The Gedankenexperiment

In the setup considered by Wigner (cf. Fig. 1), agent F carries out her measurement of S in a perfectly isolated lab L, so that the outcome z remains unknown to anyone else. The basic idea underlying the Gedankenexperiment we present here is to make some of the information about z available to the outside—but without lifting the isolation of L. Roughly, this is achieved by letting the initial state of S depend on a random value, r, which is known to another agent outside of L.

Box 1 specifies the proposed Gedankenexperiment as a step-wise procedure. The steps are to be executed by different agents—four in total. Two of them, the “friends” F and \(\overline {\mathrm{F}}\), are located in separate labs, denoted by L and \(\overline {\mathrm{L}}\), respectively. The two other agents, W and \(\overline {\mathrm{W}}\), are at the outside, from where they can apply measurements to L and \(\overline {\mathrm{L}}\), as shown in Fig. 2. We assume that L and \(\overline {\mathrm{L}}\) are, from the viewpoint of the agents W and \(\overline {\mathrm{W}}\), initially in a pure state, and that they remain isolated during the experiment unless the protocol explicitly prescribes a communication step or a measurement applied to them. Note that the experiment can be described within standard quantum-mechanical formalism, with each step corresponding to a fixed evolution map acting on particular subsystems (cf. the circuit diagram in the Methods section).

Table 1 Time evolution Full size table

Table 2 Measurements carried out by the agents Full size table

Fig. 2 Illustration of the Gedankenexperiment. In each round n = 0, 1, 2, … of the experiment, agent \(\overline{{F}}\) tosses a coin and, depending on the outcome r, polarises a spin particle S in a particular direction. Agent F then measures the vertical polarisation z of S. Later, agents \(\overline{{W}}\) and \({{W}}\) measure the entire labs \(\overline {{L}}\) and \({{L}}\) (where the latter includes S) to obtain outcomes \(\overline w\) and w, respectively. For the analysis of the experiment, we assume that all agents are aware of the entire procedure as specified in Box 1, but they are located at different places and therefore make different observations. Agent F, for instance, observes z but has no direct access to r. She may however use quantum theory to draw conclusions about r Full size image

As indicated by the term Gedankenexperiment, we do not claim that the experiment is technologically feasible, at least not in the form presented here. Like other thought experiments, its purpose is not to probe nature, but rather to scrutinise the consistency of our currently best available theories that describe nature—in this case quantum theory. (One may compare this to, say, the Gedankenexperiment of letting an observer cross the event horizon of a black hole. Although we do not have the technology to carry out this experiment, reasoning about it provides us with insights on relativity theory.)

Before proceeding to the analysis of the experiment, a few comments about its relation to earlier proposals are in order. In the case where r = tails, agent F receives S prepared in state \(\left| \to \right\rangle _{\mathrm{s}}.\) The first part of the experiment, prior to the measurements carried out by the agents \(\overline {\mathrm{W}}\) and W, is then equivalent to Wigner’s original experiment as described in the section Introduction2. Furthermore, adding to this the measurement of agent F’s lab by agent W, one retrieves an extension of Wigner’s experiment proposed by Deutsch6 (Fig. 1). The particular procedure of how agent F prepares the spin S in the first step described in Box 1, as well as the choice of measurements, is motivated by a construction due to Hardy7,8, known as Hardy’s Paradox. The setup considered here is also similar to a proposal by Brukner9, who used a modification of Wigner’s argument to obtain a strengthening of Bell’s theorem10 (cf. Discussion section).

Box 1: Experimental procedure The steps are repeated in rounds n = 0, 1, 2, … until the halting condition in the last step is satisfied. The numbers on the left indicate the timing of the steps, and we assume that each step takes at most one unit of time. (For example, in round n = 0, agent F starts her measurement of S at time 0:10 and completes it before time 0:11.) Definitions of the relevant state and measurement basis vectors are provided in Tables 1 and 2. At n:00 Agent \(\overline {\rm{F}}\) invokes a randomness generator (based on the measurement of a quantum system R in state \(\left| {{\rm{init}}} \right\rangle _{\rm{R}}\) as defined in Table 1) that outputs r = heads or r = tails with probabilities \(\frac{1}{3}\) and \(\frac{2}{3}\), respectively. She sets the spin S of a particle to \(\left| \downarrow \right\rangle _{\rm{S}}\) if r = heads and to \(\left| \to \right\rangle _{\rm{S}} \equiv \sqrt {1/2} \left( {\left| \downarrow \right\rangle _{\rm{S}} + \left| \uparrow \right\rangle _{\rm{S}}} \right)\) if r = tails, and sends it to F. At n:10 Agent F measures S w.r.t. the basis \(\left\{ {\left| \downarrow \right\rangle _{\rm{S}},\left| \uparrow \right\rangle _{\rm{S}}} \right\}\), recording the outcome \(z \in \left\{ { - \frac{1}{2}, + \frac{1}{2}} \right\}.\) At n:20 Agent \(\overline {\rm{W}}\) measures lab \(\overline {\rm{L}}\) w.r.t. a basis containing the vector \(\left| {\overline {{\rm{ok}}} } \right\rangle _{\overline {\rm{L}} }\) (defined in Table 2). If the outcome associated to this vector occurs he announces \(\overline w = \overline {{\rm{ok}}}\) and else \(\overline w = \overline {{\rm{fail}}}\). At n:30 Agent W measures lab L w.r.t. a basis containing the vector \(\left| {{{\rm{ok}}} } \right\rangle _{{\rm{L}} }\) (defined in Table 2). If the outcome associated to this vector occurs he announces w = ok and else w = fail. At n:40 If \(\overline w = \overline {{\rm{ok}}}\) and w = ok then the experiment is halted.

Analysis of the Gedankenexperiment

We analyse the experiment from the viewpoints of the four agents, \(\overline {\mathrm{F}}\), F, \(\overline {\mathrm{W}}\), and W, who have access to different pieces of information (cf. Fig. 2). We assume, however, that all agents are aware of the entire experimental procedure as described in Box 1, and that they all employ the same theory. One may thus think of the agents as computers that, in addition to carrying out the steps of Box 1, are programmed to draw conclusions according to a given set of rules. In the following, we specify these rules as assumptions (Boxes 2–4).

The first such assumption, Assumption (Q) is that any 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. For our purposes, it suffices to consider the special case where the state \(\left| \psi \right\rangle _{\mathrm{S}}\) that A assigns to S lies in the image of only one of the measurement operators \(\pi _x^{t_0}\), say the one with x = ξ. In this case, the Born rule asserts that the outcome x equals ξ with certainty; see Box 2.

Crucially, S may be a large and complex system, even one that itself contains agents. In fact, to start our analysis, we take the system S to be the entire lab L, which in any round n of the experiment is measured with respect to the Heisenberg operators \(\pi _{w = {\mathrm{ok}}}^{n:10}\) and \(\pi^{n:10}_{w=fail}\) defined in Table 2. Suppose that agent \(\overline{\mathrm{F}}\) wants to predict the outcome \(w\) of this measurement. To this aim, she may start her reasoning with a statement that describes the corresponding measurement.

Statement \(\overline {\mathrm{F}} ^{n:00}\): “The value w is obtained by a measurement of L w.r.t.\(\left\{ {\pi _{w = {\mathrm{ok}}}^{{n:10}},\pi _{w = {\mathrm{fail}}}^{{n:10}}} \right\},\) which is completed at time n:31.”

Here and in the following, we specify for each statement a time, denoted as a superscript, indicating when the agent could have inferred the statement. Agent \(\overline {\mathrm{F}}\)’s statement \(\overline {\mathrm{F}} ^{n:00}\) above does not depend on any observations, so the time n:00 we have assigned to it is rather arbitrary. This is, however, different for the next statement, which is based on knowledge of the value r. Suppose that agent \(\overline {\mathrm{F}}\) got r = tails as the output of the random number generator in round n. According to the experimental instructions, she will then prepare the spin S in state \(\left| \to \right\rangle _{\mathrm{S}}.\) Now, after completing the preparation, say at time n:01, she may make a second statement, taking into account that S remains unchanged until F starts her measurement at time n:10.

Statement \(\overline{\mathrm{F}}^{n:01}\): “The spin S is in state \(\left| \to \right\rangle _{\mathrm{S}}\) at time n:10.”

Agent \(\overline {\mathrm{F}}\) could conclude from this that the later state of the lab L, \(U_{{\mathrm{S}} \to {\mathrm{L}}}^{{\mathrm{10}} \to {\mathrm{20}}}\left| \to \right\rangle _{\mathrm{S}} = \sqrt {\frac{1}{2}} \left( {\left| { - \frac{1}{2}} \right\rangle _{\mathrm{L}} + \left| { + \frac{1}{2}} \right\rangle _{\mathrm{L}}} \right)\), will be orthogonal to \(\left| {{\mathrm{ok}}} \right\rangle _{\mathrm{L}}\). An equivalent way to express this is that the state \(\left| \to \right\rangle _{\mathrm{S}}\) has no overlap with the Heisenberg measurement operator corresponding to outcome w = ok, i.e.,

$$\left\langle \to \right| {\mathrm{\pi }}_{w = {\mathrm{fail}}}^{{n:10}}\left| \to \right\rangle = 1 - \left\langle \to \right| {\mathrm{\pi }}_{w = {\mathrm{ok}}}^{{n:10}}\left| \to \right\rangle = 1\;.$$ (4)

The two statements \(\overline {\mathrm{F}} ^{n:00}\) and \(\overline {\mathrm{F}} ^{n:01}\), inserted into (Q), thus imply that w = fail. We may assume that agent \(\overline {\mathrm{F}}\) draws this conclusion at time n:02 and, for later use, put it down as statement \(\overline {\mathrm{F}} ^{n:02}\) in Table 3. Similarly, agent F’s reasoning may be based upon a description of her spin measurement, which is defined by the operators \(\pi _{z = - \frac{1}{2}}^{n:10}\) and \(\pi _{z = + \frac{1}{2}}^{n:10}\) given in Table 2.

Table 3 The agents’ observations and conclusions Full size table

Statement \({\mathrm{F}}^{n:10}\): “The value z is obtained by a measurement of the spin S w.r.t.\(\left\{ {{\mathrm{\pi }}_{z = - \frac{1}{2}}^{{n:10}},\;{\mathrm{\pi }}_{z = + \frac{1}{2}}^{{n:10}}} \right\}\), which is completed at time n:11.”

Suppose now that agent F observed \(z = + \frac{1}{2}\) in round n. Since, by definition,

$$\left\langle \downarrow \right| {\mathrm{\pi }}_{z = - \frac{1}{2}}^{{n:10}}\left| \downarrow \right\rangle = 1$$ (5)

it follows from (Q) that S was not in state \(\left| \downarrow \right\rangle\), and hence that the random value r was not heads. This is statement \({\mathrm{F}}^{n:12}\) of Table 3. We proceed with agent \(\overline {\mathrm{W}}\), who may base his reasoning upon his knowledge of how the random number generator was initialised.

Statement \(\overline {\mathrm{W}} ^{n:21}\):“System R is in state \(\left| {{\mathrm{init}}} \right\rangle _{\mathrm{R}}\) at time n:00.”

Consider the event that \(\overline w = \overline {{\mathrm{ok}}}\) and \(z = - \frac{1}{2}\), as well as its complement. The Heisenberg operators of the corresponding measurement are given in Table 2. It is straightforward to verify that \(U_{{\mathrm{R}} \to \overline {\mathrm{L}} {\mathrm{S}}}^{{\mathrm{00}} \to {\mathrm{10}}}\left| {{\mathrm{init}}} \right\rangle _{\mathrm{R}} = \sqrt {\frac{1}{3}} \left| {\overline {\mathrm{h}} } \right\rangle _{\overline{\mathrm{L}}} \otimes \left| \downarrow \right\rangle _{\mathrm{S}} + \sqrt {\frac{2}{3}} \left| {\overline {\mathrm{t}} } \right\rangle _{\overline{\mathrm{L}}} \otimes \left| \to \right\rangle _{\mathrm{S}}\) is orthogonal to \(\left| {\overline {{\mathrm{ok}}} } \right\rangle _{\overline {\mathrm{L}} } \otimes \left| \downarrow \right\rangle _{\mathrm{S}}\), which implies that

$$\\ \left\langle {{\mathrm{init}}} \right| {\mathrm{\pi }}_{\left( {\overline w ,z} \right)

e \left( {\overline {{\mathrm{ok}}} , - \frac{1}{2}} \right)}^{{n:00}}\left| {{\mathrm{init}}} \right\rangle = 1 - \left\langle {{\mathrm{init}}} \right| {\mathrm{\pi }}_{\left( {\overline w ,z} \right) = \left( {\overline {{\mathrm{ok}}} , - \frac{1}{2}} \right)}^{{n:00}}\left| {{\mathrm{init}}} \right\rangle = 1\;.$$ (6)

Agent \(\overline {\mathrm{W}}\), who also uses (Q), can hence be certain that \(\left( {\overline w ,z} \right)

e \left( {\overline {{\mathrm{ok}}} , - \frac{1}{2}} \right)\). This implies that statement \(\overline {\mathrm{W}} ^{n:22}\) of Table 3 holds whenever \(\overline w = \overline {{\mathrm{ok}}}\). Furthermore, because agent \(\overline {\mathrm{W}}\) announces \(\overline w\), agent W can be certain about \(\overline {\mathrm{W}}\)’s knowledge, which justifies statement \({\mathrm{W}} ^{n:26}\) of the table. We have thus established all statements in the third column of Table 3.

For later use we also note that a simple calculation yields

$$\left\langle {{\mathrm{init}}} \right| {\mathrm{\pi }}_{(\overline w ,w) = (\overline {{\mathrm{ok}}} ,{\mathrm{ok}})}^{{n:00}}\left| {{\mathrm{init}}} \right\rangle = \frac{1}{{12}}$$ (7)

where \({\mathrm{\pi }}_{(\overline w ,w) = (\overline {{\mathrm{ok}}} ,{\mathrm{ok}})}^{n:00}\) is the Heisenberg operator belonging to the event that \(\overline w = \overline {{\mathrm{ok}}}\) and \(w = {\mathrm{ok}}\), as defined in Table 2. Hence, according to quantum mechanics, agent W can be certain that the outcome \((\overline w ,w) = (\overline {{\mathrm{ok}}} ,{\mathrm{ok}})\) occurs after finitely many rounds. This corresponds to the following statement (which can indeed be derived using (Q), as shown in the Methods section).

Statement W0:00: “I am certain that there exists a round n in which the halting condition at time n:40 is satisfied.”

The agents may now obtain further statements by reasoning about how they would reason from the viewpoint of other agents, as illustrated in Fig. 3. To enable such nested reasoning we need another assumption, Assumption (C); see Box 3.

Fig. 3 Consistent reasoning as required by Assumption (C). If a theory T (such as quantum theory) enables consistent reasoning (C) then it must allow any agent A to promote the conclusions drawn by another agent A' to his own conclusions, provided that A' has the same initial knowledge about the experiment and reasons within the same theory T. A classical example of such recursive reasoning is the muddy children puzzle (here T is just standard logic; see ref. 11 for a detailed account). The idea of using a physical theory T to describe agents who themselves use T has also appeared in thermodynamics, notably in discussions around Maxwell's demon12 Full size image

Agent F may insert agent \(\overline {\mathrm{F}}\)’s statement \(\overline {\mathrm{F}} ^{{\it{n}}{\mathrm{:02}}}\) into \({\mathrm{F}}^{{\it{n}}{\mathrm{:12}}}\), obtaining statement \({\mathrm{F}}^{{\it{n}}{\mathrm{:13}}}\) in Table 3. By virtue of (C), she may then conclude that statement \({\mathrm{F}}^{{\it{n}}{\mathrm{:14}}}\) holds, too. Similarly, \(\overline {\mathrm{W}}\) may combine this latter statement with his statement \(\overline {\mathrm{W}} ^{{\it{n}}{\mathrm{:22}}}\) to obtain \(\overline {\mathrm{W}} ^{{\it{n}}{\mathrm{:23}}}\). He could then, again using (C), conclude that statement \(\overline {\mathrm{W}} ^{{\it{n}}{\mathrm{:24}}}\) holds. Finally, agent W can insert this into his statement \({\mathrm{W}}^{{\it{n}}{\mathrm{:26}}}\) to obtain statement \({\mathrm{W}}^{{\it{n}}{\mathrm{:27}}}\) and, again with (C), statement \({\mathrm{W}}^{{\it{n}}{\mathrm{:28}}}\). This completes the derivation of all statements in Table 3.

Table 4 Interpretations of quantum theory Full size table

For the last part of our analysis, we take again agent W’s perspective. According to statement \({\mathrm{W}}^{{{n:00}}}\), the experiment has a final round n in which the halting condition will be satisfied, meaning in particular that agent \(\overline {\mathrm{W}}\) announces \(\overline w = \overline {{\mathrm{ok}}}\). Agent W infers from this that statement Wn:28 of Table 3 holds in that round, i.e., he is certain that he will observe w = fail at time n:31. However, in this final round, he will nevertheless observe w = ok! We have thus reached a contradiction—unless agent W would accept that w simultaneously admits multiple values. For our discussion below, it will be useful to introduce an explicit assumption, termed Assumption (S), which disallows this; see Box 4.

Box 2: Assumption (Q) Suppose that agent A has established that Statement A(i): “System S is in state \(\left| \psi \right\rangle _{\rm{S}}\) at time t 0 .” Suppose furthermore that agent A knows that Statement A(ii): “The value x is obtained by a measurement of S w.r.t. the family \(\{ \pi _x^{t_0}\} _{x \in {\cal X}}\) of Heisenberg operators relative to time t 0 , which is completed at time t.” If \(\left\langle \psi \right| \pi _\xi ^{t_0}\left| \psi \right\rangle = 1\) for some \(\xi \; \in \;\cal{X}\) then agent A can conclude that Statement A(iii): “I am certain that x = ξ at time t.”

Box 3: Assumption (C) Suppose that agent A has established that Statement A(i): “I am certain that agent A′, upon reasoning within the same theory as the one I am using, is certain that x = ξ at time t.” Then agent A can conclude that Statement A(ii): “I am certain that x = ξ at time t.”

Box 4: Assumption (S) Suppose that agent A has established that Statement A(i): “I am certain that x = ξ at time t.” Then agent A must necessarily deny that Statement A(ii): “I am certain that x ≠ ξ at time t.”

No-go theorem

The conclusion of the above analysis may be phrased as a no-go theorem.

Theorem 1. Any theory that satisfies assumptions (Q), (C), and (S) yields contradictory statements when applied to the Gedankenexperiment of Box 1.

To illustrate the theorem, we consider in the following different interpretations and modifications of quantum theory. Theorem 1 implies that any of them must violate either (Q), (C), or (S). This yields a natural categorisation as shown in Table 4 and discussed in the following subsections.

Theories that violate Assumption (Q)

Assumption (Q) corresponds to the quantum-mechanical Born rule. Since the assumption is concerned with the special case of probability-1 predictions only, it is largely independent of interpretational questions, such as the meaning of probabilities in general. However, the nontrivial aspect of (Q) is that it regards the Born rule as a universal law. That is, it demands that an agent A can apply the rule to arbitrary systems S around her, including large ones that may contain other agents. The specifier “around” is crucial, though: Assumption (Q) does not demand that agent A can describe herself as a quantum system. Such a requirement would indeed be overly restrictive (see ref. 13) for it would immediately rule out interpretations in the spirit of Copenhagen, according to which the observed quantum system and the observer must be distinct from each other14,15.

Assumption (Q) is manifestly violated by theories that postulate a modification of standard quantum mechanics, such as spontaneous16,17,18,19,20 and gravity-induced21,22,23 collapse models (cf. 24 for a review). These deviate from the standard theory already on microscopic scales, although the effects of the deviation typically only become noticeable in larger systems.

In some approaches to quantum mechanics, it is simply postulated that large systems are “classical”, but the physical mechanism that explains the absence of quantum features remains unspecified25. In the view described in ref. 3, for instance, the postulate says that measurement devices are infinite-dimensional systems whereas observables are finite. This ensures that coherent and incoherent superpositions in the state of a measurement device are indistinguishable. Similarly, according to the “ETH approach”26, the algebra of available observables is time-dependent and does not allow one to distinguish coherent from incoherent superpositions once a measurement has been completed. General measurements on systems that count themselves as measurement devices are thus ruled out. Another example is the “CSM ontology”27, according to which measurements must always be carried out in a “context”, which includes the measurement devices. It is then postulated that this context cannot itself be treated as a quantum system. Within all these interpretations, the Born rule still holds “for all practical purposes”, but is no longer a universally applicable law in the sense of Assumption (Q) (see the discussion in ref. 4).

Another class of theories that violate (Q), although in a less obvious manner, are particular “hidden-variable (HV) interpretations”28, with “Bohmian mechanics” as the most prominent example29,30,31. According to the common understanding, Bohmian mechanics is a “theory of the universe” rather than a theory about subsystems32. This means that agents who apply the theory must in principle always take an outside perspective on the entire universe, describing themselves as part of it. This outside perspective is identical for all agents, which ensures consistency and hence the validity of Assumption (C). However, because (S) is satisfied, too, it follows from Theorem 1 that (Q) must be violated (see the Methods section for more details).

Theories that violate Assumption (C)

If a theory satisfies (Q) and (S) then, by Theorem 1, it must violate (C). This conclusion applies to a wide range of common readings of quantum mechanics, including most variants of the Copenhagen interpretation. One concrete example is the “consistent histories” (CH) formalism33,34,35, which is also similar to the “decoherent histories” approach36,37. Another class of examples are subjectivistic interpretations, which regard statements about outcomes of measurements as personal to an agent, such as “relational quantum mechanics”38, “QBism”39,40, or the approach proposed in ref. 9 (see Methods section for a discussion of the CH formalism as well as QBism).

The same conclusion applies to HV interpretations of quantum mechanics, provided that we use them to describe systems around us rather than the universe as a whole (contrasting the paradigm of Bohmian mechanics discussed above). In this case, both (Q) and (S) hold by construction. This adds another item to the long list of no-go results for HV interpretations: they cannot be local10, they must be contextual41,42, and they violate freedom of choice43,44. Theorem 1 entails that they also violate (C). In particular, there cannot exist an assignment of values to the HVs that is consistent with the agents’ conclusions.

Theories that violate Assumption (S)

Although intuitive, (S) is not implied by the bare mathematical formalism of quantum mechanics. Among the theories that abandon the assumption are the “relative state formulation” and “many-worlds interpretations”6,45,46,47,48. According to the latter, any quantum measurement results in a branching into different “worlds”, in each of which one of the possible measurement outcomes occurs. Further developments and variations include the “many-minds interpretation”49,50 and the “parallel lives theory”51. A related concept is “quantum Darwinism”52, whose purpose is to explain the perception of classical measurement outcomes in a unitarily evolving universe.

While many-worlds interpretations manifestly violate (S), their compatibility with (Q) and (C) depends on how one defines the branching. If one regards it as an objective process, (Q) may be violated (cf. the example in Section 10 of ref. 53). It is also questionable whether (Q) can be upheld if branches do not persist over time (cf. the no-histories view described in ref. 54).

Implicit assumptions

Any no-go result, as for example Bell’s theorem10, is phrased within a particular framework that comes with a set of built-in assumptions. Hence it is always possible that a theory evades the conclusions of the no-go result by not fulfilling these implicit assumptions. Here we briefly discuss how Theorem 1 compares in this respect to other results in the literature.

Bell’s original work10 treats probabilities as a primitive notion. Similarly, many of the modern arguments in quantum foundations employ probabilistic frameworks55,56,57,58,59,60,61,62. In contrast, probabilities are not used in the argument presented here—although Assumption (Q) is of course motivated by the idea that a statement can be regarded as “certain” if the Born rule assigns probability-1 to it. In particular, Theorem 1 does not depend on how probabilities different from 1 are interpreted.

Another distinction is that the framework used here treats all statements about observations as subjective, i.e., they are always defined relative to an agent. This avoids the a priori assumption that measurement outcomes obtained by different agents simultaneously have definite values. (Consider for example Wigner’s original setup described in section Introduction. Even when Assumptions (C) and (S) hold, agent W is not forced to assign a definite value to the outcome z observed by agent F.) The assumption of simultaneous definiteness is otherwise rather common. It not only enters the proof of Bell’s theorem10 but also the aforementioned arguments based on probabilistic frameworks.

Nevertheless, in our considerations, we used concepts such as that of an “agent” or of “time”. It is conceivable that the conclusions of Theorem 1 can be avoided by theories that provide a nonstandard understanding of these concepts. We are, however, not aware of any concrete examples of such theories.