On testimony in scenarios with Wigner and Friend

Abstract

The paper constructs a semi-formal language suited to the analysis of Wigner’s Friend scenarios: it represents an epistemic notion of rational beliefs and perspectives, to accommodate the insights of perspectival interpretations of quantum mechanics. The language is then used to analyze a paradox put forward by Frauchiger and Renner (Nat Commun, 9(1):3711, 2018). Their argument is presented as a semi-formal derivation with specified rules of reasoning. These rules bear an affinity to some of the cherished tenets of epistemology and we argue that they are valid (one universally, and the other in experimental contexts). Since our proof is a reductio, it leaves a choice which premises are responsible for a contradiction. Our first choice is a step that appears incorrect from the point of view of the universal unitary evolution as well as the view that every measurement induces a collapse of a measured system’s state. Our second choice, brought to view by the paper’s attention to perspectives and epistemology, points to a step reporting the transmission of beliefs (testimony) about measurement results. We argue that testimony is not licensed by quantum mechanical formalism; we discuss some recent attempts to save the cogency of testimony in the context of quantum measurements.

1 Introduction

Quantum mechanics (QM) has been used to prompt a number of revisions in our views of the world. Recent arguments, starting with Brukner (2018) and Frauchiger and Renner (2018), prompt novel revisions of this kind. These arguments challenge the objectivity of knowledge, or, more exactly, our use of testimony in forming rational beliefs. Our focus in this paper is the latter argument, of Frauchiger and Renner (henceforth FR), as it is more amenable to analysis. FR purportedly derive a contradiction from the quantum mechanical principles applied to describe measurements as well as the agents performing these measurements. The argument has attracted considerable attention, evoking a variety of reactions. On the one hand, it has been argued that the argument incorrectly applies quantum mechanics. On the other, far-reaching conclusions concerning a formal theory of knowledge, or the concept of truth, have been drawn from the argument. Regrettably, the fairly informal character of the argument does not help to understand it clearly. Despite the large body of literature that surrounds it, as far as we know, no attempt has been made to present it as a formal derivation, i.e., as a sequence of steps, with passages between them licensed by specified rules of reasoning. An adequate representation of this sort is necessary for assessing the argument’s validity. Thus, the first (albeit auxiliary) task of this paper is to rewrite the FR argument as a formal derivation.

The FR argument is a generalization of the Wigner Friend paradox: it adds two more labs and two more agents to it, and proceeds in terms of epistemic notions such as knowledge or certainty. Our reconstruction of the argument will thus pay heed to these notions. We will argue that they all should be subsumed under a single category of rational beliefs, and represent them accordingly by means of a sentential operator. This raises a further issue of the epistemic rules that can justify steps in the FR argument.

It is often claimed that scenarios involving Wigner and his Friend call for accommodating perspectives. The physical situations of each Wigner and his Friend define different perspectives of these two agents. From these perspectives they differently describe a system under consideration, and produce differing predictions about it. In our notation and proof we will account for perspectives: to this end the formulas in our proof include symbols meant to indicate perspectives with respect to which they are relativized.

We will use the developed notation that accommodates both a required epistemological notion and perspectives to evaluate the FR argument. Spoiling the story, our formalisation of the FR argument is a reductio proof, so it leaves open the question of which premise(s) and/or rule(s) of reasoning is (are) to be blamed for the derived contradiction. We argue that the epistemic rules of reasoning are correct, but we identify two dubious steps in the argument. The most likely culprit (we think) is a step resulting from the questionable use of physics. Like a few researchers before us, we show that this step is wrong if quantum evolution is unitary. But we show that it is as well incorect on the view that every measurement induces a collapse of a measured system’s state.

Our derivation, by paying attention to perspectives and epistemological matters, reveals the second weak step in the derivation: this step reports on one agent learning about a result obtained by another agent from the testimony of the latter. Although we frequently rely on testimony in everyday life and sciences, the dialectics of the FR argument is that its premises should be licenced, or at least not conflict, with QM. However, as we will argue, in the context of Wigner Friend’s scenarios, it might happen that Wigner and Friend observe conflicting results on the corresponding systems that they measure. This issue has been present in the Wigner Friend paradox from its inception, though the emphasis has typically been on more elusive matters like the objectivity of facts, scientific objectivity, or the universality of truth.¹

Given the choice between the two controversial steps in our derivation, it is clearly less costly to place the blame on the first one, if it is costly at all. Still, the task remains of ensuring that QM does not undermine testimony as a source of knowledge and that observers agree on the results of their corresponding experiments. We end, therefore, with a short survey of responses to this issue.

After this preview, we return to the plan of this paper. The next Sect. 2 discusses our notation and clarifies some epistemological issues to be found in FR’s paper. The FR’s argument is re-worked and written as a semi-formal proof in Sect. 3. In this section we also probe the proof’s premises and defend the rules of reasoning employed by the proof. We then engage with selected literature on the FR paradox in Sect. 4; showing after some other researchers how the FR argument can be blocked. Section 5 discusses a fragment of the argument that uses an agent’s belief based on testimony. Our conclusions are stated in Sect. 6. The Appendix contains some auxiliary material.

2 Notation and epistemological matters

2.1 Relativisation: relational facts, perspectives, and contexts

Alleged incoherences in Nature (or inconsistencies in our theories) have often been resolved by taking a relativisation move. If successful, the move makes an inconsistency in question apparent, as the statements producing the apparent inconsistency hold, but relative to different factors. Relativisation is common to a few interpretations of quantum mechanics. QBism, as well as Healey’s pragmatic interpretation, assume that quantum states are relative to agents [see e.g, (Healey 2022a)]. Dieks’s (2019) perspectival quantum realism takes physical states and properties to be relative to what he calls perspectives, which are defined by physical systems. In a similar vein, Rovelli’s relational quantum mechanics opts for relational facts that obtain in relation (or in interaction) with physical systems [see e.g., (Di Biagio and Rovelli 2021)]. Relativisation looks promising in Wigner’s Friend scenarios: although Wigner and Friend make inconsistent state-attributions to a given system and assess the probabilities of measurement results differently, the inconsistency is apparent because both Wigner and Friend define a different perspective, and have access to different information. Relativization (to agents, perspectives, or physical systems) thus also appears promising for the resolution of the FR paradox. Indeed, proponents of these interpretations have recently offered their attempts to resolve the paradox in this spirit.²

The four interpretations, Relational QM, perspectival quantum realism, QBism and quantum pragmatism are undoubtedly different philosophical positions, each invoking a different image. Yet each of these positions takes a similar move. In relationism, instead of facts obtaining in our world, one has facts relative to a physical system, “labelled by the interacting systems” (cf. Rovelli 2022, p. 1057). In a similar vein, in perspectival realism one focuses on truths obtaining from a perspective (cf. Dieks 2022, p. 95). In QBism, quantum states are a vehicle representing an agent’s probability assignments, which “express her own personal degrees of belief [...]” (Fuchs et al. 2014). In quantum pragmatism “agents may correctly assign different quantum states to the same system in the same circumstances” (Healey 2022a). On the face of it, all four positions introduce a perspectival factor, with respect to which facts obtain and truths hold. In this very role contexts were explicitly introduced in RQM by Di Biagio and Rovelli (2021), in their explanation of what relative facts are.³

[...] whenever a system F is affected by a variable $L_S$ of another system S, the value of that variable, $L_S = x$, becomes a fact for F. [...] The interaction with F is the context in which that variable takes a specific value; we call the system F, in this role, a ‘context’: The interaction with the context determines the fact that a certain variable has a value in that context.⁴

Following this idea, I will write

for: From perspective F (in the interaction with F) the variable $L_S$ pertaining to system S takes value x. But one could equally well read it as: In context F the variable $L_S$ pertaining to S takes value x. One might prefer to mark a perspective semantically, like in $F \mid \!=(L_S = x)$, but it will not do for the present purposes, as composite formulas are needed, with its components referring to different perspectives, see e.g., Eq. 3. In some applications, a reference to time is needed as well, which is explained later. The above piece of notation, with time references added when needed, after further elaboration (in particular, adding an epistemic operator), will be used to analyze the FR argument, accommodating the insights common to the four interpretations.

There is a tradition of perspectival thinking in both general metaphysics and philosophy of science that I have no room here to discuss. It must suffice to mention (what I take to be) its founding fathers. Fine’s (2005) “Tense and Reality” has been very influential in general metaphysics; the views discussed there take “reality” to be a relational concept, i.e., it is always a reality from a perspective rather than the reality überhaupt.

Since the mid 1990s, perspectivalism was advocated by Ronald Giere as well, but the main thrust of his version is epistemological, and hence uncontroversial. Our knowledge, including scientific knowledge, is perspectival, he argues, in the sense that that “the strongest claims a scientist can legitimately make are of a qualified, conditional form: “According to this highly confirmed theory (or reliable instrument), the world seems to be roughly such and such.”” Giere (2006) As a recent example of using metaphysical perspectivalism in the philosophy of physics, I mention Slavov (2020) who appeals to perspectives (which he identifies with coordinate frames of special relativity) in an attempt to reconcile the objectivity of tenses with this theory.

Since our notation needs to capture some epistemological notions, a short excursion into matters epistemological is merited before we revisit the notation issue.

2.2 Epistemic notions in FR

Frauchiger and Renner (2018) envision four agents $F, \bar{F}, W$, and $\bar{W}$, who assess what they and their peers observe in their labs by appealing to (i) quantum mechanics, (ii) their acquaintance with the setup, (iii) their peers’ prowess in quantum physics, (iv) and direct perceptual observations of results or of someone’s announcement. To account for the agents’ epistemic states, their paper uses three kinds of phrases:

• (1) A knows that $\phi$ (ibid., p. 4) ;

• (2) W is certain that F knows that $\phi$ (ibid., p. 8) ;

• (3) A is certain that agent $A'$, upon reasoning within the same theory as the one A is using, is certain that $\phi$ (ibid., p. 5),

where $\phi$ is a sentence free of any epistemic phrases, like “$x = \xi$ at time t”.

The fact that two epistemic notions (knowledge and certainty) are invoked in the paper does not seem to reflect a difference between agents’ epistemic states, however. There is also no attempt in the paper to link the four sources, (i)–(iv), of agents’ beliefs to this two-fold terminology. Thus, rather than engaging in a little promising exegesis of the epistemological part of the FR paper, we had better develop a reading that will do justice to the FR argument.

From the epistemologists’ standpoint, the cases at hand are not knowledge, at least in the textbook sense of the concept. Knowledge is almost universally assumed to be factive, i.e., one’s knowledge that p implies that “p” is true. Yet, in case (i) for instance, an agent’s belief that p comes from their quantum mechanical calculations; however empirically adequate this theory is, caution suggests to stop short of claiming “p” to be true. Certainty does not suit the FR argument any better: on what seems to be a dominant view, certainty is identified with the highest form of knowledge (Reed 2022), so it is too strong for our purpose.⁵ The required notion seems to be that of rational belief, r-belief for short, where belief is understood as coarse-grained, that is, non-gradable. Observe that all the four cases (i)–(iv) produce beliefs based on reliable sources. Furthermore, the cases subsumed under (i) are based on Born’s rule yielding probabilities, and the argument only uses extreme probabilities, i.e., zero or one. Clearly, extreme probabilities delivered by our best theory form the basis for rational beliefs and disbeliefs. In cases not involving Born’s rule (e.g., direct observation of a measurement outcome) we will as well ascribe rational beliefs to agents, meaning they acquired them by using the best humanly achievable procedures in a given historical or social context. Given this appeal to the best available procedures, as well as to extreme probabilities of quantum mechanics, this paper takes beliefs to be a dichotomous matter, i.e., one has a belief, or does not have it. Another decision of this paper is that the differences between sources (i)–(iv) are not marked, and the same epistemic notion of r-beliefs is used in these cases. Yet, in our prose we indicate whether a belief considered in the argument is based on QM and familiarity with the setup, or an agent’s observation, or the testimony of another agent, reliable and competent in QM.

The focus is thus upon rational beliefs, which are formed by using sources (i)–(iv); the paper’s terminology is:

Agent X has a rational belief that p.

I will abbreviate it by introducing an epistemic sentential operator, $B_X$, to be read as “X has a rational belief that”, so the sentence above is rendered as $B_X p$.

2.3 On notation again

After this epistemological interlude, I return to the notation issue. To accommodate the insights of the four perspectives-friendly interpretations of QM, Relational QM, perspectival quantum realism, QBism and quantum pragmatism, the notation $L_S =_F x$ was introduced, to be read as “from perspective F the variable $L_S$ pertaining to system S takes value x”. As long as F is a physical system, I am liberal whether it is conscious, agentive, or none of these. Now, in the case of a conscious agent F, she can observe from her perspective a value taken by a variable $L_S$ and form an appropriate r-belief, which can be reported as $B_F (L_S =_F x )$. It is permitted that F has r-beliefs concerning results obtained by another agent, say G, from G’s perspective, to be denoted by $B_F (L_S =_G x )$. To read it, agent F has the r-belief (in F’s perspective) that variable $L_S$ takes value x in G’s perspective. There are even more complex r-beliefs that need to be accommodated: typically, these are based on QM predictions with extreme probabilities (like statements of perfect anti-correlations), and have a conditional form, like this one:

$$\begin{aligned} B_{X} \left( z_S =_A \frac{1}{2} \Rightarrow r_R =_B tails \right) . \end{aligned}$$

This sentence ascribes the belief to X that a specified above conditional holds, where the antecedent and the consequent of this conditional refer to variables taking values in different perspectives, A and B. In our use, a conditional of this sort describes perfectly correlated outcomes of two measurements, carried out by agents A and B. To reflect upon the language of the argument, its alphabet has a family of relational symbols (like “$=_F$”), each understood as a relativised identity, with every atomic formula having the form $t_1 =_F t_2$, where $t_1, t_2$ are terms of this language. Its logical vocabulary consists of classical connectives and a family of epistemic operators, like $B_X$. The only symbols intended to accomodate perspectival thinking are those of relativised identity.

The talk of perspectives brings us to a harder question: what differentiates perspectives (contexts)? Typically, a perspective is associated with an agent, or a measuring apparatus, but admittedly, perspectives are larger, on one extreme as large as subsets of the world that are maximal with respect to being coherent with a “small” object, i.e., an agent or a measuring apparatus. Here I do not attempt to resolve this differentiation question, I just assume that objects, like agents, measuring devices, and other physical systems define unique perspectives.⁶

I will thus use the phrase “from the perspective defined by F”, which usually, for reasons of brevity, is shortened to “from perspective F”.

Finally, a clarification about time dependence is needed. After all, a variable takes a value in interaction with a physical system at a time; before that time it did not have this value. Analogously, an agent forms a belief at a time; before that time she did not have this belief. This suggests the addition of a parameter of evaluation, namely the time of evaluation. I will thus write $t \mid \!=\phi$ for “$\phi$ is true at time t”.

We are now ready to proceed with the argument.

3 The FR argument reworked

Let us begin with a reflection on FR’s (2018) premises. They have three premises, C, Q, and S:

[C] Suppose that agent A has established [...]

“I am certain that agent $A'$, upon reasoning within the same theory as the one I am using, is certain that $x = \xi$ at time t.”

Then agent A can conclude [...]

“I am certain that $x = \xi$ at time t.”

[Q] [A]ny agent A “uses quantum theory.” By this we mean that A may predict the outcome of a measurement on any system S around him via the quantum-mechanical Born rule.

[S] Suppose that agent A has established [...]

“I am certain that $x = \xi$ at time t.”

Then agent A must necessarily deny that [...]

“I am certain that $x \ne \xi$ at time t. (ibid.)

I will use a relative of C, to be called the BB-B rule, which concerns r-beliefs rather than certainty, and requires the two agents share epistemic norms concerning the generation of r-beliefs. I will also use Q, as the Born rule is the basis of our “quantum mechanical” steps. Finally, a close variation of S will be used, to the effect that an agent cannot believe that a variable takes two values in an experiment, i.e., $B_W (w_L =_W fail)$ entails $\lnot B_W (w_L =_W ok)$.

Turning to the reconstruction of the FR argument, this is split into five subsections. I begin by recalling the setup of the FR argument and offer a preview of this argument. Next, I discuss the premises of the argument that are based on QM calculations as well as the transfer of beliefs between the agents. In the next subsection I present the FR argument as a sequence of steps, with passages between them licensed by specified rules of reasoning. The next subsection probes the steps of the argument whereas the final subsection discusses the rules of reasoning employed.

3.1 Setup

The setup of the FR experiment includes four agents (physical systems): two Friends $F, \bar{F}$ , and two “Wigners” W, and $\bar{W}$. F is in lab L, $\bar{F}$ is in lab $\bar{L}$, and each lab is initially in a pure state; the labs are considered isolated, apart from the brief period of a communication step that is described below. Outside these labs are two agents, $\bar{W}$ and W, which carry out measurements on these labs: W on L and $\bar{W}$ on $\bar{L}$.

Fig. 1

The setup of the FR experiment, as sketched in Frauchiger and Renner (2018). Image licensed under CC BY 4.0: https://www.nature.com/articles/s41467-018-05739-8/figures/2

Turning next to variables measured, in lab $\bar{L}$ friend $\bar{F}$ measures on a “quantum coin” R variable $r_R$ that takes two possible values, $r_R \in \{heads, tails \}$. $\bar{W}$ measures on lab $\bar{L}$ (which comprises $\bar{F}$ and R) a variable $\bar{w}_{\bar{L}}$ with values $\bar{w}_{\bar{L}} \in \{\bar{ok}, \bar{fail}\}$. Turning to F, she measures variable $z_S$ with values $z_S \in \{-\frac{1}{2}, \frac{1}{2} \}$ on a spin-half system S. Finally, W measures on lab L (comprising F and S) variable $w_L$ with values $w_L \in \{ok, fail \}$. For more on the variables, see Appendix A.

As the experiment is statistical, I schematically describe one run of it. Each run begins with a quantum coin R being prepared in state

$$\begin{aligned} |init\rangle _R = \sqrt{\frac{1}{3}}|heads\rangle _R + \sqrt{\frac{2}{3}}|tails\rangle _R \end{aligned}$$

and send to lab $\bar{L}$. In this lab $\bar{F}$ performs a measurement of $r_R$ on R with results $r_R \in \{tails, heads \}$. In $\bar{L}$ there is also a source of spin-half particles S, which agent $\bar{F}$ uses to prepare particles in either state $|\downarrow \rangle _S$ or state $|\rightarrow \rangle _S = \sqrt{\frac{1}{2}}(|\downarrow \rangle _s + |\uparrow \rangle _S)$, depending on the result of $r_R$ measurement on R: if $r_R = heads$, she prepares S in $|\downarrow \rangle _S$, whereas if $r_R = tails$, she prepares S in $|\rightarrow \rangle _S$, and in each case sends S, prepared as described above, to lab L.⁷

In lab L agent F performs the measurement of $z_S$ on S. Finally, from “the outside” of the two labs, $\bar{W}$ performs a measurement of $\bar{w}_{\bar{L}}$ on $\bar{L}$, whereas W performs a measurement of $w_{L}$ on L.

I now give an informal preview of FR Gedankenexperiment. Runs of this experiment can be characterized by the results that occur in them. For the purpose of the paradox, interesting runs are those with results $w_L = ok$ and $\bar{w}_{\bar{L}} = \bar{ok}$, in short $ok\!-\!\bar{ok}$ runs. One can calculate that these runs have non-zero probability (see Frauchiger and Renner 2018, p. 7), so $ok\!-\!\bar{ok}$ runs can occur. Consider now a $ok\!-\!\bar{ok}$ run. In this run $\bar{W}$ has an r-belief that $\bar{ok}$ occurs, and W has an r-belief that ok occurs. Now, the features of the FR quantum mechanical setup plus some epistemological rules license the implication that [if $\bar{W}$ has the r-belief that $\bar{ok}$ occurs, then W has the r-belief that fail occurs, i.e., W does not have the r-belief that ok occurs]. We thus end up with a contradiction: W has the r-belief that ok occurs and W does not have the r-belief that ok occurs. The $ok\!-\!\bar{ok}$ runs are contradictory, yet by QM calculations are possible, as their QM probability is $\frac{1}{12}$. From this, FR draws the conclusion that QM “cannot consistently describe the use of itself’.

The argument can be thought of as consisting of four parts. The first part draws on a quantum mechanical feature of the FR setup: by applying the Born rule, we get that the outcomes of joint measurements of some observables are perfectly anti-correlated. These anti-correlations are then taken as justifying implications of the form: if one variable takes value $\alpha$, then another variable takes value $\beta$. Importantly, each of these ani-correlations concerns a different system, with a different measurement, and a different agent. As the agents involved are assumed to be able to calculate such anti-correlations, it follows that $\bar{L}$, W, and $\bar{W}$ form the corresponding beliefs, the contents of which have the conditional form. Significantly, the contents of these beliefs, taken together with the occurrence of $\bar{w}_{\bar{L}} = \bar{ok}$, imply that $w_L = fail$ occurs. Then, in the second part, the beliefs of diverse agents, $\bar{L}$, W, and $\bar{W}$, are transformed into beliefs, with the same contents, of one agent W. In the third part it is derived that W forms the r-belief about the occurrence of ok. This brings us to the fourth and final part: at this stage there is a set of W’s beliefs, the contents of which jointly entail that fail occurs, and (it is assumed that) this entailment is known to W. To these r-beliefs is applied the principle of closure of r-beliefs with respect to the known entailment, to obtain that W has an r-belief that fail occurs. This means (since an r-belief cannot be contradictory) that W does not have an r-belief that ok, which contradicts the premise from the third part, i.e., that W has an r-belief that ok.

Let us start the discussion of the argument with the first and second parts.Then in the next subsection the argument is presented as a sequence of steps, with passages between the steps licensed by some specified rules of reasoning. After this presentation, the remaining parts of the argument are discussed. The last subsection evaluates the epistemic rules employed.

3.2 Anti-correlations, conditionals, and transfer of beliefs

Let us consider a single $ok\!-\!\bar{ok}$ run: in this run all four agents perform their measurements, each obtaining an appropriate result. The argument has three similar steps: in each, an agent forms a belief with a conditional content “if one variable is $\alpha$, another variable is $\beta$”. The beliefs in question are beliefs of different agents, however. I will do one step in detail that is more controversial than the others, and only outline the remaining two steps, which are described more fully in the Appendix.

$\bar{F}$ reflects on system $\bar{L}+L$ at time $n_0$. She is going to measure coin R, obtaining one of two results, heads or tails. Depending on her observation, she prepares a spin-half particle in one of two spin states $|\downarrow \rangle _S$ or $|\rightarrow \rangle _S = \sqrt{\frac{1}{2}}(|\downarrow \rangle _s + |\uparrow \rangle _S)$, respectively. Preparation might mean that she sets an appropriate magnetic field in a Stern-Gerlach apparatus, then lets particles from the source run through the apparatus, and finally picks a particle emerging at one location (say, up) before sending it to the lab L. In this process, the particle emerging at the other location is ignored or destroyed. For $\bar{F}$ the spin particle is thus in one of the states, $|\downarrow \rangle _S$ or $|\rightarrow \rangle _S$ rather than in the superposition of the two (so here the argument assumes state collapse rather than unitary evolution). Then $\bar{F}$ analysis is as follows: the second agent F performs measurement on S in lab L, in effect of which $S+F$ lands either in state $|\downarrow \rangle _S|F\; sees\; ``\downarrow ''\rangle$ or in state $|\rightarrow \rangle _S |F\; sees\; ``\rightarrow ''\rangle$. Since S and F are parts of lab L, we may consider these states to be states of entire L after F’s measurement and write them as $|\downarrow \rangle _L := |-\frac{1}{2}\rangle _L$ and $|\rightarrow \rangle _L = \sqrt{\frac{1}{2}}(|-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L )$. Finally, W performs a measurement on L in the basis $\{|ok\rangle _W, |fails\rangle _W\}$; as a result the composite system $L+W$ is

$$\begin{aligned} & \text { either in state } \sqrt{\frac{1}{2}} \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |fail\rangle _W + \sqrt{\frac{1}{2}} \left( |-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L \right) |ok\rangle _W \nonumber \\ \end{aligned}$$

(1)

$$\begin{aligned} & \text { or in state } \phi = \sqrt{\frac{1}{2}} \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |fail\rangle _W. \end{aligned}$$

(2)

Now, since $\bar{F}$ interprets the preparation procedure as delivering either one or the other spin state, and the prepared state following her measurement of tails evolves unitarily to the state (2), in which W observes fail, she forms the following belief about the implication:⁸

$$\begin{aligned} B_{\bar{F}} (r_R =_{\bar{F}} tails \Rightarrow w_L =_W fail). \end{aligned}$$

(3)

The state collapse indicated above is crucial for the derivation of claim (3) as Lazarovici and Hubert (2019) and Sudbery (2017) first showed: the assumption of the unitary evolution falsifies the claim. I will further discuss this issue in Section 4. Here a remark on unitary evolution is in place. Although the move assumes the state’s collapse, it can be defended. First, it can be argued that $\bar{F}$ knows that the state collapsed, because when she performed the measurement she observed exactly one state. Second, depending on her observation, $\bar{F}$ prepares S in a corresponding state and sends it to F. Then, ignoring unrealised possibilities, it can be claimed that after F’s measurement on S, the system $F +S$ is either in state $|-\frac{1}{2}\rangle _L$ or in state $|\rightarrow \rangle _L$. Yet, although this story might sounds convincing, it goes contrary to a unitary evolution. The correct unitary description is given on p. 22, see eq. 14 in particular. Another issue is that the argument above ignores an intermediate measurement made by $\bar{W}$ on $\bar{L}$. It can be calculated that unitary evolution everywhere but with $\bar{W}$’s measurement ignored deliver wrong probabilities (see the Appendix B.2).

Observe that in claim (3) the perspective in the consequent of the implication is W rather than $\bar{F}$, though it is $\bar{F}$ that carries out calculations and hypothesizes what result she might observe. However, it is the interaction with W, not with $\bar{F}$, that brings about the result fail (if it does), which is the reason for putting in the perspective W. A relationist might take a more drastic stance, arguing that the implication is badly formed. This stance, however, is pricey, as it amounts to objecting to the application of Born’s rule to composite systems, where it might yield correlations between results to be observed by different agents on the subsystems.

Note the material implication in the sub-formula: using the same alternative states above, $\bar{F}$ could equally well argue that $r_R \ne _{\bar{F}} tails$ or $w_L =_W fail$. Since this material implication holds in every scenario in which QM formalism is true, in which there is the FR setup, and a specified state collapse occurs, the material implication can be strengthened to strict conditional, restricted to the mentioned set of scenarios.⁹ The strengthening is redundant, however, as it does not bring any new aspect to the logic of the derivation.

The next two steps do not require state evolution with collapse (see Appendix B.1). The first is about agent F and systems S and R. Let us assume, says F at time $n_0$,¹⁰ that in my interaction with S, $z_S$ takes value $z_S =_F \frac{1}{2}$. F calculates the probability of the joint occurrence of [$z_S =_F \frac{1}{2}$ and $r_R =_{\bar{F}} heads$] to be 0 and the probability of $z_S =_F \frac{1}{2}$ to be non-zero (see Appendix), and hence asserts that, on the above assumption, $r_R =_{\bar{F}} tails$. She then discharges the assumption, forming at $n_0$ the r-belief:

$$\begin{aligned} n_0 \mid \!=B_{F} \left( z_S =_F \frac{1}{2} \Rightarrow r_R =_{\bar{F}} tails \right) . \end{aligned}$$

(4)

Finally comes the step with agent $\bar{W}$ who at $n_0$ calculates the state of system $\bar{L}+S$ after the coupling is established, given the initial state $|init\rangle _R$ of coin R. He finds out that in this state the joint probability of [$\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$ and $z_S =_F -\frac{1}{2}$] is 0, whereas that of $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$ is non-zero (see Appendix). On the assumption that $\bar{W}$ measures $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$, F must measure $z_S =_F \frac{1}{2}$. That is, at $n_0$ $\bar{W}$ forms the belief:

$$\begin{aligned} n_0 \mid \!=B_{\bar{W}} \left( \bar{w}_{\bar{L}}=_{\bar{W}} \bar{ok}\Rightarrow z_S =_F \frac{1}{2} \right) . \end{aligned}$$

(5)

Reflecting on the three statements of belief, (3), (4) and (5), note first that they concern different agents. Second, the contents of these beliefs have the form of a material implication. Finally, these three implications (contents of the beliefs) entail $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}\Rightarrow w_{L} =_W fail$. As things stand, this conclusion cannot be drawn because these implications are in the scope of belief operators, $B_X$, and beliefs are not factive. However, a similar conclusion will be drawn in the FR argument, based on some epistemological rules. To this end, as the first move, the beliefs of different agents are “transferred” to one agent, W, and this is done in two steps. First (i), agent W is competent in quantum mechanics, has the r-belief that his peers are also competent in this theory, and has the r-beliefs about the setup of the FR experiment. Accordingly, W reflects upon the three beliefs of F, F, and $\bar{W}$, and finds out that these beliefs are rational by the standards of each agent, respectively (which W shares with them). W thus forms r-beliefs about the above beliefs of his peers, and he does it at time $n_1$, which are reported as follows:

$$\begin{aligned} & n_1 \mid \!=B_W B_{\bar{F}} \left( r_R =_{\bar{F}} tails \Rightarrow w_L =_W fail \right) \end{aligned}$$

(6)

$$\begin{aligned} & n_1 \mid \!=B_W B_{F} \left( z_S =_F \frac{1}{2} \Rightarrow r_R =_{\bar{F}} tails \right) \end{aligned}$$

(7)

$$\begin{aligned} & n_1 \mid \!=B_W B_{\bar{W}} \left( \bar{w}_{\bar{L}}=_{\bar{W}} \bar{ok}\Rightarrow z_S =_F \frac{1}{2} \right) . \end{aligned}$$

(8)

In the second step (ii) one needs to reduce these second order beliefs to first order beliefs, and ascribe them all to agent W, which might appear controversial. It will be done by using a BB-B rule (similar to assumption C of Frauchiger and Renner 2018, p. 5) that is discussed and defended in Sect. 3.5. The result of the rule, applied to 6–8, is the following:

$$\begin{aligned} & n_1 \mid \!=B_W \left( r_R =_{\bar{F}} tails \Rightarrow w_L =_W fail \right) \end{aligned}$$

(9)

$$\begin{aligned} & n_1 \mid \!=B_W \left( z_S =_F \frac{1}{2} \Rightarrow r_R =_{\bar{F}} tails \right) \end{aligned}$$

(10)

$$\begin{aligned} & n_1 \mid \!=B_W \left( \bar{w}_{\bar{L}}=_{\bar{W}} \bar{ok}\Rightarrow z_S =_F \frac{1}{2} \right) . \end{aligned}$$

(11)

These are beliefs of W, obviously in W’s perspective, concerning correlations between outcomes obtained by different agents from their respective perspectives.

In the final move, after the pertinent beliefs are transferred to one agent, W, a rule similar to the epistemologists’ closure of knowledge under known entailment is invoked (discussed in Sect. 3.5), to obtain W’s belief concerning a consequence of the contents of his beliefs, i.e.,

$$\begin{aligned} B_W (\bar{w}_{\bar{L}}=_{\bar{W}} \bar{ok}\Rightarrow w_L =_W fail). \end{aligned}$$

(12)

3.3 The FR argument in eighteen steps

Below, the argument is broken down into eighteen steps. A short justification for each step is placed on its right-hand side. The discussion of the steps and the employed rules of reasoning are in Sects. 3.4 and 3.5, respectively.

Since $ok -\bar{ok}$ runs are possible, let us consider one. Keep in mind that in this run (as in any other) all measurements (by F, $\bar{F}$, W, and $\bar{W}$) are carried out. The assumed timeline in this run is $n_0< n_1< n_2<n_3 < n_4$. Since all the steps are taken to be true in the considered $ok-\bar{ok}$ run, we suppress this information in our notation, marking only the time dependence, as in $n \mid \!=\phi$, where n is a moment of evaluation. It is also assumed that a belief formed at one time can be retrieved later in the run.

The crux of this argument is that there can be no $ok\! -\! \bar{ok}$ runs, though such runs have non-zero probability, by the QM calculation. As in any reductio proof, we face the task of finding which premise(s) and/or rule(s) of reasoning is(are) to be blamed for the contradiction, to which we now turn.

3.4 Probing the steps of the argument

Let us now look into the steps of the argument, and the rationale behind them. The steps (a)–(i) have already been discussed, the examination of the BB-B rule being left to the next sub-section.

Following Lazarovici and Hubert (2019) and others, we argued that, if one assumes that unitary evolution is universal, then step (a) is incorrect. It requires one to invoke a collapse in the measurement of agent $\bar{F}$ on R, after which the system evolves unitarily in the remaining measurements. We return to this issue in Sect. 4, where we provide the relevant calculations.

This leads to the question (posed by Joanna Luc) of what is the result if every measurement in the FR setup induces a state collapse? A simple calculation shows that then before W’s measurement the system $L + \bar{L}+ \bar{W}$ is in one of the following four states:

$$\begin{aligned} & |\bar{ok}\rangle _{\bar{L}}|-\frac{1}{2}\rangle _L|\bar{ok}\rangle _{\bar{W}} \qquad \qquad |\bar{ok}\rangle _{\bar{L}}|+\frac{1}{2}\rangle _L|\bar{ok}\rangle _{\bar{W}} \nonumber \\ & |\bar{fail}\rangle _{\bar{L}}|-\frac{1}{2}\rangle _L|\bar{fail}\rangle _{\bar{W}} \qquad |\bar{fail}\rangle _{\bar{L}}|+\frac{1}{2}\rangle _L|\bar{fail}\rangle _{\bar{W}}. \end{aligned}$$

(13)

It is easy to check that in each state the probability of W’s obtaining result ok (or fail) in the measurement on L is non-zero. Observe also that, as a result of $\bar{W}$’s measurement, none of these states has as its component the state $|t\rangle _{\bar{L}}$ (if $\bar{L}$ is in this state, its subsystem R is in state $|tails\rangle _R$). Accordingly, there is no support for the controversial step (a) of the FR argument. (For more details, see Appendix B.3). The requirement that every measurement in the FR setup induces a state collapse thus blocks the derivation of the paradox as well.

Furthermore, if step (a) fails, then step (d) fails as well. In essence, steps (d)–(f) report on agent W reflecting on another agent correctly calculating quantum mechanical correlations, and as such they are innocuous. Yet, as we argued above, step (a) is based on $\bar{F}$’s incorrect calculations. Then step (d) is incorrect.

To continue with other steps of the argument, the sub-formulas in the scope of operator $B_W$ in (g), (h), and (i) jointly imply that $\bar{w}_{\bar{L}} =_{\bar{W}} ok \Rightarrow w_L =_W fail$. W has the r-belief that this implication holds, and hence, by the rule of the CLOSURE of r-beliefs (see next section on the rules), W has the r-belief reported by formula (j).

Next come a few factual assumptions reporting on what happens in the considered $ok-\bar{ok}$ run. Step (k) says that at time $n_2> n_1$ agent W observes that variable $w_L$ takes value ok and forms the corresponding r-belief. Step (l) is analogous, but refers to a different agent $\bar{W}$, and the belief is formed at time $n_3> n_2$. Step (m) reports on a result of $\bar{W}$’s announcement: at time $n_4 > n_3$ agent $\bar{W}$ announces that $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$, agent W hears and absorbs this information immediately, forming an appropriate r-belief $B_{W} B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$. By the BB-B rule, this second order belief is reduced to $B_{W} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$ in step (n).

Finally, as the contents of beliefs reported in (j) and (n) jointly imply that $w_L =_W fail$ and W believes this implication holds, by CLOSURE one gets that W has the r-belief stated in (o). By S this leads to formula (p), which contradicts W’s belief of (k) retrieved at $n_4$ and stated in (q).

The argument uses epistemic rules of reasoning which merit a separate section.

3.5 Rules of reasoning in FR

Apart from the Q and S assumptions of FR, our reconstruction uses one new rule, CLOSURE, and one rule, BB-B, motivated by assumption C of FR, but sufficiently different to deserve a discussion. To list the rules,

CLOSURE is similar to the epistemologists’ closure of knowledge under known entailment, or to the less discussed closure of justification under deduction, known from Gettier’s (1963) paradox. In contrast to them, it concerns rational beliefs rather than knowledge or justification and is restricted to cases of modus ponens, rather than covering the whole species of entailment.¹¹ As for BB-B, it permits one to reduce two operators ($B_WB_X$) to one operator ($B_W$) given the agreement between standards of rational beliefs in W’s milieu and X’s milieu. The rule is restricted to non-indexical beliefs, i.e., beliefs expressed by sentences containing no indexicals.

To focus first on CLOSURE, with its affinity to the closure of knowledge under known entailment, it inherits the latter’s virtues and vices. Its virtues come from the perception that without CLOSURE the body of our r-beliefs would be unreasonably frugal. To illustrate, suppose you just formed a perceptual r-belief that the National Bank is on Main St.: you just spotted the name of that bank near a street name sign reading “Main St.”. How can you progress from this r-belief to the somewhat more abstract r-belief that there is at least one bank on Main St.? It is not clear how this can be done, unless something like CLOSURE comes to the rescue. Clearly, you have an r-belief (which arguably comes from your grasp of English) that if there is a National Bank on Main St., then there is at least one bank on this street. CLOSURE then guarantees that you have an r-belief that there is at least one bank on Main St. Turning to the vices of CLOSURE, the major one is that it entails extreme skepticism, in the context of a brain-in-a-vat scenario or the like.¹² For the sake of argument, let us accept that the scenario is correct, that is, I cannot tell whether I am a brain in vat, or a flesh and blood creature. As this is an outlandish version of skepticism, the hope is that it remains limited, i.e., it does not spill over to mundane matters. Against this hope, CLOSURE dictates that skepticism spills over to (almost) every belief, so there are (almost) no r-beliefs. To illustrate, since I just had granola for breakfast, I shouldn’t be skeptical about it, it is my r-belief. However, I have an r-belief that if I have granola for breakfast, I am a flesh and blood being. Thus by CLOSURE I have an r-belief that I am a flesh and blood being. But (to recall the brain-in-a-vat scenario) I cannot have that r-belief. Thus, I must have been wrong about my granola r-belief: as a psychological fact, I have the belief that I just had granola, but it is not rational. This outlandish skepticism infects (almost) every matter.

The moral is that if one wants to fend off extreme skepticism (and have no means to question the cogency of the brain-in-a-vat scenarios), one needs to reject CLOSURE. However, as Dretske (2005) noted, extreme skepticism is generated by applying closure to particular implications (heavyweight implications as he called them), like the implication from having granola for breakfast to being a flesh and blood being. Given that there is a distinction between heavyweight implications and lightweight implications,¹³ one might restrict CLOSURE to be applicable only to lightweight implications, and not rely on it if applied to heavyweight implications. Now, turning to CLOSURE as used in our version of the FR argument, the implications involved concern measurement results, like $\bar{w}_{\bar{L}} = \bar{ok}\rightarrow z_S = \frac{1}{2}$ (see steps (i), (h), (g), or (n) and (j)). These are definitely not heavyweight implications, so the objection to CLOSURE motivated by fending-off extreme skepticism, has no force.¹⁴

Turning to the BB-B rule, it emphasizes rational beliefs understood as being regulated by certain rules concerning the grounds for their acceptance. What counts as grounds for accepting a given r-belief is not up to an individual’s decision; rather, it is determined by the individual’s peer group or milieu. Such grounds are clearly different for different beliefs. More interestingly, they might also vary with the context in which a belief is formed. In some contexts, the grounds for accepting a given r-belief are more demanding or of a specific kind than in other contexts. Rules for belief acceptance dictate what counts as sufficient evidence for r-beliefs, and such rules might vary from milieu to milieu. Often, procedures that counted as providing evidence for an r-belief for scientists in the past do not count as evidence for this belief for current scientists working in the same field. In a simplified epistemic picture that is sufficient for our purposes, whether agent X forms the r-belief that $\phi$ depends on two factors: what evidence for $\phi$ she has and whether the rules for accepting beliefs, as present among her peers, decide that this evidence suffices for accepting the belief that $\phi$. As rules of belief acceptance might vary with peer groups, the rationality of one’s belief is relativized to an agent’s peer group. For an agent’s belief to be rational, its formation should comply with the rules of belief acceptance as they are in the agent’s milieu.

The BB-B rule has two premises. The first one, $B_WB_X \phi$, says that W has a belief, rational by the rules of belief acceptance in W’s milieu, that X has a belief $\phi$, rational by the rules in X’s milieu. To illustrate, a forensic analyst W currently examines the belief of inspector X, held twenty years ago, that John was absent at a crime scene ($\equiv \phi$). W finds John’s alibi convincing though not impeccable. He also examines records of forensic analysis from the crime scene 20 years ago to learn that no samples of John’s DNA were found, and, importantly, given W’s current expertise, that all the procedures were performed in compliance with the Quality Assurance Standards of 2004. W thus forms the rational belief that X was rational in believing that $\phi$, i.e., $B_WB_X \phi$. To come to this r-belief, W had to learn what evidence for $\phi$ inspector X had and what the rules were for accepting the belief that $\phi$ in X’s peer group. Note also that for the truth of this claim it is irrelevant whether W’s and X’s rules of belief acceptance agree. This issue, however, is crucial for passing to the conclusion of the BB-B rule and is the subject of the second premise. To continue with the story, W reflects that in contrast to today, in 2004 for technological or legal reasons, low-level DNA samples (aka touch DNA) had not been collected and used in legal procedures in this jurisdiction. As a result, W recognizes that his and X’s rules for accepting the belief that $\phi$ disagree. Accordingly, W abstains from forming an r-belief that $\phi$.

To justify the BB-B rule, consider what happens when it fails, i.e., consider two agents, W and X, such that W has the r-belief that the rules for belief acceptance of $\phi$ in X’s milieu match those of her milieu. Assume further that (i) W has a second order r-belief, $B_W B_X \phi$, but (ii) W’s belief that $\phi$ is not rational, i.e., it is not an r-belief by the standards of W’s milieu. However, W’s second-order r-belief says that, as judged by W, X’s belief that $\phi$ is rational by X’s standards. But if W assesses X’s belief that $\phi$ as rational, then, since standards of r-beliefs of X and W agree, it must be that W’s belief that $\phi$ is rational (by W’s standards). A failure of the BB-B rule is thus on the verge of a contradiction. It is not a contradiction simpliciter because, as a psychological fact, W might fail to form a propositional attitude towards $\phi$; however, if he forms it, it must be the r-belief that $\phi$. This claim neatly tallies with Corta et al’s (2023) assessment of the C assumption of FR (which is stronger than BB-B) as “solid”, to which they arrived in a possible-worlds semantic for epistemic logic. It should also be noted that the BB-B rule stands if the content of X’s belief depends on a perspective, say X’s perspective: the rule simply requires keeping the content intact, so the perspective is unchanged.

I also claim that the BB rule is correctly used in our version of the FR argument. It is used four times, its premise being stated in (d), (e), (f), and (m). In each application to (d), (e), (f), the content of the second-order belief is a conditional based on quantum mechanical calculations. If these calculations are correct, they can be performed by anyone who has competence in QM and is familiar with the setup. Now, it is assumed that W and his peers have competence in QM and are familiar with the setup. Thus, W and his peers have the same standards for r-beliefs based on quantum mechanical calculations concerning the setup of the FR experiment. The “normative” premise of the rule is thus satisfied in these three cases.¹⁵

The fourth application of the rule, to (m), looks different. Although W and $\bar{W}$ are assumed to be equally competent in QM and equally familiar with the experiment’s setup, the outcome $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$ was obtained in $\bar{W}$’s perspective since $\bar{W}$ observed this outcome and on this observation formed the r-belief, $B_{\bar{W}}(\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$. In contrast, the second agent, W, did not make measurements on lab $\bar{L}$ so he could not observe $\bar{ok}$. His belief is thus not based on the observation, but on $\bar{W}$’s testimony, and (emphatically) concerns the belief of $\bar{W}$. This belief $B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$ is to serve as the premise of the BB-B rule. Now, W knows what evidence for $(\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$ agent $\bar{W}$ had ($\bar{W}$ told him this). W has also an r-belief that his and $\bar{W}$’s rules for belief acceptance for QM observation reports agree, and that these rules permitted $\bar{W}$ to form r-belief $B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$. Putting all this together, W has the r-belief $B_W (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$. We take this application of the BB-B rule as uncontroversial, despite the fact that its conclusion refers to two perspectives, W and $\bar{W}$, as it reports on W’s belief concerning results obtained by a different agent. For it is impossible to transform this sentence into a non-perspectival claim, since there is no way to remove or change the subscript at the identity $=$.

Yet, there is a controversy in the vicinity, and it concerns the viability of forming the second order beliefs, which serve as premises of the BB-B rule. Needless to say, in the absence of its premises, the BB-B rule is void. These second order beliefs are based on a successful communication between agents involved. In the case under discussion, W should be able to learn from $\bar{W}$ what result the latter observed (and whether their rules of belief acceptance agree). Now, the idea that QM permits non-concordant results of corresponding measurements endangers the successful communication of agents; an example of reports on non-concordant results is displayed below:

$$\begin{aligned} B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}) \qquad B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{fail}) \end{aligned}$$

In short, $\bar{W}$’s experiment on L yields result $\bar{ok}$, yet W’s experiment on joint system $L+\bar{W}$ yields result “$\bar{W}$ is r-believing $\bar{fail}$”. Given discordant results, one had better abstain from accepting the second order beliefs.

The discordant results form the core of the testimony problem of QM. We discuss the problem and attempts to solve it in Section 5.

To sum up, I argued that BB-B rule is universally valid and is correctly applied in the FR argument. I also argued that although CLOSURE might fail to be universally valid, it can be reliably used in the FR argument. Before we return to the issue of testimony, it will be conducive for our discussion to first relate this reworking of the FR argument to the literature.

4 Engaging with the literature

There is a large body of literature on the FR paradox, part of which need to be related to, first to point to a few ways the present version improves upon the FR argument and, second, to explain why some attempts to resolve the FR paradox do not apply to the present version.

The paradox was presented here as a semi-formal proof, i.e., a sequence of formulas in a given language, with each formulas being either a premise or derived from the earlier formulas by means of the assumed rules of reasoning (which were specified as well). The proof is stated in a notation that pays heed to epistemic notions inherent to the FR argument. Although commentators on the FR paradox typically notice the importance of epistemic notions, only a few of them provide a systematic treatment of them. Our focus were interpretations of QM that take a relativization move, like QBism, relational QM, perspectivalism, and quantum pragmatism, which are claimed to resolve the FR paradox. This claim was evaluated by using a notation intended to capture the idea common to these interpretations, that facts (or truths) are relative to a perspective.

The introduction of perspectives sets our work apart from the other logic- and epistemology-informed projects of Boge (2019), Nurgalieva and del Rio (2019), and Corti et al. (2023) that investigate the paradox from the standpoint of possible-worlds semantics for an epistemic logic with multiple agents. The picture that emerges from the first two publication is that of a nuanced conflict between an epistemic logic (based on possible-worlds semantics) and quantum reasonings. The third publication belongs to the same semantics-based approach; it arrives at some results similar to ours, like the reliability of the C assumption of FR (similar to our BB-B rule), or the heavy price of abandoning the transmissibility of knowledge (which is similar to what epistemologists call “testimony” and is discussed here in Sect. 5). Now, the sentiment behind our paper is that perhaps the time is not yet ripe enough to attack the FR argument with the toolbox of multiple-agent epistemic logic with its possible-worlds semantics since, quite simply, the argument is not yet fully understood. A necessary stage at arriving at its understanding, I take it, is to write the argument down as a semi-formal proof ending with a contradiction, with steps and rules of reasoning justifying the passages between the steps. In a sense, priority is given to a proof-theoretic approach over a semantical approach. Furthermore, as some steps or rules used in the argument touch upon matters epistemological, and seem to be in conflict with our usual ways of thinking about knowledge or beliefs, such alleged conflicts should be analyzed, I think, by putting epistemology into perspective (as a philosophical discipline) rather than epistemic logic. As an epistemic logic attempts to systematize what epistemology attempts to establish, a conflict with some principles cherished by epistemology will be more telling, informative, and dramatic. Given these two differences, our way with the FR argument has little in common with the three papers mentioned above. The three papers were not helpful for the present project as they fail short of exhibiting the FR paradox as a semi-formal proof, leaving it at an intuitive level.

Our derivation takes care to improve on a perhaps minor problem which nevertheless hinders the understanding of the original FR argument: a failure to discharge premises. The content of the r-belief of step (g) is the conditional, $r_R =_{\bar{F}} tails \rightarrow w_L =_W fail$, which stands in contrast to FR’s (p. 4) move that says: “Suppose that agent $\bar{F}$ got $r = tails$ as the output of the random number generator in run n”. This premise (supposition) is not discharged in the FR argument. Accordingly, it should be present in the conclusion, so the conclusion should assert the impossibility of a $tails - ok - \bar{ok}$ runs rather than the impossibility of $ok - \bar{ok}$ runs. Incidentally, since the former runs have a non-zero quantum mechanical probability as well, the FR-style argument can be produced for such runs as well. Yet, to stick with the original argument, with its focus on $ok - \bar{ok}$ runs, one needs to discharge the premise. This is done by drawing the conditional conclusion $r_R =_{\bar{F}} tails \rightarrow w_L =_W fail$, as in the present argument.¹⁶

I argued that the beliefs based on the quantum mechanical calculations of anti-correlations should be read as beliefs that specific material implications or strict conditionals, rather than counterfactual conditionals, obtain. There are thus steps like $n \mid \!=B_W (r_R =_{\bar{F}} tails \rightarrow w_L =_W fail)$ in our proof. This is in contrast with Healey’s (2018) way with the FR argument, who works with counterfactual conditionals. For instance, he reads the sub-formula occurring in our step (g) as the counterfactual $(r_R =tails\; \square \!\!\rightarrow w_L = fail)$¹⁷ and that is essential to his analysis. His crucial move is the objection to what he calls the assumption of “intervention insensitivity”, and the objection only makes sense if the discussed dependence between values taken by variables is rendered by counterfactual conditionals. The intervention insensitivity assumption says that

[t]he truth-value of an outcome-counterfactual is insensitive to the occurrence of a physically isolated intervening event. An outcome-counterfactual is a statement of the form $O_{t_1} \square \!\!\rightarrow O_{t_2}$ where $O_t$ states the outcome of a quantum measurement at t, $t_1 < t_2$, and $A \; \square \!\!\rightarrow B$ means “If A had been the case then B would have been the case”: An event then intervenes just if it occurs in the interval $(t_1, t_2)$, and it is physically isolated if it occurs in a laboratory that is then physically isolated from laboratories where $O_{t_1}, O_{t_2}$ occur. (ibid., p. 1577)

A case of intervention sensitivity thus needs a true counterfactual $O_{t_1} \square \!\!\rightarrow O_{t_2}$ together with a false counterfactual $(O_{t_1} \wedge O_{t*} ) \square \!\!\rightarrow O_{t_2}$, where $O_{t*}$ is the proposition stating the occurrence of an intervening event in the sense explained above. Now, intervention sensitivity is logically possible because counterfactual conditionals do not respect the rule of strengthening of antecedent. As a result, strengthening the antecedent $O_{t_1}$ to $O_{t_1} \wedge O_{t*}$ can produce a counterfactual with a different truth-value than the initial counterfactual. This is not true, however, about material or strict conditionals. Each of these conditionals satisfy the strengthening of antecedents and so does not leave room for intervention sensitivity. Now, Section 3.4 argued that the appropriate beliefs occurring in the FR argument are beliefs that some material implications or strict conditionals obtain. Even if the argument fails to be fully convincing, the presented version of the FR does not use counterfactuals. This version thus leaves no room for the objection to intervention insensitivity, as understood above. However, despite our objections to the above counterfactual analysis of intervention sensitivity, we believe that the concept sheds light on the FR argument. The underlying issue is whether the intermediate measurement $\bar{W}$ on $\bar{L}$ significantly changes the state of $\bar{L}+ L$, where “significantly” means that the probability of the joint result “$r_R =_{\bar{F}} tails$ and $w_L =_W ok$” changes from zero to non-zero. As we will see below, the answer depends on whether or not collapse is assumed: there is no such significant change of state on a collapse interpretation, in contrast to there being a significant change on the assumption of non-collapse, i.e., of the universal unitary evolution of states.

A different diagnosis of the FR argument comes from the QBism camp. DeBrota et al. (2020) identify the problem with an agent assigning a quantum state to a composite system containing the agent herself and a particle.¹⁸ Their argument is that such an assignment provides probability for results of any measurement that the agent performed on the particle and herself, which (allegedly) compromises the agent’s freedom of responses to questions concerning the results measured on the particle, like “what was the result of the spin measurement of the particle?”. DeBrota et al. (2020) believe that free agents exercise this kind of freedom, to quote, “since she is a free agent, she has control over the answer to this question. It is up to her whether she replies “up”, “down”, or by sticking her tongue out.”¹⁹ I will not dwell here on whether the argument is correct (this seems to be a tangential issue), focusing instead on the allegation that the FR argument appeals to the controversial state assignment. The authors point to Friend’s (F) conclusion “I am certain that W will observe $w = fail$ at time ...” (in Table 3 of the FR paper). In their view, this shows that F makes a prediction about W’s measurement on herself (ibid., p. 1871). Under closer scrutiny, this objection relies on a mistake, however. The quoted claim is derived by the C rule and is not based on the controversial state assignment. The required state assignments are discussed in subsection 3.2, and in none of them does an agent assign a state to a system she/he is a part of.

Turning next to relational QM, its proponents put the blame on the assumption of the absolute nature of facts in the FR argument (see e.g., Di Biagio and Rovelli 2021, p. 29), which (they say) underlies the consistency (C) assumption of FR. Their objection is that the absolute nature of facts (or assumption (C)) mandates the following implication: “If W, applying quantum theory, can be certain that $L_S = a$ relative to F, then W can reason as if $L_S = a$ also relative to W” (ibid.). To put this in our notation, if $B_W (L_S =_F a)$, then $B_W (L_S =_W a)$. (Note different subscripts at $=$). This implication cannot be licensed by the BB-B rule, because the rule does not permit changing the content of the beliefs, so it leaves intact subscripts to the identity sign. So, the move targeted by Di Biagio and Rovelli (2021) is not present in our version of the argument, though some steps report on beliefs with a conditional content and with different perspectives, like ($\star$) $B_{\bar{W}} (\bar{w}_{\bar{L}}=_{\bar{W}} \bar{ok}\Rightarrow z_S =_F \frac{1}{2})$. Some proponents of relational QM might object to ($\star$), on the basis that nobody can ever infer anything about anybody else’s experiences. For us, this objection goes beyond the prudence that perspectivalism requires. Given that the results are dichotomic and $\Rightarrow$ is the material implication, the sub-formula of ($\star$) above is equivalent to $\lnot (\bar{w}_{\bar{L}}=_{\bar{W}} \bar{ok}\wedge z_S =_F -\frac{1}{2})$. $\bar{W}$ can form the r-belief in the above by calculating from Born’s rule that the two results, measured by him and F, are anti-correlated, which moreover can be read in an epistemic way: I believe (says $\bar{W}$) that if I compared my report on what I observed and F’s report on what he observed, the combination $\bar{ok}$ and $- \frac{1}{2}$ would be absent (it is a different matter if this comparison can be accomplished). Besides, statements like ($\star$) are innocuous in our formalisation, since there are no rules permitting the change of perspectives in the belief operator as well as inside the sub-formula.

We end this (highly selective) survey with (what we take to be) a successful answer: the assumption of the universal validity of unitary evolution blocks the derivation of the FR paradox, i.e., the contradiction (r) in our proof is not derived. (Note that on p. 14 we argued that the paradox is also blocked by the assumption that every measurement induces a state collapse). The argument was put forward by Sudbery (2017), Lazarovici and Hubert (2019) and more recently, by Mucino and Okon (2020). We follow here the second authors, as their analysis is more complete. To begin with, the coin is prepared in state $\sqrt{\frac{1}{3}}|heads\rangle _R + \sqrt{\frac{2}{3}}|tails\rangle _R$. After the spin-half particle is prepared and sent to lab L, the system $\bar{L}+ S$ is in state $\sqrt{\frac{1}{3} }|h\rangle _{\bar{L}} |\downarrow \rangle _S + \sqrt{\frac{2}{3}}|t\rangle _{\bar{L}} |\rightarrow \rangle _S$. The subsequent measurements carried out by F, $\bar{W}$ and W lead, by unitary evolution, to the following state of $\bar{L}+ L +\bar{W}+ W$:

$$\begin{aligned} \phi= & \sqrt{\frac{3}{16}} \left( |h\rangle _{\bar{L}} + |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{fail}\rangle _{\bar{W}} |fail\rangle _W\;+ \; \nonumber \\ & \quad + \sqrt{\frac{1}{48}} \left( |h\rangle _{\bar{L}} + |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L \right) |\bar{fail}\rangle _{\bar{W}} |ok\rangle _W\; - \nonumber \\ & \quad - \sqrt{\frac{1}{48}} \left( |h\rangle _{\bar{L}} - |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{ok}\rangle _{\bar{W}} |fail\rangle _W\;+ \nonumber \\ & \quad + \sqrt{\frac{1}{48}} \left( |h\rangle _{\bar{L}} - |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L \right) |\bar{ok}\rangle _{\bar{W}} |ok\rangle _W \end{aligned}$$

(14)

Given this state, the probability of the joint result “tails and ok” is $\langle \phi \mid P\phi \rangle$, where P is the projection on the vector defined by $|tails\rangle _R |ok\rangle _W$.²⁰ The result is $\frac{1}{12}$, rather than 0. Accordingly, the anti-correlation-based argument for step (a) is incorrect: since the probability of “tails and ok” is non-zero, it is simply not true that the observation of tails implies the observation of fail. However, the anti-correlation-based arguments for two similar steps in the FR argument, (b) and (c), are correct, since the relevant joint probabilities are indeed zero, as Lazarovici and Hubert (2019) observed. For completeness, we include these calculations in the Appendix B.1.

In calculating state (14) we assumed that it results from the unitary evolution induced by subsequent measurements of $\bar{F}$, F, $\bar{W}$ and W. One might be interested in calculating how the state evolved if $\bar{W}$’s measurement did not occur, to learn whether this measurement significantly changes the state of $L +\bar{L}+\bar{W}+ W$. Indeed, it does: this state is calculated in Appendix B.2, and the probability of the joint result of tails and ok, given this state, is shown to be zero, in contrast to the non-zero value of this probability calculated in state (14). This zero probability outcome should not be seen, however, as providing support for step (a). After all, this result comes from ignoring one measurement in the calculation of the final state of the FR experiment, and this omission should not be attributed to any agent $\bar{F}, F, \bar{W}$, or W, since each is assumed to be familiar with the setup of the experiment, including the familiarity with the sequence of measurements performed.

Let us sum up our progress thus far. We argued that a proponent of the universality of unitary evolution as well as a proponent of measurement-induced state collapse have convincing ways to block the derivation of the FR paradox. Each of the two schools questions the validity of step (a). One can thus put the FR paradox to rest by opting for one of the two options concerning the evolution of states. Interestingly, one does not need perspectivalism to block the paradox.

However, the problem is that there is yet another way to block the FR argument, which our focus on perspectives and beliefs brings to the fore: can agents communicate? The FR argument relies on one agent, $\bar{W}$, conveying the information about his result to the other agent (see step (m)). One response is that there is not any problem here, since obviously people communicate and frequently base their rational beliefs (or knowledge) on testimony of others. However, the logic of the FR paradox, as we understand it, is to use only those steps in the argument that are supported by QM, or at the very least, are not in conflict with QM. And, as we will argue, QM tells against testimony as a source of r-beliefs or knowledge. This issue is not merely limited to a decision on how to respond to the FR paradox: even if one chooses to put the blame on step (a), which we think is the most reasonable option, the issue of QM and testimony needs to be clarified and resolved. This is a task to which we now turn.

5 Challenging testimonial-based beliefs

A step of the FR argument invokes the announcement of observed results: agent $\bar{W}$ measures on lab $\bar{L}$, observing either $\bar{ok}$ or $\bar{fail}$, and announces “$\bar{ok}$” if she observes $\bar{ok}$, and “$\bar{fail}$” if she observes $\bar{fail}$. In our version of the FR argument this is captured by formation of rational beliefs, described by the passage from step (l) to step (n). This passage has two intermediate steps: the formation of the second-order belief (from (l) to (m)) and the application of the BB-B rule (from (m) to (n)). To recall,

$$\begin{aligned} n_3 \mid \!=B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}) \qquad \bar{W}\text { forms the r-belief about her result } \end{aligned}$$

(l)

$$\begin{aligned} n_4 \mid \!=B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}) \qquad W'\text {s belief after } \bar{W}'\text {s announcement } \end{aligned}$$

(m)

$$\begin{aligned} n_4 \mid \!=B_W (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}) \qquad \text { by BB-B from (m) } \end{aligned}$$

(n)

Note that W acquires the r-belief that $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$ despite not being able to observe the result, since W does not interact with system $\bar{L}$. W’s r-belief that $\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok}$ is based on the testimony of $\bar{W}$, it is a testimonial-based r-belief. Testimony consists of one agent acquiring knowledge or r-belief on the basis of the information she receives from another agent. Typically, testimony-based formation of knowledge (or r-beliefs) is not analyzed by breaking it into two steps, as we did above. However, perspectivalism naturally encourages this two-steps analysis, and it will pay off in shedding light on testimony in quantum Gedankenexperiments.

A short reflection on testimony is needed here. Although there are a few philosophical questions about testimonial-based knowledge,²¹ the consensus has it that a large portion of our knowledge is testimonial-based. Clearly, the consensus carries over to r-beliefs, as they are somewhere between knowledge and mere beliefs. Typically, testimony involves announcing and listening to (or, sending and receiving) and it is part of our common sense view of communication. On this view, we say things about ourselves or the external world and others listen to our reports, and form beliefs. By default, a listener accepts what she has heard or read, but if in doubt, she can check whether the conveyed information is correct. Given some additional factors, the testimony-based beliefs are uplifted to r-beliefs or instances of knowledge. Similar ideas underlie classical physical communication. For instance, temperature readings are gathered at a remote ski area and relayed over electromagnetic waves to my computer, which receives them. The notion that the information gathered by one agent / system can be reliably shared with other agents / systems seems to form the core of a common concept of objectivity. Apparently, science, in particular experimental science, is inconceivable without there being a reliable network of information sharing between involved agents.²²

Returning to the steps under discussion, I argued that BB-B rule (which justifies step from (m) to (n)) is universally valid since its negation verges upon a contradiction. We turn thus to the other step, from (l) to (m), asking whether it can be represented in the formalism of QM, or, more mildly, whether this formalism can be supplemented by a postulate justifying this step.

Now, testimony-based r-beliefs are not licensed by quantum formalism of unitary evolution.²³ To be more in line with this formalism, instead of analyzing $\bar{W}$’s announcement, we had better focus on how agent W actively learns about a result obtained by $\bar{W}$. Clearly, W should carry out a measurement on the composite system $\bar{W}+ \bar{L}$, shortly after $\bar{W}$ registered a result, either $\bar{ok}$ or $\bar{fail}$. Suppose that $\bar{W}$ observed $\bar{ok}$, so she has formed an r-belief, to be written as $B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$. The state of $\bar{W}+ \bar{L}$ is given by the superposition, of the following form:

$$\begin{aligned} \psi = a_1 |\bar{ok}\rangle _{\bar{L}} |\bar{W}sees ``\bar{ok}"\rangle + a_2 |\bar{fail}\rangle _{\bar{L}} |\bar{W}sees ``\bar{fail}"\rangle . \end{aligned}$$

(15)

Now, given that coefficients $a_1$ and $a_2$ are non-zero, one gets a probabilistic answer that each result, one corresponding to $\bar{W}sees ``\bar{ok}"$ as well as one corresponding to $\bar{W}sees ``\bar{fail}"$ can be observed with non-zero probabilities. In particular, despite $\bar{W}$ observing $\bar{ok}$, W might learn that $\bar{W}sees ``\bar{fail}"$. Accordingly, $\bar{W}$ and W might form conflicting r-beliefs: $B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$ and $B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{fail})$, contradicting step (l) $\Rightarrow$ (m) above. Although these r-beliefs are conflicting, there is no actual contradiction; furthermore, adding a third observer will not reveal a contradiction. Still, the result is troubling, since there appears to be no fact about which measurement outcome occurs that agents can share.

To discuss this predicament in more detail, let us assume that $|\bar{W}sees ``\bar{ok}"\rangle$ and $|\bar{W}sees ``\bar{fail}"\rangle$ are orthogonal vectors, spanning the 2-dimensional Hilbert space, and representing the states of $\bar{W}$ together with his device. We abbreviate this notation by writing $|\bar{ok}\rangle _{\bar{W}} := |\bar{W}sees ``\bar{ok}"\rangle$ and $|\bar{fail}\rangle _{\bar{W}} := |\bar{W}sees ``\bar{fail}"\rangle$. In the process of $\bar{W}$’s measurement on $\bar{L}$ these states become coupled with states $|\bar{ok}\rangle _{\bar{L}}$ and $|\bar{fail}\rangle _{\bar{L}}$ of $\bar{L}$, respectively, as displayed in (15). Consider now two observables

$$\begin{aligned} A_1= & \frac{a_1}{a_2} |\bar{ok}\rangle _{\bar{L}}|\bar{ok}\rangle _{\bar{W}} \langle \bar{fail}|_{\bar{W}}\langle \bar{fail}|_{\bar{L}} + \frac{a_2}{a_1} |\bar{fail}\rangle _{\bar{L}}|\bar{fail}\rangle _{\bar{W}}\langle \bar{ok}|_{\bar{W}}\langle \bar{ok}|_{\bar{L}} \end{aligned}$$

(16)

$$\begin{aligned} A_2= & |\bar{ok}\rangle _{\bar{L}}|\bar{ok}\rangle _{\bar{W}} \langle \bar{ok}|_{\bar{W}}\langle \bar{ok}|_{\bar{L}} - |\bar{fail}\rangle _{\bar{L}}|\bar{fail}\rangle _{\bar{W}}\langle \bar{fail}|_{\bar{W}}\langle \bar{fail}|_{\bar{L}}. \end{aligned}$$

(17)

These observables satisfy the eigenvalue equations:

$$\begin{aligned} & A_1 \psi = 1 \psi \qquad A_1 \psi ' = -1 \psi ' \end{aligned}$$

(18)

$$\begin{aligned} & A_2 |\bar{ok}\rangle _{\bar{L}} |\bar{ok}\rangle _{\bar{W}} = 1 |\bar{ok}\rangle _{\bar{L}} |\bar{ok}\rangle _{\bar{W}} \quad A_2 |\bar{fail}\rangle _{\bar{L}} |\bar{fail}\rangle _{\bar{W}} = - 1 |\bar{fail}\rangle _{\bar{L}} |\bar{fail}\rangle _{\bar{W}}, \end{aligned}$$

(19)

where $\psi = a_1 |\bar{ok}\rangle _{\bar{L}} |\bar{ok}\rangle _{\bar{W}} + a_2 |\bar{fail}\rangle _{\bar{L}} |\bar{fail}\rangle _{\bar{W}}$ is the superposition state (15) and $\psi ' = a_1 |\bar{ok}\rangle _{\bar{L}} |\bar{ok}\rangle _{\bar{W}} - a_2 |\bar{fail}\rangle _{\bar{L}} |\bar{fail}\rangle _{\bar{W}}$.

Accordingly, $A_1$ can be used to ascertain that system $\bar{L}+ \bar{W}$ is in state (15), that is, ascertain whether $\bar{W}$ successfully performed the measurement on $\bar{L}$ that resulted in the coupling of states of $\bar{L}$ and of $\bar{W}$, to be seen in the state (15). If the observed eigenvalue is 1, the measurement was performed, and if the observed eigenvalue is $-1$, the measurement was not performed. Note also that since $\psi$ is an eigenstate of $A_1$, the measurement of $A_1$ does not change the state of the system $\bar{W}+\bar{L}$. One might hope that $A_2$ can be used to ascertain which outcome $\bar{W}$ was observed: if W obtains eigenvalue 1, $\bar{W}$’s outcome was $\bar{ok}$ and if W obtains eigenvalue $-1$, $\bar{W}$’s outcome was $\bar{fail}$.

Yet, there are two problems with this answer. First, $A_1$ and $A_2$ do not commute, i.e. $A_1A_2 \ne A_2 A_1$. They cannot thus share a common eigenstate, which means that W cannot ascribe to $\bar{L}+ \bar{W}$ a state that would (1) represent a definite outcome obtained by $\bar{W}$ and at the same time (2) represent $\bar{L}+ \bar{W}$ as being in the entangled state (15). The measurement of $A_2$ will change the state of $\bar{L}+ \bar{W}$, which means that the coupling in state (15) that underlies $\bar{W}$’s observation of a given outcome is modified by W’s measurement of $A_2$. Accordingly, W cannot learn that $\bar{W}$ performed the experiment and observed a given result.²⁴

Secondly, and more dramatically, nothing prohibits W from obtaining an eigenvalue $-1$ (i.e., corresponding to vector $|\bar{W}sees ``\bar{fail}"\rangle _{\bar{W}} |\bar{fail}\rangle _{\bar{L}}$), on the basis of which she forms the r-belief $B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{fail})$, whereas $\bar{W}$’s r-belief is $B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$, as (on our assumption) $\bar{W}$ observed $\bar{ok}$.

The unitary formalism of QM has thus a consequence that agents can have conflicting beliefs concerning measurement results. As we emphasized, the conflict is not an outright contradiction, but vigilance is required to keep clear of it. For instance, by promoting r-beliefs to knowledge, we will get, given factivity of knowledge, the contradiction $(\bar{w}_{\bar{L}} =_{\bar{W}} \bar{fail})$ $\wedge \; (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$.

Before we discuss responses to this predicament, let us reflect on the situation. To recall, the logic of the FR argument requires its steps to be supported, or at least not in conflict, with QM. Thus, an appeal to everyday practice is non sequitur: conflicting beliefs concerning measurement results in QM merely show that QM’s predictions do not agree with our everyday practice. Similarly, pointing to the fact that science is a social enterprise, based to a large extend on testimonial knowledge, is non sequitur: such arguments merely show that QM predictions are in conflict with the social character of sciences. After this detour, we turn to two recent approaches attempting to ensure that observers form consistent beliefs concerning measurement results: by Adlam and Rovelli (2023) and Healey (2024).

To begin with Adlam and Rovelli (2023), while working in the framework of Relational QM, they propose supplementing this interpretation by (what they call) the postulate of cross-perspective links:

In a scenario where some observer Alice measures a variable V of a system S, then provided that Alice does not undergo any interactions which destroy the information about V stored in Alice’s physical variables, if Bob subsequently measures the physical variable representing Alice’s information about the variable V, then Bob’s measurement result will match Alice’s measurement result.

Clearly, the postulate aligns W and $\bar{W}$’s beliefs about measurement results in the FR argument. We may safely assume, given the FR setup, that between W and $\bar{W}$’s measurement, the system $\bar{W}+ \bar{L}$ does not undergo any interaction destroying information about $\bar{W}$’s result. It follows then by the postulate that W observes the eigenvalue corresponding to $|\bar{W}sees~''\bar{ok}"\rangle _{\bar{W}}|\bar{ok}\rangle _{\bar{L}}$ iff $\bar{W}$ observes the eigenvalue corresponding to $|\bar{ok}\rangle _{\bar{L}}$. Consequently, the beliefs of the two agents match, i.e., we either have $B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{fail})$ and $B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{fail})$ or $B_W B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$ and $B_{\bar{W}} (\bar{w}_{\bar{L}} =_{\bar{W}} \bar{ok})$.

Notwithstanding the postulate’s ability to reconcile the conflicting beliefs, one might ask what support can be offered for it. The authors point first to the fact that the information about Alice’s outcome is “necessarily stored physically in some of the variables of Alice”. Then they invoke physicality of information, to claim that “part of what it means for that information to be ‘physical’ is that it should be accessible to other observers who have access to Alice and the ability to perform appropriate measurements”. But this line does not strike us as correct. Of course, on the everyday understanding of physicality, Alice’s results should be accessible to other qualified observers. Yet, the worry is that in the realm of quantum experiments the physical Nature might not behave in the way we think it should. As explained above, unitary QM permits that W obtains an eigenvalue corresponding to vector $|\bar{W}sees~''\bar{fail}"\rangle _{\bar{W}} |\bar{fail}\rangle _{\bar{L}}$), whereas $\bar{W}$ observes an eigenvalue corresponding to $|\bar{ok}\rangle _{\bar{L}}$. The cross-perspective link exactly prohibits this to happen, but what supports this prohibition?

This question receives more attention in the second approach, namely that of Healey (2022b, 2024). The main novelty of this approach is a semantic idea that a report on a quantum measurement is not true or false simpliceter, i.e., in the world (or a model of a language considered), but rather is true or false in the world and at an assessment parameter (more generally, in a model of a given language and at a value taken by the parameter). The idea is known from tense logic (Belnap 2002) or MacFarlane’s (2014) work in the philosophy of language. Here we leave it aside whether the motivations underlying these two projects extend to sentences about quantum measurements, and just assume after Healey that there is a parameter, called “decoherence environment” at which such sentences are evaluated. We take it (without having a textual evidence though) that a report on result r of measurement M might be true, false, or indeterminate at a decoherence environment. The latter case occurs if a decoherence environment is not proper for the reported measurement.

To explain the assessment parameter, consider an isolated laboratory, in which a magnitude M pertaining to a given system takes a particular value, e, in a spatiotemporal region. To quote (Healey 2022, p. 18), “An M-decoherence environment E of a quantum event e [...] is a region $R_E$ of spacetime that includes the region where e occurs, together with physical processes in $R_E$ that can be modeled by robust decoherence of that system’s states in a “pointer basis” associated with different values of M”. Clearly, a decoherence environment for one quantum event might fail to be a decoherence environment for another quantum event.

In region $R_E$ decoherence processes induce a reduced state of the system. This state approximates a “classical” (aka proper) mixture in a pointer basis associated with eigenvalues of M. It is classical in the minimal sense that multiplication coefficients sum up to one, so one can view such a mixture as describing multiple alternative possible outcomes, each with the attached probability. Then one of these outcome is indeterministically selected to be actualised in the measurement. Since at an M-decoherence environment of quantum event e the sentence “The measurement of M yields result e” is to be evaluated as true, this decoherence environment should be thought as comprising two kinds of processes: (i) the creation of reduced state of the system, approximating a proper mixture in a pointer basis corresponding to possible values of M, and (ii) the indeterministic selection of one of these values.

Let us follow (Healey 2022, p. 15) in applying these ideas to the FR scenario. $\bar{F}$ and coin R, placed in an isolated laboratory, undergo a measurement interaction. The processes culminate with producing a decoherence environment $E_{\bar{F}+R}$ for the event of measuring the magnitude $r_R$ (“face of coin R)”. First, the diagonal terms of the reduced state, in the $r_R$ basis, become equal to:

$$\begin{aligned} |\bar{F}sees\; h\rangle |h\rangle \langle h|\langle \bar{F}sees\; h| \qquad |\bar{F}sees\; t\rangle |t\rangle \langle t|\langle \bar{F}sees\; t|. \end{aligned}$$

The off-diagonal terms become negligible. Second, an indeterministic process actualizes one of values corresponding to diagonal elements, say h. Result h is displayed, the memory of it is created in $\bar{F}$, as are its traces in multiple pieces of the laboratory. Accordingly, in decoherence environment $E_{\bar{F}+R}$, the sentence “$\bar{F}$ measures the result h of measurement $r_R$” is evaluated as true, whereas the sentence “$\bar{F}$ measures the result t of measurement $r_R$” is evaluated as false. Note that $E_{\bar{F}+R}$ is not a decoherence environment for results measured by $\bar{W}$, since at this stage $\bar{W}$ is outside of the isolated system formed by $\bar{F}+R$. Accordingly, the reports on $\bar{W}$’s measurement, as assessed at $E_{\bar{F}+R}$, are indeterminate.

Let us thus turn to the measurement of $\bar{W}$ on $\bar{F}+ R$. This measurement requires breaking the isolation of the $\bar{F}+R$’s laboratory. As a result, the decoherence environment $E_{\bar{F}+R}$ is destroyed, traces of result h are removed from pieces of the equipment and the memory of $\bar{F}$. To consider a decoherence environment for $\bar{W}$’s measurement, one needs to focus on a larger isolated system, comprising R, $\bar{F}$, W, and some devices. As above, in the new decoherence environment $E_{\bar{W}+ \bar{F}+R}$ the reduced state has negligible off-diagonal terms, its diagonal terms being

$$\begin{aligned} & |\bar{W}sees\;''\bar{F}sees\; h''\rangle |\bar{F}sees\; h\rangle |h\rangle \langle h|\langle \bar{F}sees\; h|\langle \bar{W}sees\;''\bar{F}sees\; h''| \end{aligned}$$

(20)

$$\begin{aligned} & |\bar{W}sees\;''\bar{F}sees\; t''\rangle |\bar{F}sees\; t\rangle |t\rangle \langle t|\langle \bar{F}sees\; t|\langle \bar{W}sees\;''\bar{F}sees\; t''|. \end{aligned}$$

(21)

Indeterministically, a value corresponding to one of diagonal terms is actualised in decoherence environment $E_{\bar{W}+ \bar{F}+R}$. For instance, it happens that $\bar{F}$ sees t and the result is t. Accordingly, in $E_{\bar{W}+ \bar{F}+R}$ the sentence “$\bar{F}$ sees t and the result is t” is evaluated as true, whereas the sentence “$\bar{F}$ sees h and the result is h” is evaluated as false. Depending on the decoherence environment, different sentence are evaluated as true/false.

Clearly, the evaluation of sentences in appropriate decoherence environments helps to avoid contradictions. However, it is controversial whether it helps with the testimony issue. To see the problem, we summarize the above scenario in three steps, indicating an appropriate decoherence environment on the left-hand side of the turnstile symbol:

Steps (l’) and (m’) state what $\bar{F}$ and $\bar{W}$ see in their respective decoherence environments, and step (n’) comes from applying the BB-B rule to (m’). Apparently, $\bar{F}$ and $\bar{W}$ have conflicting rational beliefs in (l’) and (n’), though they are equally competent in QM. The conflict is apparent, though, since their beliefs are in different decoherence environments (or, more precisely, the sentences stating these beliefs are true at different decoherence environments). However, it is hard to resist the conclusion that the conflict is avoided only at the cost that $\bar{F}$ and $\bar{W}$ cannot communicate.

This uneasiness is mitigated in Healey’s second move that points to the practical impossibility of Wigner Friend’s scenarios. Such scenarios require the existence of the right kind of decoherence processes in Friend’s lab together with a perfect isolation of this lab from decoherence processes outside, where Wigner and his devices are located. This combination of decoherence and isolation can hardly be maintained given how fast and wide decoherence processes spread. Wigner thus becomes a part of decoherence environment for Friend’s measurement.²⁵

Similarly, in the FR setup W quickly becomes part of decoherence environment for $\bar{W}$’s measurement. As a result, the reduced state of $W+\bar{W}+ \bar{L}$ approximates a proper mixture, whose summands represent two outcomes, (1) result $\bar{ok}$, $\bar{W}$ seeing $\bar{ok}$ and W seeing that $\bar{W}$ is seeing $\bar{ok}$, and (2) result $\bar{fail}$, $\bar{W}$ seeing $\bar{fail}$ and W seeing that $\bar{W}$ is seeing $\bar{fail}$. In this enlarged decoherence environment, when the reduced state approximates a proper mixture, an indeterministic event occurs. Either one or the other diagonal element is realized. Either outcome (1) or outcome (2) is actualized.

The most important factor in this picture is the emergence of a single decoherence environment in which a single actualization event occurs. To quote Healey (2024), "[w]e all share a single decoherence environment and so we can reasonably expect to be able to reach agreement on the outcomes of any quantum measurements that can ever actually be performed." Still, the lingering worry is how the two concepts–the emergence of a single decoherence environment, powered by the unitary evolution, and the indeterministic actualization of an outcome–are to be combined. In particular, why in a single decoherence environment is a single outcome actualized? One might think that this is a matter of logic: contradictories cannot be realized at the same spatiotemporal location. But exactly, the question at issue is whether outcomes like $\bar{ok}$ and $\bar{fail}$ are contradictories, i.e., the conjunction of reports on the occurrence of one and the other implies a contradiction. As in the case of the cross-perspective link, it would be better to derive the rule “one decoherence environment, then one outcome actualized" from physics, but this is not forthcoming.

The discussed approach prompts a reflection on the status of the postulate that agents should be able to form beliefs and learn from the testimony of other agents. In contrast to more objective-looking postulates (such as non-contradictoriness), this postulate is directed at a specific human aim, namely, securing human communication. There are several other postulates that have a similar status. Their focus is on us, human beings, and not necessarily on some other animals or constructs of Gedankenexperiments. It is (typically) human beings who engage in experimenting and theorizing. From this perspective, the argument that conflicting experimental results are highly improbable (because they require a subtle combination of isolation and decoherence) is a substantial step towards securing the postulate. One might demand more, that conflicting results be impossible rather than highly improbable. Given that it is known in which quantum regime the conflicting outcome might occur, the probabilistic answer seems sufficient, given the postulate’s status. The message is that in the quantum realm, the conflicting outcomes of competent observers are not logically excluded, yet their appearance can be curbed.

To sum up, we have two approaches in Healey (2022b and 2024). The first is based on a novel semantic theory; however, we do not see that it helps to resolve the testimony problem. The second approach fares better in our view, though some unclarity remain. Its essence is the macrophysical impossibility, due to the pervasiveness of decoherence, of scenarios with conflicting measurement results.

6 Conclusions

We developed a semi-formal language that pays heed to both perspectives and epistemic matters and applied it to the analysis of Wigner’s Friend scenarios. In this language we re-wrote the FR argument in a linear form: as a sequence of formulas, some of which are premises, whereas the others are derived from the former by the specified rules of reasoning. In our reconstruction the argument ends with deriving a contradiction, as it should. We then identified some weak spots in the derivation. The most natural culprit for the contradiction is (we think) a QM-based conditional concerning the results of measurements: step (a). Interestingly, this step is incorrect from the point of view of the universal unitary evolution, as well as the view that every measurement induces a state collapse of a measured system. Some proponents of relational QM might also object to conditionals involving more than one perspective, like steps (a)–(c).

There are a few morals to our findings. First, the FR argument is based on incorrect calculations. Secondly, in contrast to the claims of some proponents of perspectival approaches to QM, perspectives are not needed to block the FR argument. In particular, it turns out that the two perspective-friendly epistemic rules used in our reconstruction are valid. The value of the developed perspective-friendly notation, we think, is that it offers insight into what it takes to be a perspectivalist; this insight might facilitate a more formal analysis of perspectivalism in the future.

Furthermore, the notation (with its focus on perspectives) draws attention to the problem of testimony, permitting a precise statement of what the problem is (see, e.g., p. 25). It is not that perspectivalism or our notation created this problem; at least since the inception of the Wigner’s Friend paradox, the issue of whether the results of corresponding measurements agree has been present. And the issue requires a resolution, independently of whether it is responsible for the FR paradox (which we strongly doubt) or not. We argued that the QM formalism with unitary evolution of states, when applied to Wigner’s Friend scenarios, permits agents to have conflicting beliefs about the results of corresponding measurements. This means that testimony is a problem for every interpretation of QM that assumes unitary evolution only.

We discussed at length Healey’s (2022, 2024) attempts to resolve the problem. His semantic approach permits that in different decoherence environments conflicting reports on measurement results are true. However, there is no decoherence environment in which both conflicting reports are true. Healey’s second argument adds that reports on real experiments would typically match, as experimenters share a common decoherence environment. In the macroscopic worlds in which agents carry out experiments and compare their results, conflicting beliefs that result from the unitary evolution of states are thus extremely unlikely. We pointed to a problematic status of the assumption that a single decoherence environments permits a single actualisation event, but we left this issue for a future research.

Appendices

Appendix A States of $\bar{L}$ and L

States $|\bar{h}\rangle _{\bar{L}}$ and $|\bar{t}\rangle _{\bar{L}}$ are states of the entire system $\bar{L}$, depending on whether its subsystem R is in state $|heads\rangle _R$ or $|tails\rangle _R$, respectively.

States $|-\frac{1}{2}\rangle _{L}$ and $|+\frac{1}{2}\rangle _{L}$ are states of the whole system L, depending on whether its subsystem S is in state $|\downarrow \rangle _S$ or $|\uparrow \rangle _S$, respectively.

Other pertinent states of $\bar{L}$ and L are defined as follows:

$$\begin{aligned} & |\bar{ok}\rangle _{L} = \sqrt{\frac{1}{2}} \left( |h\rangle _{L} - |t\rangle _{L} \right) \qquad |\bar{fail}\rangle _{L} = \sqrt{\frac{1}{2}} \left( |h\rangle _{L} + |t\rangle _{L} \right) \end{aligned}$$

(1)

$$\begin{aligned} & |ok\rangle _L = \sqrt{\frac{1}{2}} \left( |-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L \right) \quad |fail\rangle _L = \sqrt{\frac{1}{2}} \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) \nonumber \\ \end{aligned}$$

(2)

These states are associated with states of $\bar{W}$ and W, respectively, since $\bar{W}$ makes a measurement on $\bar{L}$ in basis $\{|\bar{ok}\rangle _{\bar{L}}, |\bar{fail}\rangle _{\bar{L}} \}$ and W makes a measurement on L in basis $\{|ok\rangle _{L}, |fail\rangle _{L} \}$. The states of $\bar{W}$ are $|\bar{ok}\rangle _{\bar{W}} := |\bar{W}\text { sees } ``\bar{ok}"\rangle$ and $|\bar{fail}\rangle _{\bar{W}} := |\bar{W}\text { sees } ``\bar{fail}"\rangle$, whereas the states of W are $|ok\rangle _{W} := |W \text { sees } ``ok "\rangle$ and $|fail\rangle _{W} := |W \text { sees } ``fail "\rangle$.

Appendix B Descriptions of the system’s evolution

1.1 B.1 Unitary evolution everywhere and all interactions accommodated

We describe the unitary evolution of the composite system $\bar{L}+ L + \bar{W}+W$, accounting for the full sequence of interactions, in the following order: $\bar{F}$’s measurement on coin R, $\bar{F}$’s preparation of system S, F’s measurement on S, $\bar{W}$’s measurement on $\bar{L}$, and W’s measurement on L.

$$\begin{aligned} & \text {State of coin } R:\quad \sqrt{\frac{1}{3}}|heads\rangle _R + \sqrt{\frac{2}{3}}|tails\rangle _R \end{aligned}$$

(3)

$$\begin{aligned} & \text {State of } \bar{L}\!+\!S \text { after } \bar{F}\text { prepares } S: \quad \sqrt{\frac{1}{3}} |h\rangle _{\bar{L}} |\downarrow \rangle _S + \sqrt{\frac{2}{3}}|t\rangle _{\bar{L}} |\rightarrow \rangle _S \end{aligned}$$

(4)

$$\begin{aligned} & \text {State of } \bar{L}+L \text { after } F's \text { measurement on } S: \nonumber \\ & \sqrt{\frac{1}{3} }|h\rangle _{\bar{L}} |-\frac{1}{2}\rangle _L + \sqrt{\frac{1}{3}}|t\rangle _{\bar{L}} |-\frac{1}{2}\rangle _L + \sqrt{\frac{1}{3}}|t\rangle _{\bar{L}} |+\frac{1}{2}\rangle _L \end{aligned}$$

(5)

$$\begin{aligned} & \text {State of } \bar{L}+L+\bar{W}\text { after } \bar{W}'s \text { measurement on } \bar{L}: \nonumber \\ & \quad \sqrt{\frac{1}{3}}( (|h\rangle _{\bar{L}} + |t\rangle _{\bar{L}})(|-\frac{1}{2}\rangle _L |\bar{fail}\rangle _{\bar{W}}) \;+ \; \sqrt{\frac{1}{12}}( (|h\rangle _{\bar{L}} + |t\rangle _{\bar{L}})(|+\frac{1}{2}\rangle _L |\bar{fail}\rangle _{\bar{W}}) \; - \nonumber \\ & \quad - \; \sqrt{\frac{1}{12}}( (|h\rangle _{\bar{L}} - |t\rangle _{\bar{L}})(|+\frac{1}{2}\rangle _L |\bar{ok}\rangle _{\bar{W}}) \end{aligned}$$

(6)

$$\begin{aligned} & \text {State of } \bar{L}+L+\bar{W}+W \text { after } W 's \text { measurement on } L: \nonumber \\ & \quad \sqrt{\frac{3}{16}}(|h\rangle _{\bar{L}} + |t\rangle _{\bar{L}})(|-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L) |\bar{fail}\rangle _{\bar{W}} |fail\rangle _W\;+ \; \nonumber \\ & \quad + \sqrt{\frac{1}{48}} (|h\rangle _{\bar{L}} + |t\rangle _{\bar{L}})(|-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L) |\bar{fail}\rangle _{\bar{W}} |ok\rangle _W\; - \nonumber \\ & \quad - \sqrt{\frac{1}{48}} (|h\rangle _{\bar{L}} - |t\rangle _{\bar{L}})(|-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L) |\bar{ok}\rangle _{\bar{W}} |fail\rangle _W\;+ \nonumber \\ & \quad + \sqrt{\frac{1}{48}} (|h\rangle _{\bar{L}} - |t\rangle _{\bar{L}})(|-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L) |\bar{ok}\rangle _{\bar{W}} |ok\rangle _W \end{aligned}$$

(7)

We now calculate the probabilities relevant for steps (c), (b), and (a). The probability that F observes $+\frac{1}{2}$ on system S whereas $\bar{F}$ observes heads on her coin, we calculate by applying projection $P_{|h\rangle _{\bar{L}} |+\frac{1}{2}\rangle _L }$ to state (5), i.e. schematically, $\langle B5| P_{|h\rangle _{\bar{L}} |+\frac{1}{2}\rangle _L } |B5\rangle$; this is zero.

Probability of the joint measurement: of $\bar{W}$ obtaining $\bar{ok}$ on $\bar{L}$ and of F obtaining $-\frac{1}{2}$ on S, is calculated by applying the corresponding projection to state (6). This probability is zero as well. Probability that $\bar{F}$ observes tails and W observes ok we calculate by applying an appropriate projection operator to state (7) above, getting the value $\frac{1}{12}$ rather than 0.

1.2 B.2 Unitary evolution everywhere but with one measurement ignored

Here we ignore $\bar{W}$’s measurement on $\bar{L}$ and calculate the evolution of system $\bar{L}+L +W$:

$$\begin{aligned} & \text {State of } \bar{L}+L \text { after } F's \text { measurement on } S: \\ & \qquad \sqrt{\frac{1}{3}} |h\rangle _{\bar{L}} |-\frac{1}{2}\rangle _L s+ \sqrt{\frac{1}{3}}|t\rangle _{\bar{L}} |-\frac{1}{2}\rangle _L \\ & \quad + \sqrt{\frac{1}{3}}|t\rangle _{\bar{L}} |+\frac{1}{2}\rangle _L\\ & \quad \text {State of } \bar{L}+ L + W \text { after W}'s \text { measurement on } L \\ & \quad \sqrt{\frac{1}{12}} |h\rangle _{\bar{L}} (|-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L ) |fail\rangle _W + \sqrt{\frac{1}{12}} |h\rangle _{\bar{L}} (|-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L )|ok\rangle _W \\ & \quad \sqrt{\frac{1}{12}}|t\rangle _{\bar{L}} (|- \frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L)|fail\rangle _W + \sqrt{\frac{1}{12}}|t\rangle _{\bar{L}} (|- \frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L) |ok\rangle _W + \\ & \quad \sqrt{\frac{1}{12}}|t\rangle _{\bar{L}} (|- \frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L)|fail\rangle _W - \sqrt{\frac{1}{12}}|t\rangle _{\bar{L}} (|- \frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L) |ok\rangle _W = \\ & \quad \sqrt{\frac{1}{12}} |h\rangle _{\bar{L}} (|-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L ) |fail\rangle _W + \sqrt{\frac{1}{12}} |h\rangle _{\bar{L}} (|-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L )|ok\rangle _W \\ & \quad \sqrt{\frac{1}{12}}|t\rangle _{\bar{L}} (|- \frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L)|fail\rangle _W + \sqrt{\frac{1}{12}}|t\rangle _{\bar{L}} (|- \frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L)|fail\rangle _W . \end{aligned}$$

In this state the probability that $\bar{F}$ observes tails and W observes ok is zero, as summands on the right of lines 2 and 3 cancel.

1.3 B. 3 Evolution with a state collapse

1.3.1 B. 3. 1 One collapse, then unitary evolution only.

This is a scenario with just one state collapse: $\bar{F}$’s measurement on R induces a collapse of R’s state, but then each state evolves unitarily. After the first measurement R is either in state $|heads\rangle _R$ or state $|tails\rangle _R$. Given each state, $\bar{F}$ prepares S in a different state. Accordingly, $\bar{L}+S$ is in one of these states: $|h\rangle _{\bar{L}} |\downarrow \rangle _S$ (option A) or $|t\rangle _{\bar{L}} |\rightarrow \rangle _S$ (option B). We describe how each of these states evolve, given subsequent measurements.

Option A ($\bar{L}+L$ in state $|h\rangle _{\bar{L}} |-\frac{1}{2}\rangle _L$ after F’s measurement on S).

$$\begin{aligned} & \text { After } \bar{W}'s \text { measurement on } \bar{L}: \nonumber \\ & \quad \frac{1}{2} \left( \left( |h\rangle _{\bar{L}} + |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L |\bar{fail}\rangle _{\bar{W}} \right) \;+ \; \frac{1}{2} \left( \left( |h\rangle _{\bar{L}} - |t\rangle _{\bar{L}} \right) \left( |- \frac{1}{2}\rangle _L |\bar{ok}\rangle _{\bar{W}} \right) \; \right. \right. \nonumber \\\end{aligned}$$

(8)

$$\begin{aligned} & \text {After } W's \text { measurement on } L: \nonumber \\ & \quad \frac{1}{4} \left( |h\rangle _{L} + |t\rangle _{L} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{fail}\rangle _{W} |fail\rangle _W +\nonumber \\ & \frac{1}{4} \left( \left( |h\rangle _{\bar{L}} + |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L - |+\frac{1}{2}\rangle _L \right) |\bar{fail}\rangle _{\bar{W}}|ok\rangle _W +\right. \nonumber \\ & \frac{1}{4} \left( \left( |h\rangle _{\bar{L}} - |t\rangle _{\bar{L}} \right) \left( |- \frac{1}{2}\rangle _L + |+ \frac{1}{2}\rangle _L \right) |\bar{ok}\rangle _{\bar{W}} |fail\rangle _W\; + \right. \nonumber \\ & \quad + \frac{1}{4} \left( \left( |h\rangle _{L} - |t\rangle _{L} \right) \left( |- \frac{1}{2}\rangle _L - |+ \frac{1}{2}\rangle _L \right) |\bar{ok}\rangle _{\bar{W}} |ok\rangle _W\right. \end{aligned}$$

(9)

Option B $\left( \bar{L}+L\right.$ in state $\sqrt{\frac{1}{2}}|t\rangle _{\bar{L}} \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right)$ after F’s measurement on $\left. S \right)$

$$\begin{aligned} & \text {After } \bar{W}'s \text { measurement on } \bar{L}\nonumber \\ & \quad \sqrt{\frac{1}{8}} \left( |h\rangle _{\bar{L}} + |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{fail}\rangle _{\bar{W}} \nonumber \\ & \quad - \sqrt{\frac{1}{8}} \left( |h\rangle _{\bar{L}} - |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{ok}\rangle _{\bar{W}} \end{aligned}$$

(10)

$$\begin{aligned} & \text {After } W's \text { measurement on } L \nonumber \\ & \quad \sqrt{\frac{1}{8}} \left( |h\rangle _{\bar{L}} + |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{fail}\rangle _{\bar{W}}|fail\rangle _W \nonumber \\ & \quad -\sqrt{\frac{1}{8}} \left( |h\rangle _{\bar{L}} - |t\rangle _{\bar{L}} \right) \left( |-\frac{1}{2}\rangle _L + |+\frac{1}{2}\rangle _L \right) |\bar{ok}\rangle _{\bar{W}} |fail\rangle _W. \end{aligned}$$

(11)

The “collapse argument” runs as follows. $\bar{F}$ observed $tails_R$, the state collapsed to $|tals\rangle _R$. The system $\bar{L}+ L +\bar{W}+W$ evolved according to option B; so W observed fail. For, in state (11) probability of joint result, tails and ok is zero.

Note, however, that in state (9) the probability of this joint result is non-zero. Accordingly, the following probability-based argument for step (a) is invalid: I do not know in which state, (11) or (9), the composite system is, but since in each state the probability of the joint result “tails and ok” is zero, I know that if tails is measured, then fails is measured.

1.3.2 B. 3. 2 Every measurement induces a state collapse.

After $\bar{F}$’s measurement $\bar{L}+S$ is in one of these states:

$$\begin{aligned} |h\rangle _{\bar{L}} |\downarrow \rangle _S \qquad |t\rangle _{\bar{L}} |\rightarrow \rangle _S = \frac{1}{\sqrt{2}}(|t\rangle _{\bar{L}} |\downarrow \rangle _S + |t\rangle _{\bar{L}} |\uparrow \rangle _S) \end{aligned}$$

After F’s measurement on S the system $\bar{L}+L$ is in one of these states

$$\begin{aligned} |h\rangle _{L} |-\frac{1}{2}\rangle _L \qquad |t\rangle _{L} |-\frac{1}{2}\rangle _L \qquad |t\rangle _{L} |+\frac{1}{2}\rangle _L \end{aligned}$$

Then $\bar{W}$ measures on $\bar{L}$ in the basis $\{|\bar{ok}\rangle _{\bar{L}}, |\bar{fail}\rangle _{\bar{L}} \}$, the result being the following four alternative states of $L+ \bar{L}+\bar{W}$:

$$\begin{aligned} & |\bar{ok}\rangle _{L}|-\frac{1}{2}\rangle _L|\bar{ok}\rangle _{\bar{W}} \qquad |\bar{ok}\rangle _{L}|+\frac{1}{2}\rangle _L|\bar{ok}\rangle _{\bar{W}} \nonumber \\ & |\bar{fail}\rangle _{L}|-\frac{1}{2}\rangle _L|\bar{fail}\rangle _{\bar{W}} \qquad |\bar{fail}\rangle _{L}|+\frac{1}{2}\rangle _L|\bar{fail}\rangle _{\bar{W}}. \end{aligned}$$

(12)