensibly ask whether someone knows that p. This includes propositions that have the logical form of a disjunction, and, more generally, all propositions whose contents are not fully precise. However, the view proposed in the currently discussed objection is that the knowledge operators should be applicable only to propositions that assign an exact value of some property (even if we do not want to say that I know that my sister is either in Indonesia or Cambodia, a fortiori, we would not want to say that I do not know that my sister is either in Indonesia or Cambodia, so we need to conclude that the operator “I know that...” is not applicable to p = “my sister is either in Indonesia or Cambodia”). This seems an unparalleled restriction of knowledge operators. It is even more far-reaching than it seems at first glance. To show this, let me develop the example with my sister further: Assume that she is in fact in Cambodia and I learned this from her e-mail a moment ago, so now I know that she is in Cambodia. This looks fine because “my sister is in Cambodia” does not have the logical form of a disjunction. However, this is still not a precise property since my sister is much smaller than Cambodia, so the truth of the proposition “my sister is in Cambodia” is consistent with very many exact locations of my sister. In some sense, this proposition is a hidden disjunction (“my sister is in Cambodia” = “my sister is in Phnom Penh or in Siem Reap or...”, where, in this reformulation, any disjunct is itself a long disjunction). Therefore, saying that I know that my sister is in Cambodia is still not accurate if we follow the line of reasoning suggested in the objection: “I know that my sister is in X” would be well-formed only if X is a description of her (possible) exact location. However, using such a restrictive view on knowledge, we would need to conclude that we never or almost never know anything. Some philosophers would endorse this conclusion, but it is surely revisionary with respect to the common way of using the word “knowledge”.

The move that has been used to motivate this restriction is to ask (in the original example with my sister) whether I know where she is located. To such a question, the disjunctive answer might seem unsatisfactory. However, thanks to the broad applicability of knowledge operators, one can reasonably ask another question: Is it the case that I know that my sister is either in Indonesia or Cambodia? I see no obstacles to answering this question in the affirmative. This does not mean that my sister is “smeared out” because “being smeared out” is not the meaning of the classical disjunction. Quite the contrary, the classical disjunction requires that at least one of the disjuncts must be true for the whole disjunction to be true, so my sister as a whole must be located in one of these countries to make true the proposition that she is either in Indonesia or Cambodia.

What about the disjunctive facts, then? Are we forced to admit them in our ontology if we accept the unrestricted applicability of knowledge operators and the factivity of knowledge? This is not quite clear to me because the acceptance of both of the mentioned assumptions is common, whereas the acceptance of disjunctive facts is not. In the theory of truthmakers, which considers what kinds of facts make true various kinds of propositions, disjunctive propositions are often regarded as made true not by disjunctive facts, but by the facts expressed by some of their disjuncts (see, e.g., Mulligan et al. 1984, p. 314; cf. also MacBride 2021). However, this does not make these propositions being about our knowledge instead of being about reality; they are made true by the facts in the world, not by the facts about the agents asserting them. Another approach might be to accept disjunctive facts as real but non-fundamental (cf. the last paragraph in Sect. 3.7).

Finally, and most importantly, one should ask whether the proposed restrictive view on knowledge would help justify HS’s distinction between “ontic” and “epistemic” states. I think that the opposite is the case: if we restrict the word “knowledge” only to the cases where the exact value of some property is known, then a belief that a system is in a state $\psi$, where $\psi$ is “epistemic” in HS’s sense, cannot under any circumstances qualify as knowledge because such $\psi$ is consistent with many complete states. Even “ontic” states are not guaranteed to capture the exact value of some property by their definition alone (see Sect. 3.7), a fortiori this concerns “epistemic” states. Therefore, it seems that the proposed defense of HS’s terminology eventually undermines it, even though in a different way than my original reasoning.

Let me now turn to the fourth and the last objection concerning the mass example (based on my discussion with Stephen Leeds, although he did not formulate it in exactly these words). HS’s reasoning for the epistemic nature of “epistemic” states might be reconstructed as follows: Given the method of preparation, the quantum state of the system is uniquely determined; therefore, if the same $\lambda$ is compatible with two different quantum states, there must be some factor additional to $\lambda$ that determines which state the system is in; since this factor does not come from $\lambda$ and $\lambda$ is a complete state (i.e., there is no physical information about the system that is not captured by $\lambda$), this factor cannot be ontic—so it must be our knowledge. This uniqueness of the assignment of a state is not shared by the mass example; for instance, an object of a mass of 3 kg can be ascribed any of the following E_M states: (0 kg, 4 kg], (1 kg, 5 kg] and (2 kg, 6 kg]. Therefore, the mass example is disanalogous to the quantum case in the aspect that is crucial for the above reasoning.

I think that if the HS’s argument for the epistemic nature of their “epistemic” states is understood as in the above paragraph, then its weakest point is the move from the quantum state not being uniquely determined by $\lambda$ to the epistemic nature of the additional factor needed to determine it. In fact, the first step in this reasoning—namely, the observation that the preparation procedure uniquely determines the quantum state—suggests what kind of thing this additional factor is: it captures the way in which the system has been prepared, which is an aspect of the history of the system. Since the history of the system is an objective physical process, I see no reason for regarding this factor as epistemic. Therefore, under this reading, “ontic” states encode some information about the complete state of the system (at a given moment), whereas “epistemic” states in addition encode some information about its history (which is not captured by $\lambda$). The main difference is that if quantum states are “epistemic”, then the same $\lambda$ can be prepared in different ways, whereas if they are “ontic”, then, for each $\lambda$, there is exactly one way of preparing the system in this complete state. However, this cannot be a basis for attributing to “ontic” and “epistemic” states radically different ontological status. It is also not the case that if different people knew different things about the system, then they might attribute to it different quantum states: either one knows how the system has been prepared and attributes to it the quantum state correctly, or one does not know the method of preparation, in which case that person must remain agnostic about the quantum state of this system.

Coming back to the mass example, it is indeed disanalogous in that it lacks the reference to the preparation of the state. However, the importance of the notion of preparation is not that obvious: if the criticism of the previous paragraph is correct, then the appeal to the notion of preparation does not help in justifying that “epistemic” states are indeed epistemic. The definitional difference between “epistemic”/“ontic” states concerns their overlapping/non-overlapping supports, and this difference is preserved in the case of E_M/O_M. It is also imaginable that one can add some procedure that makes the choice of an element of E_M unique in a given context (and acts trivially on the elements of O_M); it seems implausible that the addition of such a procedure would change the ontological status of the elements of O_M and E_M.

3.6 Epistemic “Informational Holes” in States and the Epistemic Changes of States

We have seen that statements such as “the quantum state represents mere observer’s knowledge” or “the quantum state is a representation of an observer’s knowledge of reality rather than reality itself” are problematic in light of the factivity of knowledge. Are there any claims in the vicinity of these that are more reasonable? I think there are at least two. Even though one cannot say that some state is merely epistemic, one can reasonably say that some lack of information (or lack of knowledge) is merely epistemic or that some change of a state is merely epistemic. Let us look at these in turn.

Whenever non-trivial (i.e., different from 0 and 1) probabilities are used in the representation of the physical state of an individual system, there is some indeterminateness involved, and one can reasonably ask: Is the physical reality itself indeterminate in a given respect or is this only our lack of knowledge? It is commonly believed that, in the case of classical statistical mechanics, the latter holds: the classical particles have precise positions and momenta, but we do not know them. However, this does not mean that the states of classical statistical mechanics represent “mere knowledge” about individual systems—rather, they represent the states of physical particles in the world, albeit in an incomplete way.¹⁵ What is merely epistemic here is the “informational hole” in the theoretical state, not the positive informational content of it. This “informational hole” (e.g., the missing information about the exact positions and momenta in the case of classical statistical mechanics) does not have any counterpart in reality—the real physical state has this hole “filled in”, it is only a hole in our knowledge. However, the positive informational content of the state (e.g., the known information about positions and momenta in the case of classical statistical mechanics) is not merely our knowledge, but it also captures (incompletely) the real physical state.¹⁶

It should be clear that this sense of “merely epistemic” does not underlie the distinction between “epistemic” and “ontic” made by HS, as all incomplete states involve such informational holes, not only “epistemic” ones. If quantum states have overlapping supports, then they cannot be complete,¹⁷ so overlapping supports are indicators of epistemicity in the sense of the presence of epistemic informational holes, but this is not HS’s intended meaning of epistemicity. States with non-overlapping supports can also have informational holes (cf. $O_M$ in Sect. 3.4).

The second type of claims, concerning the nature of the changes of states, can be illustrated by the following example. Assume that I know for sure that my keys are somewhere in my house and that this house can be divided into a number of places. The set of such places (closed under the union and intersection) is the set of possible states of my keys. Are these states ontic or epistemic? As we should expect, they are both. On the one hand, these states can represent the actual location of the keys—a fact about the physical reality. On the other hand, they can represent my beliefs concerning the location of my keys.¹⁸ This is a synchronic level: at any given time, the states can represent both the location of my keys and my beliefs concerning their location. However, there might be a difference at the diachronic level, that is, in how the states evolve in time. Suppose that I wake up in the morning and start looking for my keys. Initially, I have no idea where they are located, so my beliefs are represented by the state that is the union of all places in my house. However, when I start checking place by place, then I exclude more and more places as possible locations of my keys. My beliefs change—they are no longer represented by the most encompassing state; the consecutive states representing my beliefs correspond to smaller and smaller places. Therefore, the epistemic evolution is here non-trivial. Meanwhile, during this entire process of my looking for my keys, the keys themselves stay wherever they have been initially. Therefore, at least up to the point when I find them and pick them up, the ontic evolution is trivial.

In the example with the keys, there is a change in my beliefs without any change in the relevant aspect of the world (i.e., the actual state of my keys).¹⁹ However, this should not be conceived as a difference between two types of states because the same set of states is used to represent the possible locations of my keys and my possible beliefs about the location of my keys. It is the change of states that can be said to be merely epistemic here, not the states themselves. The same intuition can be applied to the case of quantum mechanics (Sect. 3.8).

3.7 Exact and Inexact Values of Properties

An anonymous reviewer objected to the argument from the factivity of knowledge (Sect. 3.3) and to my simple example (Sect. 3.4) that “getting things ‘more or less right’ ($\lambda$, in this case) is not enough for onticity since onticity depends on $\psi$ univocally capturing $\lambda$”. In response, let us observe that for $\psi$ to univocally capture $\lambda$, there must be a one-to-one correspondence between wave functions and complete states, which amounts to the claim that the wave function is itself complete. Therefore, under the most natural reading of this objection, it identifies the concept of an ontic state with the concept of a complete state, contrary to HS’s intention. In HS’s definition of “onticity”, it is $\lambda$ that univocally determines the “ontic” state, not the other way around (because there might be more than one $\lambda$ in the support of $\psi$), so the intuition invoked in this objection is not satisfied here.

A way of modifying this objection is to say that “getting things ‘more or less right’ is not enough for onticity since it requires getting something exactly right, even though not necessarily the entire $\lambda$”. One can continue this train of thought by assuming that $\lambda$ consists in a set of the exact values of certain properties (more than one), and a state is ontic if it captures the exact value of at least one of these properties. In fact, this is how HS seem to think about $\lambda$’s—as consisting of $\psi$ and (perhaps) several hidden variables (HS 2010, pp. 129–130). A similar idea is expressed by PBR (2012, pp. 475–476). Let us grant this understanding of complete states as consisting of exact values of several properties and call one of them f with values belonging to ${\mathbb {R}}$. Then, for each $\lambda$, the value of f is unique (so it can be written as $f (\lambda )$). Consider two different values of this property, denoted by $f_1$ and $f_2$. Each of them determines the set of all complete states whose value of f is $f_i$ (for $i = 1, 2$)—that is, the set $F_i := \{ \lambda \in \Lambda | f(\lambda ) = f_i \}$. What is more, $F_1$ and $F_2$ are disjoint sets because every $\lambda$ corresponds to exactly one value of f.²⁰ Therefore, if every wave function uniquely determines the exact value of f and these values are different for different wave functions, then the supports of such wave functions must be disjoint, so they are “ontic” states in HS’s sense.

This relation might be used to motivate HS’s definition of “ontic” states in the following way: a state can be said to be “ontic” only if it exactly captures the value of at least one of the properties that constitute complete states. However, for this way of motivating the definition to have a chance to work, the relation would need to hold in both ways; that is, it should be the case not only that capturing the exact value of some property implies disjoint supports, but also that disjoint supports imply capturing the exact value of some property. But this is not the case: the implication in the latter direction is in general false, as is shown by the following example. Assume that every complete state consists in the specification of the values of two properties, f and g. Consider sets of complete states $X_1 := \{ \lambda _1, \lambda _2 \}$ and $X_2 := \{ \lambda _3, \lambda _4 \}$, where $\lambda _1 = \langle f_1, g_1 \rangle , \lambda _2 = \langle f_2, g_2 \rangle , \lambda _3 = \langle f_1, g_2 \rangle$ and $\lambda _4 = \langle f_2, g_1 \rangle$. These sets provide a sought-for counterexample because $X_1$ and $X_2$ are disjoint, but they do not determine an exact value of either of the properties f or g.

One can try to prevent situations of this kind by defining properties in a way that makes such counterexamples impossible. In fact, this seems to be done by PBR (2012, p. 476, description of Fig. 1, notation changed), who define a physical property in the following way: they consider a collection of probability distributions over the set of complete states labelled by $l \in L$ (i.e., $\{ \mu _l (\lambda ) \}_{l \in L}$) and say that if, in such a collection, every pair of distributions have disjoint supports, then the label $l \in L$ is uniquely determined by $\lambda$ and, therefore, is called “a physical property”. It is not clear whether a probability distribution over complete states is an object of the right category to be called a property (especially in light of the intuition mentioned earlier that complete states are the specifications of the values of physical properties), but at least it looks reasonable to say that a given value of a given physical property might correspond to a certain probability distribution (namely, the one that is non-zero for all and only complete states that, for this particular property, determine this particular value). However, there is a more serious problem with this approach to defining properties: being a physical property depends here on what the collection of probability distributions we started with is. The same probability distribution will then correspond to a physical property when “immersed” in some collections of probability distributions, but not when “immersed” in others. This seems to be too high a level of arbitrariness in specifying what a physical property is.

Another way of objecting to my counterexample to the implication from disjoint supports (i.e., “onticity” in HS’s sense) to capturing the exact value of some property is to say that even though such counterexamples cannot be excluded in general, they do not hold in the particular case we are interested in. What is needed here is the assumption that quantum mechanics is such that either (i) probability distributions over complete states associated with wave functions are not disjoint or (ii) they are disjoint and each wave function captures an exact value of some property. Using our former notation and assuming again, for simplicity, that there are only two properties, case (ii) might be realised as follows: the wave function captures the exact value of f, whereas the value of g is a hidden variable. Then, every wave function $\psi _{f_i}$²¹ would correspond to the set $\{\lambda \in \Lambda | f (\lambda ) = f_i \}$, which is equal to $\{ \langle f_i, g_j \rangle | \exists _{\lambda \in \Lambda } f(\lambda ) = f_i \wedge g (\lambda ) = g_j \}$. The wave functions $\psi _{f_i}$ correspond to disjoint probability distributions and capture the value of one property (i.e., the value of f). Of course, both f and g can be replaced by any number of properties. Currently, I believe this is the closest to HS’s (2010) understanding of this issue. However, some additional argument is needed to support the hypothesis that either (i) or (ii) holds. In other words, we need to exclude the remaining option that (iii) probability distributions over complete states associated with wave functions are disjoint but wave functions do not capture the exact value of any property. That is, we need to show that the supports of probability distributions associated with wave functions are not similar to sets $X_1$ and $X_2$ above. I am not aware of any argument for this hypothesis.

The remaining question is whether providing such an argument would be sufficient to justify HS’s definitions of “ontic” and “epistemic” states.²² Associating the concept of onticity with the exact values of properties seems to be supported by the following intuition: what is real in the strict sense is the exact values of properties, and their inexact specifications do not form a part of our ontology, but statements about them might be true because they are made true by these exact values being exemplified in the world.²³ However, in this way, we do not gain any support for calling states with partially overlapping supports “epistemic” (unless we assume that whatever is not ontic is thereby epistemic, which is rather dubious). In particular, the fallaciousness of saying that something represents “merely knowledge” is not dismissed. What is more, the equivalence between disjoint supports of states and capturing of the exact value of some property by these states is, at best, accidental: if it holds in quantum mechanics, this is because of the particular features of this theory, not because this equivalence is analytically true for any collection of probability distributions (we have seen that it is easy to construct counterexamples to it).

Perhaps, in the considerations about the exact vs. non-exact values of properties, a more useful distinction (than ontic vs. epistemic) would be between what is ontologically fundamental vs. everything that supervenes on it²⁴ (where both categories are treated as ontic). The exact values of physical quantities might then be said to be ontologically fundamental, whereas the inexact values of these physical quantities might be said to supervene on them (i.e., in some sense, they are still real properties—albeit non-fundamental ones and fully determined by the fundamental ones). Such an ontology might seem outrageously generous, but it all depends on our view on supervenience; if, following Armstrong (1997, pp. 12–13), one believes that “what supervenes is no addition of being” (he calls this doctrine “ontological free lunch”), then accepting such entities does not seem at all costly. What HS call complete states would then be fundamental, together with those incomplete states that amount to specifying the exact value of some properties of the system (and nothing more); those incomplete states that amount to the inexact specification of some properties are supervening; and those incomplete states that involve both the information about the exact values of some properties and the information about the inexact values of some other properties have a hybrid status (i.e., they involve some information about fundamental properties, but also some information about supervenient properties). This does not match with HS’s distinction between “epistemic” and “ontic” states. Although “epistemic” states must be either supervening or hybrid, “ontic” states might belong to any of the three categories (which follows from the considerations found in the preceding paragraphs).²⁵

3.8 A Reformulation of the Problem

In light of the above arguments, I would like to propose a reformulation of the problem of onticity vs. epistemicity regarding quantum mechanics. Instead of asking, “Is a given state ontic or epistemic?” (a synchronic question), one can ask, “Is a given change of a state ontic or epistemic?” (a diachronic question). The meaning of the questions of the latter kind is explained by means of my toy example with the location of my keys in Sect. 3.6.

Perhaps, at least to some extent, this diachronic way of thinking was a guiding intuition for HS’s definitions of $\psi$-ontic and $\psi$-epistemic models. This is suggested, for example, by the following quote: “By our definitions, $\psi$ has an ontic character if and only if a variation of $\psi$ implies a variation of reality and an epistemic character if and only if a variation of $\psi$ does not necessarily imply a variation of reality” (HS 2010, p. 132). However, it is unlikely that “variation” here should be understood in a dynamical way (as a change of a state) because the predicates “ontic” and “epistemic” are attributed by HS to states, not to the changes of states. Instead, “variation” here seems to mean only “difference”. Therefore, even if the underlying intuition was somewhat similar, the difference between the two formulations (i.e., the synchronic one and diachronic one) is conceptually important and deserves a strong emphasis.

One can object here that whether a change of a state is ontic or epistemic should depend (at least to some extent) on whether the state itself is ontic or epistemic. However, my question is posed for states that are both ontic and epistemic (in the sense of Def. 1). The case under consideration is precisely of this kind: both “ontic” and “epistemic” states in HS’s sense (defined in terms of non-overlapping vs. overlapping supports, see Def. 2) are both ontic and epistemic in the philosophical sense (i.e., any of them can be used to represent the state of the system and the knowledge of an observer that the system is in that state; see Def. 1). Depending on the way in which they are used in a particular context, their change might be either ontic or epistemic. The question is now whether some particular ways of changing these states (such as measurement-induced collapse or unitary evolution) are an instance of the former or latter type of change. Therefore, the reformulated debate about the changes of states presupposes a particular answer to the debate about the status of the states themselves: namely, that they are both ontic and epistemic in the philosophical sense. And this must be so: to ask the question I am interested in, it must be possible to use the states under consideration to represent something in the world and someone’s knowledge about it; only then does it make sense to ask whether a given change was a change in the world or the relevant aspect of the world remained the same and only someone’s beliefs about it changed.

4 Can the Collapse of the Quantum State be Interpreted as a Successful Change of Partial Beliefs?

In Sect. 3, I argued that as long as we do not consider the evolution of states, any state that represents knowledge (and, as such, is epistemic) by the factivity of knowledge is also ontic. The appropriate question is then not whether a given state represents the reality or our knowledge about reality, but whether a given change of a state is a change in reality or only a change in our beliefs about reality. This leads to a question whether, by merely looking at the formal features of the evolution of states, we can tell what kind of change we are dealing with. I believe that, in general, the answer is negative, in the sense that there is no universal formal criterion distinguishing ontic change from merely epistemic change (especially if we allow partial beliefs that do not have the actual complete state in their support, i.e., assign to it zero probability). However, I believe that we can conclude something about particular cases, including the quantum mechanical case we are interested in.

First, the PBR theorem is relevant for our modified question about changes of states, despite the fact that it was formulated as an answer to the original question about states. Assume that at time $t_i$ an observer S ascribes a quantum state $\psi _i$ to a system Q and at a later time $t_f$ the same observer S ascribes to Q a different state $\psi _f$. Could the change from $\psi _i$ to $\psi _f$ be merely epistemic, assuming that the PBR theorem is true, that is, assuming that $\psi _i$ and $\psi _f$ have non-overlapping supports? The answer depends on whether we allow quantum states to represent partial beliefs that are entirely false in the sense of assigning zero probability to the actual complete state of the system. If we allow this, then the answer is “yes”. For example, if Q was in the same complete state $\lambda$ throughout the whole interval $[t_i, t_f]$, $\psi _i$ assigns zero probability to $\lambda$ and $\psi _f$ assigns non-zero probability to $\lambda$, then S ascribed wrongly $\psi _i$ to Q at $t_i$ and ascribed rightly $\psi _f$ to Q at $t_f$. In this case, the change from $\psi _i$ to $\psi _f$ is merely a change of S’s beliefs about the complete state of the system. However, if we use quantum states to represent only partial beliefs that assign non-zero probability to the actual complete state of the system, then the change from $\psi _i$ to $\psi _f$ cannot be merely epistemic. If the system was at $t_i$ in $\lambda$, then it cannot be at $t_f$ still in $\lambda$ because in $t_f$ its quantum state is $\psi _f$, which, as the PBR theorem tells us, assigns zero probabilities to all complete states that are assigned non-zero probabilities by $\psi _i$, including $\lambda$.

Second, below, I will provide a new and independent argument that the measurement-induced collapse of the quantum state cannot be regarded as a successful change of a partial belief state. On the one hand, my argument is more limited than that of PBR in the sense that it concerns only the quantum state collapse, whereas the PBR theorem is relevant for any change of the quantum state. However, it is usually the collapse, not the Schrödinger evolution, that is conjectured to be a merely epistemic change.²⁶ On the other hand, my argument does not assume anything about the structure of the prepared state, whereas the PBR theorem relies on the Preparation Independence Postulate (or some weakened version of it; see Myrvold 2018), which makes my argument more general in this respect. Additionally, it is interesting on its own because the plot of the argumentation is entirely different than in the PBR theorem.

How can our belief states change? First, consider full belief states. I will make two assumptions (hopefully not very controversial, at least in the context of the debate to which the current paper contributes). Every belief has a certain logical value, and I will assume that there are only two such values, truth and falsity. Every belief also has some degree of specificity or informativeness: for example, a belief that the mass of the object is between 1 kg and 4 kg is less specific (less informative) than the belief that the mass of the object is between 1 kg and 2 kg.

When is a change of a full belief state successful? In terms of our two parameters characterising beliefs (i.e., the logical value and the degree of informativeness), one can distinguish two such cases. The most obvious one is if the initial belief is false and the final belief is true. In this case, the informativeness of these beliefs does not matter: even if the initial belief was more informative, we surely want to replace it with a true belief, even if the latter is less informative. The subtler case is when the initial belief is true and the final belief is also true but more informative. It seems that at least if we are concerned only with these two parameters (i.e., logical value and informativeness), these are the only cases of a successful change of full beliefs.

In the case of partial beliefs, the issue becomes much more complex. Instead of asking whether a partial belief is true or false, we should be asking how close it comes to the truth. Let me use an example to explain what difficulties we are encountering here. Assume that there are only three possible complete states, that is, $\Lambda = \{ \lambda _1, \lambda _2, \lambda _3 \}$, and that the actual state of the system is $\lambda _2$. S’s partial beliefs about the state of the system are represented by $Pr (\lambda _i) = p_i$, where $0 \le p_i \le 1$ and $\sum _{i=1}^3 p_i = 1$. If $p_2 \ne 0$ and $p_j \ne 0$ for at least one $j \ne 2$, then we cannot say that S’s partial beliefs concerning the state of the system are simply true or false because a non-zero number is assigned to the actual state and a non-zero number is assigned to at least one of the non-actual states. However, if $p_2 \gg p_1 + p_3$, then it seems reasonable to assert that S is closer to the truth than to falsity and the reverse for $p_2 \ll p_1 + p_3$. Can this proximity to the truth be determined in a systematic way or even measured? This issue is investigated in formal epistemology, and various measures (called inaccuracy measures or epistemic utility measures) have been proposed.²⁷

However, it is debatable which of the proposed measures (if any) is adequate. Furthermore, one might question the idea that the closeness to the truth of our partial beliefs admits a linear order at all, that is, whether we indeed can say for any two partial belief states whether one of them is closer to the truth than the other (and, if there is no such linear order, then a fortiori there could be no measure). To see this, assume the same $\Lambda$ as before and consider two subjects, S and $S'$, with partial beliefs given by $(\frac{1}{8}, \frac{3}{4}, \frac{1}{8})$ and $(\frac{2}{5}, \frac{3}{5}, 0)$ (cf. Fallis and Lewis 2016, p. 577 for a similar example). The former probability distribution is more “peaked” over the actual complete state, but it does not exclude any of the non-actual complete states, whereas the latter is less “peaked” but entirely excludes one of the non-actual complete states. Therefore, S is closer to truth in the sense that his credence in the true hypothesis (that the system is in the state $\lambda _2$) is higher than that of $S'$, whereas $S'$ is closer to truth in the sense that he eliminated one of the false hypotheses altogether, which S did not do.

For our purposes, the safest move is to assume that closeness to the truth can be captured by a parameter that is at least partially ordered and to stay silent on whether this order is also linear (and whether this parameter is a measure, as many formal epistemologists want it to be). Are there any other parameters that we should take into account in assessing which of the two partial belief states is better than the other? In the analysis of the full belief case, we have mentioned informativeness and, perhaps, there are also some other parameters. Fortunately, we do not need to decide what and how many such parameters there are. We can afford to proceed in an entirely general way and assume that there are $n \ge 1$ parameters of epistemic goodness of partial belief states (denoted by $G^j$, $j = 1, \ldots , n$), each of which is partially ordered. One can define a function $g^j : B \rightarrow G^j$ that assigns one of the elements of $G^j$ to all partial belief states belonging to the set B; in short, we will write $g^j_i := g^j (b_i)$. Given two belief states, $b_i$ and $b_{i'}$, there are four possible ways in which their j-th goodness parameters may be related: (1) $g^j_i > g^j_{i'}$, (2) $g^j_i < g^j_{i'}$, (3) $g^j_i = g^j_{i'}$, and (4) $g^j_i \not \sim g^j_{i'}$ (incomparable).

What amounts to a successful change of partial belief states? We need to take into account all our parameters $G^j$ and the fact that each of them might be only partially ordered. A transition from a partial belief state $b_{i}$ to $b_{i+1}$ is a clear epistemic success when at least one of these parameters increases (in the sense of going up in the partial order) and the rest either increase or do not change, that is: $g^j_i < g^j_{i+1}$ for some $j \in \{ 1, \ldots , n\}$ and $g^k_i \le g^k_{i+1}$ for all $k \in \{ 1, \ldots , n\}$ such that $k \ne j$.

Arguably, however, the cases of clear epistemic success are not the only cases of epistemic success. If some of the parameters increase, some stay the same, and some others change into incomparable ones, then this still seems to be a case of epistemic success, albeit not a clear one. An even more subtle case would be that some of the parameters decrease, but the increase of some others compensates this. For this compensation, we need some numerical measure that tells us how large a given increase or decrease was. If each of our parameters has a numerical value, then the measure of increase/decrease could just be the difference $g^j_{i+1} - g^j_i$. However, in general, it could be the case that even though our parameters are only partially ordered, a measure of increase/decrease (call it v) is locally defined so that it enables the mentioned compensation.

What should v look like? It can be defined for some pairs of the form $\langle g^j_i, g^j_{i'} \rangle$ but not necessarily for all of them. Formally, v should be a partial function on $(G^1 \times G^1) \cup \ldots \cup (G^n \times G^n)$, and its values should be real numbers. Additionally, v should satisfy the following conditions:

• if $g^j_i < g^j_{i'}$, then $v(g^j_i \rightarrow g^j_{i'}) > 0$ or $v(g^j_i \rightarrow g^j_{i'})$ is undefined²⁸ (i.e., v is larger than zero for the change from a parameter that is lower in the partial order to the parameter that is higher);

• if $g^j_i > g^j_{i'}$, then $v(g^j_i \rightarrow g^j_{i'}) < 0$ or $v(g^j_i \rightarrow g^j_{i'})$ is undefined (i.e., v is smaller than zero for the change from a parameter that is higher in the partial order to the parameter that is lower);

• $v(g^j_i \rightarrow g^j_{i'}) = - v(g^j_{i'} \rightarrow g^j_i)$ for any $g^j_i$ and $g^j_{i'}$ for which $v(g^j_i \rightarrow g^j_{i'})$ is defined (i.e., v should have the same absolute value and the opposite sign for two transitions that are reversals of each other).

To sum up, in the most general case of an epistemically successful transition of partial belief states $b_i \rightarrow b_{i+1}$ (for a given i), we require that some parameters increase and allow that some parameters decrease (as long as this is compensated by an increase of other parameters), some do not change and some change into incomparable ones. Therefore, a transition $b_i \rightarrow b_{i+1}$ is epistemically successful iff one can renumber the parameters $G^j$ so that the following five conditions hold:

• $g^j_i < g^j_{i+1}$ for $1 \le j \le m_1$ (where $0 < m_1 \le n$) and

• $g^j_i > g^j_{i+1}$ for $m_1 < j \le m_2$ (where $m_1 \le m_2 \le n$) and

• $g^j_i = g^j_{i+1}$ for $m_2 < j \le m_3$ (where $m_2 \le m_3 \le n$) and

• $g^j_i \not \sim g^j_{i+1}$ for $m_3 < j \le n$ and

• if $m_2 > m_1$, then $\sum _{j=1}^{m_1} \left| v(g^j_i \rightarrow g^j_{i+1})\right| > \sum _{j=m_1+1}^{m_2} \left| v(g^j_i \rightarrow g^j_{i+1})\right|$ (the increase of parameters in the first group compensates for the decrease of the parameters in the second group).

There are $m_1$ increasing parameters, $m_2 - m_1$ decreasing, $m_2 - m_3$ unchanging and $n - m_3$ changing into incomparable ones; out of these numbers, only $m_1$ is required to be greater than zero (because, otherwise, there would be no improvement in our partial beliefs, so the transition would not be successful).

Having developed this abstract account of what a successful change of partial belief states might amount to (which is not a particular theory but a scheme that encompasses many possible theories, hopefully all reasonable ones), we have tools to address the main question of this section: Can the measurement-induced collapse of the quantum state be interpreted as a successful change of a partial belief state? To show that this is not the case, we will consider a particular experiment.

Assume we have three spin-measuring devices and a beam of electrons in a superposition state:

$$\begin{aligned} \psi _1 = \alpha |z \uparrow \rangle + \beta |z \downarrow \rangle . \end{aligned}$$

(1)

We set the devices so that the first one measures spin in the z-direction, the second one measures spin in the x-direction, and the third one again measures spin in the z-direction. We assume that the measurements are very fast, one after another, so that the Schrödinger evolution between them can be ignored. One of the courses of events allowed by quantum mechanics for this experimental setup is the following: we perform the measurement on some electron and get the answers “up”, “right” and again “up”. Therefore, the state of the electron changes first from $\psi _1$ to

$$\begin{aligned} \psi _2 = |z \uparrow \rangle = \frac{1}{\sqrt{2}} |x \rightarrow \rangle + \frac{1}{\sqrt{2}} |x \leftarrow \rangle ; \end{aligned}$$

(2)

then to

$$\begin{aligned} \psi _3 = |x \rightarrow \rangle = \frac{1}{\sqrt{2}} |z \uparrow \rangle + \frac{1}{\sqrt{2}} |z \downarrow \rangle ; \end{aligned}$$

(3)

and finally to

$$\begin{aligned} \psi _4 = |z \uparrow \rangle . \end{aligned}$$

(4)

Importantly, $\psi _4 = \psi _2$; that is, in this series of measurements we attribute to the system the same state twice, although not in consecutive measurements but with some measurement leading to a different collapsed state in between. If we interpret the quantum states $\psi _i$ as representing S’s partial belief states $b_i$, then this amounts to the following series of changes of partial beliefs being allowed by quantum mechanics: $b_1 \rightarrow b_2 \rightarrow b_3 \rightarrow b_4$ with $b_2 = b_4$.²⁹ However, one can show that no possible series of successful changes of partial belief states can be like this. The intuition is as follows: if we improve in the change from $b_2$ to $b_3$, then we cannot improve in the reverse change from $b_3$ to $b_2$, but the latter is exactly the same as the change from $b_3$ to $b_4$ because $b_4 = b_2$. The intuition behind the argument is as simple as this, but because our conditions defining a successful change of a partial belief state are quite complex, showing this in detail requires more effort.

Let us start by observing that $b_2 = b_4$ entails that $g^j_2 = g^j_4$ for all $j = 1, \ldots , n$. This will be used in the following lemma:

Lemma 1

For every $j \in \{ 1, \ldots , n \}$, if $b_2 = b_4$, then a parameter $G^j$ increases in one of the transitions $b_2 \rightarrow b_3$or $b_3 \rightarrow b_4$ iff it decreases in the other.

Proof

Assume that $G^j$ increases in the first transition so that $g^j_2 < g^j_3$. Our assumption $b_2 = b_4$ leads to $g^j_2 = g^j_4$, so we have $g^j_4 < g^j_3$, which means that in the second transition this parameter decreases (i.e., the other two options, $g^j_3 = g^j_4$ and $g^j_3 \not \sim g^j_4$, are excluded). Analogously, if $G^j$ decreases in the first transition, it needs to increase in the second transition. Now, assume that $G^j$ increases in the second transition so that $g^j_3 < g^j_4$. Our assumption $b_2 = b_4$ leads to $g^j_2 = g^j_4$, so we have $g^j_3 < g^j_2$, which means that in the first transition this parameter decreases. Analogously, if $G^j$ decreases in the second transition, it needs to increase in the first transition. $\square$

The above lemma will help us in proving the theorem that excludes the scenario we are investigating:

Theorem 1

If, in a series of changes of partial belief states $b_2 \rightarrow b_3 \rightarrow b_4$ , the first and the last belief state is the same (i.e., $b_2 = b_4$ ), then it cannot be the case that each transition $b_i \rightarrow b_{i+1}$ is an instance of clear epistemic success or unclear epistemic success.

Proof

Consider a series of changes of partial belief states $b_2 \rightarrow b_3 \rightarrow b_4$ such that $b_2 = b_4$. For reductio, assume that each transition $b_i \rightarrow b_{i+1}$ in this series is an instance of epistemic success (clear or unclear). Assume without loss of generality that in the first transition the parameters $G^j$ with $j \le m_1$ increase, the parameters $G^j$ with $m_1 < j \le m_2$ decrease, the parameters $G^j$ with $m_2 < j \le m_3$ do not change, and the parameters $G^j$ with $m_3 < j \le n$ change into incomparable ones. We know that $m_1 \ne 0$ (because, for any kind of success, at least one parameter must increase) and that $m_2 > m_1$ (the lemma together with the fact that in the second transition at least one parameter increases entail that in the first transition at least one parameter decreases). To compensate for the decrease of the parameters $G^j$ with $m_1 < j \le m_2$, we need to have

$$\begin{aligned} \sum _{j=m_1+1}^{m_2} \left| v(g^j_2 \rightarrow g^j_{3})\right| < \sum _{j=1}^{m_1} \left| v(g^j_2 \rightarrow g^j_{3})\right| . \end{aligned}$$

(5)

However,

$$\begin{aligned} \sum _{j=1}^{m_1} \left| v(g^j_2 \rightarrow g^j_{3})\right| = \sum _{j=1}^{m_1} \left| v(g^j_4 \rightarrow g^j_{3})\right| = \sum _{j=1}^{m_1} \left| v(g^j_3 \rightarrow g^j_{4})\right| \end{aligned}$$

(6)

(the first equality comes from $g^j_2 = g^j_4$ and the second from the properties of the partial function v). Similarly,

$$\begin{aligned} \sum _{j=m_1+1}^{m_2} \left| v(g^j_2 \rightarrow g^j_{3})\right| = \sum _{j=m_1+1}^{m_2} \left| v(g^j_4 \rightarrow g^j_{3})\right| = \sum _{j=m_1+1}^{m_2} \left| v(g^j_3 \rightarrow g^j_{4})\right| . \end{aligned}$$

(7)

This implies that

$$\begin{aligned} \sum _{j=m_1+1}^{m_2} \left| v(g^j_3 \rightarrow g^j_{4})\right| < \sum _{j=1}^{m_1} \left| v(g^j_3 \rightarrow g^j_{4})\right| , \end{aligned}$$

(8)

that is, in the second transition the sum of the values of the partial function v for the first $m_1$ parameters is larger than the sum of the values of the partial function v for the parameters $G^j$ with $m_1 < j \le m_2$. However, this is precisely the opposite of what we wanted because in the second transition the first $m_1$ parameters decrease, whereas parameters $G^j$ with $m_1< j \le m_2$ increase (and there are no other increasing or decreasing parameters, which follows from the lemma and from the fact that these are the only increasing or decreasing parameters in the first transition). Therefore, Eq. (8) means that in the second transition the increasing parameters do not compensate for the decreasing ones, which contradicts the assumption that both transitions are cases of epistemic success. $\square$

Before closing this section, let us notice that the above argument has been formulated for a particular sequence of outcomes, which is possible according to quantum mechanics, although its realisation is not guaranteed. For example, in the last of the three measurements, one could obtain $\psi '_4 = |z \downarrow \rangle$, in which case we would have $\psi _2 \ne \psi '_4$. In fact, there is a $50\%$ chance that we will obtain the outcome that I have used (i.e., $|z \uparrow \rangle$) and a $50\%$ chance that we will obtain the other outcome (i.e., $|z \downarrow \rangle$). Does this weaken the force of my argument? I do not think so: my intention was to just consider this particular sequence of outcomes and to ask whether this particular sequence could be interpreted as consisting of successful epistemic changes. If the answer is “no” (as I argued for), then we rejected in this way a hypothesis (with the universal quantifier) that all measurement-induced collapses can be understood as successful epistemic changes. This leaves open the possibility that measurement-induced collapses are epistemic changes that are not always successful; indeed, this is one of the options I acknowledge as being left for a supporter of the epistemic view in the next section (i.e., option (3)).

Still, one can ask whether there is something to be said about the alternative outcome, that is, $\psi '_4 = |z \downarrow \rangle$. If we assume that the supports of $\psi '_4 = |z \uparrow \rangle$ and $\psi _2 = |z \downarrow \rangle$ overlap emptily, then if the changes due to the collapse are merely epistemic, it must be the case that at some point our beliefs about the complete state of the system were entirely wrong: either the actual complete state was not in the support of $\psi _2$ or it was not in the support of $\psi '_4$. This means that either we were entirely wrong in our initial state ascription or some of the changes of our beliefs were not epistemically successful. However, the mentioned assumption about supports follows from the PBR (2012) theorem, so if the premises of this theorem are satisfied, then this assumption is true. (This shows again the relevance of the PBR theorem for the reformulated debate.)

5 The Remaining Options for the Epistemic View

I have argued that because knowledge is factive, those states that represent someone’s knowledge about the physical reality thereby also represent something about the physical reality itself. This led me to the conclusion that, instead of asking whether a given state is ontic or epistemic, we should rather ask whether a given change of state is ontic or epistemic. In particular, the question worth asking in the context of quantum mechanics is whether the measurement-induced collapse of the quantum state can be interpreted as an epistemically successful change of our partial beliefs about the complete state of the system. I have argued for the negative answer to this question. However, this does not mean that no options are left for the defenders of the epistemic view on the change of quantum states in the measurement-induced collapse.

I think that they can take one of the following positions:

• (1) Deny the existence of complete states.

• (2) Accept the existence of complete states but treat quantum states as only associated with probability distributions over measurement results, not over the space of complete states.

• (3) Assume that the changes of our partial beliefs in accordance with the collapse rule are not always successful; that is, they are sometimes changes from better partial beliefs to worse or incomparable partial beliefs.

• (4) Deny the assumption that the change of quantum state between consecutive measurements in accordance with the Schrödinger equation can be ignored.

• (5) Interpret the collapse as partially epistemic and partially ontic.

Let us look more closely at each of these positions in turn.

The first option amounts to rejecting the whole framework of ontological models, as put forward by HS (2010) and reviewed in Sect. 2. This seems to be the most popular position among the defenders of the epistemic view on quantum mechanics (cf. Leifer 2014,p. 72, where the list of proponents of this option, called by him “neo-Copenhagen”, is much longer than the list of defenders of the epistemic view who would accept the framework of ontological models).³⁰ This is a significant limitation for both my argument and the PBR theorem (as well as HS’s framework of ontological models in general) because they crucially rely on the assumption of the existence of set of complete states $\Lambda$, which means that the most popular variety of epistemic positions is entirely immune to these arguments.

Similarly, the approach of PBR and my own assume that a quantum state can be understood as a probability distribution over the set of complete states, which means that a defender of the second option on our list can also ignore these arguments. However, this position seems to be unstable because, if there is a complete state the system is in, then this state should constrain what the possible results of measurements conducted on this system are and do this at least as precisely as the quantum state does; but if both the quantum state and complete state constrain the possible measurement results, it is difficult to imagine that the quantum state is not related in any way to the complete state.

To comment on the third option, it might seem dubious at first glance: if using the collapse rule renders our partial beliefs worse than they were before (or neither better nor worse), why would we use it at all? It seems that, in such a case, it would be more epistemically profitable to abandon this rule. However, if we did not “update” the state after our first measurement from $\psi _1$ to $\psi _2$, then our predictions would be empirically less adequate, so the “updating” clearly contributes to the predictive success of quantum mechanics. Despite its initial appeal, this argumentation does not entirely exclude the position with number (3) on my list. Even if using the collapse rule always improves our predictions concerning the measurement results, this does not automatically mean that it always improves our partial beliefs about the complete state of the system. Perhaps in this regard, the collapse rule sometimes leads to changes of our partial beliefs that are not epistemically successful but in a way that is impossible for us to recognise, so that, despite this drawback, quantum mechanics is still the best theory of quantum phenomena available to us.

The fourth option is based on the observation that our decision to ignore the Schrödinger evolution between measurements was an idealisation. However, I find it implausible that it is this idealisation that was responsible for the contradiction at which we have arrived. If the Schrödinger evolution was non-negligible in this scenario, then we should get an improvement of our predictions of measurement results by using it, but this is not the case—if the temporal distance between the measurements is very small, then it can be safely ignored.

The last option might seem to be similar to what is proposed by Hance et al. (2022), who claim that a wave function can have, in some aspects, an epistemic nature and, in some aspects, an ontic nature. However, recall that what we are considering here is not the ontic or epistemic character of a wave function (which I argued is both ontic and epistemic in the philosophical sense of these words given by Def. 1) but of the changes of a wave function. The approach to epistemic change presented here (as well as the standard Bayesian epistemology) is not suitable for dealing with such a mixed view. This is because, for a change to be a change of someone’s beliefs about a certain state of affairs, this state of affairs itself must remain the same—otherwise, what would these beliefs be about, and how could we compare them with respect to how accurate they are in capturing this state of affairs? Changes of beliefs are implicitly understood as changes of beliefs on the same subject. Of course, the improvement of our knowledge about some state of affairs does not require that this state will still be present in the world: this might be an improvement of our knowledge about the state of affairs that belongs to the past. What is required to not change is only the specification of the subject of our beliefs. However, the mixture of epistemic and ontic change seems to amount to a change of beliefs together with what these beliefs are about; therefore, it is precisely this combination that is problematic. I do not want to claim that one cannot build an account to deal with epistemic change of this kind, but for this, some new formal tools would be needed.³¹

Summing up, my argumentation surely does not rule out all epistemic views on the nature of measurement-induced quantum collapse but puts restrictive constraints on the class of available options. This argumentation is relevant only to approaches assuming that there are complete states of quantum systems, so one might say that it threatens less extreme variants of the epistemic view and does not have any bearing on more extreme ones, such as QBism.

6 Remarks Concerning Theories Without Measurement-Induced Collapse

One can object to my analysis that proving a statement whose application is limited only to theories based on measurement-induced collapse is not philosophically interesting because such theories are not the most philosophically interesting versions of quantum mechanics currently available.³² I do not agree that this is uninteresting because the concept of a measurement-induced collapse has been central, not marginal, in the discussion about quantum mechanics throughout its history, and it is still commonly used in the introductory expositions of quantum mechanics nowadays. What is more, the defenders of the interpretations of quantum mechanics that do not refer to measurement-induced collapse usually regard the changes of a wave function as ontic.

In any case, one could ask whether my ideas can be extended in some way to cover other interpretations (or versions) of quantum mechanics.³³ My argument in Sect. 4 could not be straightforwardly carried forward to this case as it crucially relies on the assumption that the collapse is associated with a measurement because, without that, we would not obtain the effect of coming back to the same state $|\psi _2\rangle$. However, I believe that the epistemic understanding of the change of a wave function is unattractive for many interpretations (or versions) of quantum mechanics that do not postulate measurement-induced collapse, which I will argue for below, using some elements of the framework developed earlier.

Consider the following three classes of interpretations (or versions) of quantum mechanics without measurement-induced collapse (i.e., assuming that there is a single evolution rule for the wave function):

• class 1: the wave function is complete and evolves deterministically (mathematically speaking³⁴);

• class 2: the wave function is incomplete and evolves deterministically (mathematically speaking); and

• class 3: the wave function is complete and evolves indeterministically (mathematically speaking).

Taken together, these three classes cover many of the known interpretations (or versions) of quantum mechanics. For example, classes 1 and 3 encompass all interpretations according to which the ontology of the physical world is exhausted by the wave function, such as the Everettian interpretation, wave function realism,³⁵ and some versions of GRW. Bohmian mechanics, which is another widely discussed approach, belongs to class 2. Below, I will formulate three arguments, showing that it is better not to combine interpretations (or versions) of quantum mechanics belonging to these three classes with the epistemic view on the change of a quantum state because this leads to some very implausible consequences.

Argument 1 (for classes 1 and 3): the wave function is complete and evolves either deterministically or indeterministically. Assume that this evolution is epistemic. This leads us to an absurd conclusion that nothing changes in the physical world: from the completeness assumption, all conceivable changes in the physical world could only be changes of the wave function, but its changes are, by assumption, epistemic, so no place is left for any ontic changes whatsoever.

Argument 2 (for classes 1 and 2): the wave function is either complete or incomplete and evolves deterministically. Assume that this evolution is epistemic. Consider an isolated system and assume that we attribute to it the wave function $\psi _1$ at $t_1$ (e.g., as a result of performing some measurements on it or because we know in which way it has been prepared). Then, we can use the equations of evolution to compute that at $t_2$ the wave function of the system will be $\psi _2$, at $t_3$ it will be $\psi _3$, and so on for k different times $t_i$, $i = 1, \ldots , k$. Now, looking back, which wave function should we attribute to our system at $t_1$? A natural response here is $\psi _1$ as it was supposed to be the state of the system at $t_1$. However, recall that we assumed that the change due to our deterministic equations is only epistemic. If this change is an instance of clear epistemic success, then we should attribute to the system at $t_1$ some state $\psi _i$ with $i > 1$ (and if it is an instance of unclear epistemic success, we at least do not lose anything by doing so). This is because our computations of the wave function for consecutive $t_i$’s are not supposed to track any changes in the physical system that we investigate, but only how our beliefs about it should change. The most reasonable thing to do (under the assumptions of this argument) seems to be to attribute to our system at $t_1$ the wave function $\psi _{\infty }$ (if it is well-defined). However, this is clearly in disagreement with how the formalism of quantum mechanics is actually used.

Argument 3 (for class 3): the wave function is complete and evolves indeterministically so that, given the wave function at $t_1$, the probability distribution over the space of wave functions at $t_2 > t_1$ is uniquely determined (where these probabilities might be non-trivial, that is, different from 0 and 1). Assume that this evolution is epistemic. Consider an isolated system and assume that we attribute to it the wave function $\psi _1$ at $t_1$, and then we compute its evolution according to the appropriate equations, concluding that at $t_2$ it will be in $\psi _2$ with probability $p_2$, in $\psi _3$ with probability $p_3$, and so on. If our description at $t_1$ was adequate at all, then it was fully adequate because of the completeness of the wave function. However, if the probabilities calculated for $t_2$ are non-trivial, then our description at $t_2$ is not fully adequate (because unless we attribute a probability of 1 to exactly one state, our closeness to truth cannot be perfect). But recall that the change is assumed to be epistemic. Therefore, it was either a change to a less adequate description, or our initial attribution of the state to the system was inadequate.³⁶

7 Summary

I have argued that HS’s terminology of “epistemic” vs. “ontic” models/states is conceptually inadequate and that the debate could be reformulated in terms of the ontic/epistemic character of the changes of states rather than the states themselves. I have also shown that the epistemic understanding of the change of a quantum state under the measurement-induced collapse has certain consequences that are undesirable for those who adopt the framework of ontological models. For interpretations of quantum mechanics that do not use the concept of measurement-induced collapse, the situation is less clear because they are varied, but it has been argued that—at least for some classes of them—the epistemic nature of the change of a quantum state is also implausible.

It should be stressed that even though both in the case of the ontic-epistemic distinction with respect to quantum states and with respect to the changes of quantum states my conclusion is, in some sense, negative, it is very different in the former case than in the latter case. Concerning the ontic–epistemic distinction between states, I have argued that its version formulated by HS within the framework of ontological models (i.e., the “ontic”–“epistemic” distinction in my notation; see Def. 2) is conceptually problematic or even conceptually inconsistent. As a result, any question formulated in terms of this distinction will inherit its problematic status—I would just say that it will be ill-posed. This does not mean that the philosophical ontic-epistemic distinction (see Def. 1) is itself problematic, but it fails to coincide with HS’s one and fails to distinguish two classes of states—under some mild assumptions (that are satisfied by quantum states as understood in the framework of ontological models), any state should be regarded as both ontic and epistemic (cf. Sect. 3.3). Therefore, I reject the distinction by HS as philosophically irrelevant (which does not undermine its formal validity—one can prove theorems in terms of it, etc.).

In contrast, I do not see any conceptual problems with the ontic–epistemic distinction at the level of the changes of states (within the framework of ontological models and in many other contexts). Because of its conceptual viability, one can formulate well-posed questions using this distinction. The question I am interested in is whether various rules of the change of the quantum state capture a merely epistemic change, and my proposed answer to this question is negative (unless one is willing to accept some of rather unwelcome consequences of this thesis listed in Sect. 5 and Sect. 6). However, I am far from rejecting the distinction itself.