quantum theory cannot consistently describe the use of itself

Quantum theory cannot consistently describe the use of itself

Daniela Frauchiger and Renato Renner∗

Institute for Theoretical Physics, ETH Zurich, 8093 Zurich; Switzerland

Abstract

Quantum theory provides an extremely accurate description of fundamental processes in physics.It thus seems likely that the theory is applicable beyond the, mostly microscopic, domain in which ithas been tested experimentally. Here we propose a Gedankenexperiment to investigate the questionwhether quantum theory can, in principle, have universal validity. The idea is that, if the answer wasyes, it must be possible to employ quantum theory to model complex systems that include agentswho are themselves using quantum theory. Analysing the experiment under this presumption, wefind that one agent, upon observing a particular measurement outcome, must conclude that anotheragent has predicted the opposite outcome with certainty. The agents’ conclusions, although allderived within quantum theory, are thus inconsistent. This indicates that quantum theory cannotbe extrapolated to complex systems, at least not in a straightforward manner.

Introduction

Direct experimental tests of quantum theory are mostly restricted to microscopic domains. Nevertheless,quantum theory is commonly regarded as beging (almost) universally valid. It is not only used todescribe fundamental processes in particle and solid state physics, but also, for instance, to explain thecosmic microwave background or the radiation of black holes.

The presumption that the validity of quantum theory extends to larger scales has remarkable con-sequences, as noted already in 1935 by Schrodinger [1]. His famous example consisted of a cat that isbrought into a state corresponding to a superposition of two macroscopically entirely different states,one in which it is dead and one in which it is alive. Schrodinger pointed out, however, that suchmacroscopic superposition states do not represent anything contradictory in themselves.

This view was not shared by everyone. In 1967, Wigner proposed an argument, known as theWigner’s Friend Paradox, which should show that “quantum mechanics cannot have unlimited valid-ity” [2]. His idea was to consider the views of two different observers in an experiment analogous tothe one depicted by Fig. 1. One observer, called agent F, measures the vertical polarisation z of a spinone-half particle S, such as a silver atom. Upon observing the outcome, which is either z = − 1

2 orz = + 1

2 , agent F would thus say that S is in state

ψS = |↓〉S or ψS = |↑〉S , (1)

respectively. The other observer, agent W, has no direct access to the outcome z observed by hisfriend F. Agent W could instead model agent F’s lab as a big quantum system, L ≡ S⊗D⊗ F, whichcontains the spin S as a subsystem, another subsystem, D, for the friend’s measurement devices andeverything else connected to them, as well as a subsystem F that includes the friend herself. Supposethat, from agent W’s perspective, the lab L is initially in a pure state and that it remains isolated duringagent F’s spin measurement experiment. (One may object that these assumptions are unrealistic [3],but, crucially, the laws of quantum theory do not preclude that they be satisfied to arbitrarily good

∗[email protected]

1

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Figure 1: Wigner’s and Deutsch’s arguments. Agent F measures the spin S of a silver atom,obtaining outcome z. From F’s perspective, S is then in one of the two pure states ψS given in (1).Agent W, who is outside of F’s lab, may instead regard that lab, including the agent F, as a big quantumsystem L (orange box). Wigner argued that, having no access to z, he would assign a superpositionstate ΨL of the form (3) to L [2]. Deutsch later noted that agent W could in principle test this stateassignment by applying a carefully designed measurement to L [6].

approximation [4].) Translated to quantum mechanics, this means that the dependence of the finalstate of L on the initial state of S is described by a linear map of the form

US→L =

{|↓〉S 7→ |− 1

2 〉L ≡ |↓〉S ⊗ |“z=− 12

”〉D⊗ |“ψS= |↓〉”〉F

|↑〉S 7→ |+ 12 〉L ≡ |↑〉S ⊗ |“z=+ 1

2”〉

D⊗ |“ψS= |↑〉”〉F .

(2)

Here |“z=− 12

”〉D

and |“z=+ 12

”〉D

denote states of D depending on the measurement outcome z shownby the devices within the lab. Analogously, |“ψS= |↓〉”〉F and |“ψS= |↑〉”〉F are states of F, which dependon the friend’s own knowledge ψS ; cf. (1). Now, suppose agent W knew that the spin was initialisedto |→〉S ≡

√1/2(|↓〉S + |↑〉S) before agent F measured it. Then, by linearity, the final state that agent W

would assign to L is

ΨL =√

1/2(|− 1

2 〉L + |+ 12 〉L), (3)

i.e., a linear superposition of the two macroscopically distinct states defined in (2). To compare this toagent F’s view (1), one must consider the restriction of (3) to S. The latter is a maximally mixed state,and thus obviously different from agent F’s pure state assignment (1). But, crucially, the difference canbe explained by the two agents’ distinct level of knowledge: Agent F has observed z and hence knows thefinal spin direction, whereas agent W is ignorant about it [5]. Consequently, although the superpositionstate (3) may appear “absurd” [2], it does not contradict (1). For this reason, the Wigner’s FriendParadox cannot be regarded as an argument that rules out quantum mechanics as a universally validtheory.

In this work we propose a Gedankenexperiment that extends Wigner’s setup. It consists of agentswho are using quantum theory to reason about other agents who are also using quantum theory. Ourmain finding is that such a self-referential use of the theory yields contradictory claims. This resultcan be phrased as a no-go theorem (Theorem 1). It asserts that three natural-sounding assumptions,(Q), (C), and (S), cannot all be valid. Assumption (Q) captures the universal validity of quantumtheory (or, more specifically, that an agent can be certain that a given proposition holds whenever thequantum-mechanical Born rule assigns probability 1 to it). Assumption (C) demands consistency, inthe sense that the different agents’ predictions are not contradictory. Finally, (S) is the requirementthat, from the viewpoint of an agent who carries out a particular measurement, this measurement hasone single outcome. The theorem itself is neutral in the sense that it does not tell us which of thesethree assumptions is wrong. However, it implies that any specific interpretation of quantum theory,when applied to the Gedankenexperiment, will necessarily conflict with at least one of them. This givesa way to test and categorise interpretations of quantum theory.

2

Results

The Gedankenexperiment

In the setup considered by Wigner (cf. Fig. 1), agent F carries out her measurement of S in a perfectlyisolated lab L, so that the outcome z remains unknown to anyone else. The basic idea underlyingthe Gedankenexperiment we present here is to make some of the information about z available to theoutside — but without lifting the isolation of L. Roughly, this is achieved by letting the initial state ofS 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 beexecuted by different agents — four in total. Two of them, the “friends” F and F, are located in separatelabs, denoted by L and L, respectively. The two other agents, W and W, are at the outside, from wherethey can apply measurements to L and L, as shown in Fig. 2. We assume that L and L are, from theviewpoint of the agents W and W, initially in a pure state, and that they remain isolated during theexperiment unless the protocol explicitly prescribes a communication step or a measurement appliedto 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 circuitdiagram in the appendix).

Experimental Procedure

The steps are repeated in rounds n = 0, 1, 2, . . . until the halting condition in the last step is satisfied.

At n:00 Agent F invokes a randomness generator (based on the measurement of a quantum system R instate |init〉R as defined in Table 1) that outputs r = heads or r = tails with probabilities 1/3 and 2/3,respectively. She sets the spin S of a particle to |↓〉S if r = heads and to |→〉S ≡

√1/2(|↓〉S + |↑〉S) if

r = tails, and sends it to F.

At n:10 Agent F measures S w.r.t. the basis {|↓〉S, |↑〉S}, recording the outcome z ∈ {− 12,+ 1

2}.

At n:20 Agent W measures lab L w.r.t. a basis containing the vector |ok〉L (defined in Table 2). If theoutcome associated to this vector occurs he announces w = ok and else w = fail.

At n:30 Agent W measures lab L w.r.t. a basis containing the vector |ok〉L (defined in Table 2). If theoutcome associated to this vector occurs he announces w = ok and else w = fail.

At n:40 If w = ok and w = ok then the experiment is halted.

Box 1: Steps of the Gedankenexperiment. 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 Fstarts her measurement of S at time 0:10 and completes it before time 0:11. Definitions of the relevantstate and measurement basis vectors are provided in Tables 1 and 2.

As indicated by the term Gedankenexperiment, we do not claim that the experiment is technologicallyfeasible, at least not in the form presented here. Like other thought experiments, its purpose is notto probe nature, but rather to scrutinise the consistency of our currently best available theories thatdescribe nature — in this case quantum theory. (One may compare this to, say, the Gedankenexperimentof letting an observer cross the event horizon of a black hole. Although we do not have the technologyto 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 earlierproposals are in order. In the case where r = tails, agent F receives S prepared in state |→〉S. Thefirst part of the experiment, prior to the measurements carried out by the agents W and W, is thenequivalent to Wigner’s original experiment as described in the introduction [2]. Furthermore, adding tothis the measurement of agent F’s lab by agent W, one retrieves an extension of Wigner’s experimentproposed by Deutsch [6] (Fig. 1). The particular procedure of how agent F prepares the spin S in thefirst step described in Box 1 is motivated by a construction due to Hardy [7, 8]. The setup consideredhere is also similar to a proposal by Brukner [9], who used a modification of Wigner’s argument to

3

r = tails<latexit sha1_base64="Ttxwd5kPt1Rw/cxxIeWK31NCJU4=">AAACD3icbVDLSgMxFL3js9bXqEs3wSK4KjNudKFYcOOygn1AO5RMmmlDk8yQZApl6NIPcO9Wf8Gd6NJP8A8Ef8JM24W2HggczrmvnDDhTBvP+3SWlldW19YLG8XNre2dXXdvv67jVBFaIzGPVTPEmnImac0ww2kzURSLkNNGOLjO/caQKs1ieWdGCQ0E7kkWMYKNlTquqy7bApu+EpnBjOtxxy15ZW8CtEj8GSldvX/fIwCodtyvdjcmqaDSEI61bvleYoIMK8MIp+NiO9U0wWSAe7RlqcSC6iCbXD5Gx1bpoihW9kmDJurvjgwLrUcitJX5lXrey8V/PckIjRQmc/tNdB5kTCapoZJM10cpRyZGeTioyxQlho8swUQx+wNE+tiOMTbCoo3Gnw9ikdRPy75X9m/9UuUCpijAIRzBCfhwBhW4gSrUgMAQHuEJnp0H58V5dd6mpUvOrOcA/sD5+AHrAJ+x</latexit><latexit sha1_base64="uUiu1LChDPkWGtuxrvOgKdBdHaY=">AAACD3icbVDLSgMxFM34rPU16lKQYBFclRk3ulAsuHHZgn1AO5RMeqcNzWSGJFMoQ5d+gHu3+gu6El36Cf6BoB9hpu1CWw8EDufcV44fc6a043xYC4tLyyurubX8+sbm1ra9s1tTUSIpVGnEI9nwiQLOBFQ10xwasQQS+hzqfv8q8+sDkIpF4kYPY/BC0hUsYJRoI7VtW160QqJ7Mkw1YVyN2nbBKTpj4HniTknh8u3r9uC58l1u25+tTkSTEISmnCjVdJ1YeymRmlEOo3wrURAT2iddaBoqSAjKS8eXj/CRUTo4iKR5QuOx+rsjJaFSw9A3ldmVatbLxH89wSgEktCZ/To481Im4kSDoJP1QcKxjnAWDu4wCVTzoSGESmZ+gGmPmDHaRJg30bizQcyT2knRdYpuxS2UztEEObSPDtExctEpKqFrVEZVRNEA3aMH9GjdWU/Wi/U6KV2wpj176A+s9x+weaGk</latexit><latexit sha1_base64="uUiu1LChDPkWGtuxrvOgKdBdHaY=">AAACD3icbVDLSgMxFM34rPU16lKQYBFclRk3ulAsuHHZgn1AO5RMeqcNzWSGJFMoQ5d+gHu3+gu6El36Cf6BoB9hpu1CWw8EDufcV44fc6a043xYC4tLyyurubX8+sbm1ra9s1tTUSIpVGnEI9nwiQLOBFQ10xwasQQS+hzqfv8q8+sDkIpF4kYPY/BC0hUsYJRoI7VtW160QqJ7Mkw1YVyN2nbBKTpj4HniTknh8u3r9uC58l1u25+tTkSTEISmnCjVdJ1YeymRmlEOo3wrURAT2iddaBoqSAjKS8eXj/CRUTo4iKR5QuOx+rsjJaFSw9A3ldmVatbLxH89wSgEktCZ/To481Im4kSDoJP1QcKxjnAWDu4wCVTzoSGESmZ+gGmPmDHaRJg30bizQcyT2knRdYpuxS2UztEEObSPDtExctEpKqFrVEZVRNEA3aMH9GjdWU/Wi/U6KV2wpj176A+s9x+weaGk</latexit><latexit sha1_base64="k6+GrEFVe5CgbeCEWWSVGpJXw70=">AAACD3icbVDLSsNAFL2pr1pfUZduBovgqiRudKFQcOOygn1AW8pketMOnUzCzKRQQj/CvVv9BXfi1k/wD/wMJ20WWj0wcDjnvuYEieDaeN6nU1pb39jcKm9Xdnb39g/cw6OWjlPFsMliEatOQDUKLrFpuBHYSRTSKBDYDia3ud+eotI8lg9mlmA/oiPJQ86osdLAddVNL6JmrKLMUC70fOBWvZq3APlL/IJUoUBj4H71hjFLI5SGCap11/cS08+oMpwJnFd6qcaEsgkdYddSSSPU/Wxx+ZycWWVIwljZJw1ZqD87MhppPYsCW5lfqVe9XPzXk5xhqChb2W/Cq37GZZIalGy5PkwFMTHJwyFDrpAZMbOEMsXtDwgbUzvG2AgrNhp/NYi/pHVR872af+9X69dFSGU4gVM4Bx8uoQ530IAmMJjCEzzDi/PovDpvzvuytOQUPcfwC87HNztZnSE=</latexit> w 6= ok

<latexit sha1_base64="ll+M5lzoRbA2FFrb7/hPEYm9d/k=">AAACEXicbVC7TgJBFL3rE/GFmNjYTCQmVmTXRgsLEhtLTOSRACGzw12YMDu7zsyqZMNX2NvqL9gZW7/AztLPcBYoFDzJJCfn3NccPxZcG9f9dJaWV1bX1nMb+c2t7Z3dwl6xrqNEMayxSESq6VONgkusGW4ENmOFNPQFNvzhZeY37lBpHskbM4qxE9K+5AFn1FipWyjek7bEW9IOqRmoMI2G426h5JbdCcgi8WakVDmoiy8AqHYL3+1exJIQpWGCat3y3Nh0UqoMZwLH+XaiMaZsSPvYslTSEHUnndw+JsdW6ZEgUvZJQybq746UhlqPQt9WZifqeS8T//UkZxgoyub2m+C8k3IZJwYlm64PEkFMRLJ4SI8rZEaMLKFMcfsDwgbUjjE2xLyNxpsPYpHUT8ueW/auvVLlAqbIwSEcwQl4cAYVuIIq1IDBAzzBM7w4j86r8+a8T0uXnFnPPvyB8/EDAqSfkg==</latexit><latexit sha1_base64="kJgLoN6WKiZ69kXiYDk0feckoqs=">AAACEXicbVC7TsMwFHXKq5RXKBILA4YKiYUqYYGBoRILY5HoQ2qjynFvWquOE2wHqKJ+BTsr/AIbYuUL+AM+gA/AaTtAy5EsHZ1zXz5+zJnSjvNp5RYWl5ZX8quFtfWNzS17u1hXUSIp1GjEI9n0iQLOBNQ00xyasQQS+hwa/uAy8xt3IBWLxI0exuCFpCdYwCjRRurYxXvcFnCL2yHRfRmm0WDUsUtO2RkDzxN3SkqV3Tr/3j84qXbsr3Y3okkIQlNOlGq5Tqy9lEjNKIdRoZ0oiAkdkB60DBUkBOWl49tH+MgoXRxE0jyh8Vj93ZGSUKlh6JvK7EQ162Xiv55gFAJJ6Mx+HZx7KRNxokHQyfog4VhHOIsHd5kEqvnQEEIlMz/AtE/MGG1CLJho3Nkg5kn9tOw6ZffaLVUu0AR5tIcO0TFy0RmqoCtURTVE0QN6Qs/oxXq0Xq03631SmrOmPTvoD6yPH7oqoBM=</latexit><latexit sha1_base64="kJgLoN6WKiZ69kXiYDk0feckoqs=">AAACEXicbVC7TsMwFHXKq5RXKBILA4YKiYUqYYGBoRILY5HoQ2qjynFvWquOE2wHqKJ+BTsr/AIbYuUL+AM+gA/AaTtAy5EsHZ1zXz5+zJnSjvNp5RYWl5ZX8quFtfWNzS17u1hXUSIp1GjEI9n0iQLOBNQ00xyasQQS+hwa/uAy8xt3IBWLxI0exuCFpCdYwCjRRurYxXvcFnCL2yHRfRmm0WDUsUtO2RkDzxN3SkqV3Tr/3j84qXbsr3Y3okkIQlNOlGq5Tqy9lEjNKIdRoZ0oiAkdkB60DBUkBOWl49tH+MgoXRxE0jyh8Vj93ZGSUKlh6JvK7EQ162Xiv55gFAJJ6Mx+HZx7KRNxokHQyfog4VhHOIsHd5kEqvnQEEIlMz/AtE/MGG1CLJho3Nkg5kn9tOw6ZffaLVUu0AR5tIcO0TFy0RmqoCtURTVE0QN6Qs/oxXq0Xq03631SmrOmPTvoD6yPH7oqoBM=</latexit><latexit sha1_base64="D5mKxgBWJYdhD/H4BgOs9OEsfk4=">AAACEXicbVC7TsMwFHV4lvIKZWSxqJCYqoQFBoZKLIxFog+pqSrHvWmt2k6wHaCK+hXsrPALbIiVL+AP+AycNgO0HMnS0Tn35RMmnGnjeV/Oyura+sZmaau8vbO7t+8eVFo6ThWFJo15rDoh0cCZhKZhhkMnUUBEyKEdjq9yv30PSrNY3ppJAj1BhpJFjBJjpb5becCBhDscCGJGSmTxeNp3q17NmwEvE78gVVSg0Xe/g0FMUwHSUE607vpeYnoZUYZRDtNykGpICB2TIXQtlUSA7mWz26f4xCoDHMXKPmnwTP3dkRGh9USEtjI/US96ufivJxmFSBG6sN9EF72MySQ1IOl8fZRybGKcx4MHTAE1fGIJoYrZH2A6InaMsSGWbTT+YhDLpHVW872af+NX65dFSCV0hI7RKfLROaqja9RATUTRI3pGL+jVeXLenHfnY1664hQ9h+gPnM8fL42dmg==</latexit>

z = + 12

<latexit sha1_base64="fII6jZtHN6rFb1EEv78zS59uEV0=">AAACC3icbVDLSsNAFL2pr1pfVZdugkEQhJLpRhcKBTcuq9gHtLFOppN26GQSZiZCDfkE17rVX3Anbv0I/8DPcNK60NYDA4dz7mPu8WPOlHbdT6uwsLi0vFJcLa2tb2xulbd3mipKJKENEvFItn2sKGeCNjTTnLZjSXHoc9ryR+e537qjUrFIXOtxTL0QDwQLGMHaSDf3Z0fdQGKSoiytZr2y41bcCex5gn6IU3Mer24BoN4rf3X7EUlCKjThWKkOcmPtpVhqRjjNSt1E0RiTER7QjqECh1R56eTXmX1glL4dRNI8oe2J+rsjxaFS49A3lSHWQzXr5eK/nmCE5kfN7NfBiZcyESeaCjJdHyTc1pGdB2P3maRE87EhmEhmLrDJEJsx2sRXMtGg2SDmSbNaQW4FXSKndgpTFGEP9uEQEBxDDS6gDg0gIOEJnuHFerBerTfrfVpasH56duEPrI9vbWGdDA==</latexit><latexit sha1_base64="ta+BI70a+/J0GS+UFX6Ck4zc1a4=">AAACC3icbVDLSsNAFJ34rPVVdekmGARBKJludKFQcOOySl/Q1jqZ3rRDJ5MwMxFqyCe41q0u3boTt36Ef+BnOGm70NYDA4dz7mPu8SLOlHbdL2thcWl5ZTW3ll/f2NzaLuzs1lUYSwo1GvJQNj2igDMBNc00h2YkgQQeh4Y3vMj8xh1IxUJR1aMIOgHpC+YzSrSRbu7Pj9u+JDTBaVJKuwXHLbpj2PMET4lTdh6vq7evUOkWvtu9kMYBCE05UaqF3Uh3EiI1oxzSfDtWEBE6JH1oGSpIAKqTjH+d2odG6dl+KM0T2h6rvzsSEig1CjxTGRA9ULNeJv7rCUYhO2pmv/ZPOwkTUaxB0Ml6P+a2Du0sGLvHJFDNR4YQKpm5wKYDYsZoE1/eRINng5gn9VIRu0V8hZ3yGZogh/bRATpCGJ2gMrpEFVRDFEn0hJ7Ri/VgvVnv1sekdMGa9uyhP7A+fwBmJp5v</latexit><latexit sha1_base64="ta+BI70a+/J0GS+UFX6Ck4zc1a4=">AAACC3icbVDLSsNAFJ34rPVVdekmGARBKJludKFQcOOySl/Q1jqZ3rRDJ5MwMxFqyCe41q0u3boTt36Ef+BnOGm70NYDA4dz7mPu8SLOlHbdL2thcWl5ZTW3ll/f2NzaLuzs1lUYSwo1GvJQNj2igDMBNc00h2YkgQQeh4Y3vMj8xh1IxUJR1aMIOgHpC+YzSrSRbu7Pj9u+JDTBaVJKuwXHLbpj2PMET4lTdh6vq7evUOkWvtu9kMYBCE05UaqF3Uh3EiI1oxzSfDtWEBE6JH1oGSpIAKqTjH+d2odG6dl+KM0T2h6rvzsSEig1CjxTGRA9ULNeJv7rCUYhO2pmv/ZPOwkTUaxB0Ml6P+a2Du0sGLvHJFDNR4YQKpm5wKYDYsZoE1/eRINng5gn9VIRu0V8hZ3yGZogh/bRATpCGJ2gMrpEFVRDFEn0hJ7Ri/VgvVnv1sekdMGa9uyhP7A+fwBmJp5v</latexit><latexit sha1_base64="ij40H2LA/MFL+fJOtlOFymeoRrU=">AAACC3icbVDLSgMxFM3UV62vqks3wSIIQpl0owuFghuXFewD2loy6Z02NJMZkoxQh/kE9271F9yJWz/CP/AzzLSz0NYDgcM595F7vEhwbVz3yymsrK6tbxQ3S1vbO7t75f2Dlg5jxaDJQhGqjkc1CC6habgR0IkU0MAT0PYm15nffgCleSjvzDSCfkBHkvucUWOl+8ers56vKEtImtTSQbniVt0Z8DIhOamgHI1B+bs3DFkcgDRMUK27xI1MP6HKcCYgLfViDRFlEzqCrqWSBqD7yezXKT6xyhD7obJPGjxTf3ckNNB6Gni2MqBmrBe9TPzXk5xBdtTCfuNf9BMuo9iAZPP1fiywCXEWDB5yBcyIqSWUKW4vwGxM7Rhj4yvZaMhiEMukVasSt0puSaV+mYdUREfoGJ0igs5RHd2gBmoihhR6Ri/o1Xly3px352NeWnDynkP0B87nDwZym2A=</latexit>

r = tails<latexit sha1_base64="Ttxwd5kPt1Rw/cxxIeWK31NCJU4=">AAACD3icbVDLSgMxFL3js9bXqEs3wSK4KjNudKFYcOOygn1AO5RMmmlDk8yQZApl6NIPcO9Wf8Gd6NJP8A8Ef8JM24W2HggczrmvnDDhTBvP+3SWlldW19YLG8XNre2dXXdvv67jVBFaIzGPVTPEmnImac0ww2kzURSLkNNGOLjO/caQKs1ieWdGCQ0E7kkWMYKNlTquqy7bApu+EpnBjOtxxy15ZW8CtEj8GSldvX/fIwCodtyvdjcmqaDSEI61bvleYoIMK8MIp+NiO9U0wWSAe7RlqcSC6iCbXD5Gx1bpoihW9kmDJurvjgwLrUcitJX5lXrey8V/PckIjRQmc/tNdB5kTCapoZJM10cpRyZGeTioyxQlho8swUQx+wNE+tiOMTbCoo3Gnw9ikdRPy75X9m/9UuUCpijAIRzBCfhwBhW4gSrUgMAQHuEJnp0H58V5dd6mpUvOrOcA/sD5+AHrAJ+x</latexit><latexit sha1_base64="uUiu1LChDPkWGtuxrvOgKdBdHaY=">AAACD3icbVDLSgMxFM34rPU16lKQYBFclRk3ulAsuHHZgn1AO5RMeqcNzWSGJFMoQ5d+gHu3+gu6El36Cf6BoB9hpu1CWw8EDufcV44fc6a043xYC4tLyyurubX8+sbm1ra9s1tTUSIpVGnEI9nwiQLOBFQ10xwasQQS+hzqfv8q8+sDkIpF4kYPY/BC0hUsYJRoI7VtW160QqJ7Mkw1YVyN2nbBKTpj4HniTknh8u3r9uC58l1u25+tTkSTEISmnCjVdJ1YeymRmlEOo3wrURAT2iddaBoqSAjKS8eXj/CRUTo4iKR5QuOx+rsjJaFSw9A3ldmVatbLxH89wSgEktCZ/To481Im4kSDoJP1QcKxjnAWDu4wCVTzoSGESmZ+gGmPmDHaRJg30bizQcyT2knRdYpuxS2UztEEObSPDtExctEpKqFrVEZVRNEA3aMH9GjdWU/Wi/U6KV2wpj176A+s9x+weaGk</latexit><latexit sha1_base64="uUiu1LChDPkWGtuxrvOgKdBdHaY=">AAACD3icbVDLSgMxFM34rPU16lKQYBFclRk3ulAsuHHZgn1AO5RMeqcNzWSGJFMoQ5d+gHu3+gu6El36Cf6BoB9hpu1CWw8EDufcV44fc6a043xYC4tLyyurubX8+sbm1ra9s1tTUSIpVGnEI9nwiQLOBFQ10xwasQQS+hzqfv8q8+sDkIpF4kYPY/BC0hUsYJRoI7VtW160QqJ7Mkw1YVyN2nbBKTpj4HniTknh8u3r9uC58l1u25+tTkSTEISmnCjVdJ1YeymRmlEOo3wrURAT2iddaBoqSAjKS8eXj/CRUTo4iKR5QuOx+rsjJaFSw9A3ldmVatbLxH89wSgEktCZ/To481Im4kSDoJP1QcKxjnAWDu4wCVTzoSGESmZ+gGmPmDHaRJg30bizQcyT2knRdYpuxS2UztEEObSPDtExctEpKqFrVEZVRNEA3aMH9GjdWU/Wi/U6KV2wpj176A+s9x+weaGk</latexit><latexit sha1_base64="k6+GrEFVe5CgbeCEWWSVGpJXw70=">AAACD3icbVDLSsNAFL2pr1pfUZduBovgqiRudKFQcOOygn1AW8pketMOnUzCzKRQQj/CvVv9BXfi1k/wD/wMJ20WWj0wcDjnvuYEieDaeN6nU1pb39jcKm9Xdnb39g/cw6OWjlPFsMliEatOQDUKLrFpuBHYSRTSKBDYDia3ud+eotI8lg9mlmA/oiPJQ86osdLAddVNL6JmrKLMUC70fOBWvZq3APlL/IJUoUBj4H71hjFLI5SGCap11/cS08+oMpwJnFd6qcaEsgkdYddSSSPU/Wxx+ZycWWVIwljZJw1ZqD87MhppPYsCW5lfqVe9XPzXk5xhqChb2W/Cq37GZZIalGy5PkwFMTHJwyFDrpAZMbOEMsXtDwgbUzvG2AgrNhp/NYi/pHVR872af+9X69dFSGU4gVM4Bx8uoQ530IAmMJjCEzzDi/PovDpvzvuytOQUPcfwC87HNztZnSE=</latexit>

z = + 12


w = ok<latexit sha1_base64="kamY2cjvZb9ReC5tCIE37+IeXK8=">AAACDnicbVDLSgMxFL1TX7U+OurSTbAIrsqMG10oFty4rGAf0JaSSTNtaCYZkoxShi7du3erv+BOxJ2/4B8I/oSZtgttPRA4nHNfOUHMmTae9+nklpZXVtfy64WNza3toruzW9cyUYTWiORSNQOsKWeC1gwznDZjRXEUcNoIhpeZ37ilSjMpbswopp0I9wULGcHGSl23eIfOUTvCZqCiVA7HXbfklb0J0CLxZ6R08f59jwCg2nW/2j1JkogKQzjWuuV7semkWBlGOB0X2ommMSZD3KctSwWOqO6kk8PH6NAqPRRKZZ8waKL+7khxpPUoCmxldqKe9zLxX08wQkOFydx+E552UibixFBBpuvDhCMjUZYN6jFFieEjSzBRzP4AkQG2Y4xNsGCj8eeDWCT147Lvlf1rv1Q5gynysA8HcAQ+nEAFrqAKNSCQwCM8wbPz4Lw4r87btDTnzHr24A+cjx8iJZ6p</latexit><latexit sha1_base64="Dfq6iTzXZhde6iJPe2zGdIq4Pco=">AAACDnicbVC7SgNBFJ2NrxgfWbUUZDAIVmHXRgvFgI1lAuYBSQizk9lkyDyWmVklLCnt7W31F8RGxM5f8A8E/QhnkxSaeGDgcM59zQkiRrXxvA8ns7C4tLySXc2trW9s5t2t7ZqWscKkiiWTqhEgTRgVpGqoYaQRKYJ4wEg9GFykfv2aKE2luDLDiLQ56gkaUoyMlTpu/gaewRZHpq94Igejjlvwit4YcJ74U1I4f/u63XuufJc77merK3HMiTCYIa2bvheZdoKUoZiRUa4VaxIhPEA90rRUIE50OxkfPoIHVunCUCr7hIFj9XdHgrjWQx7YyvREPeul4r+eoJiECuGZ/SY8aSdURLEhAk/WhzGDRsI0G9ilimDDhpYgrKj9AcR9ZMcYm2DORuPPBjFPakdF3yv6Fb9QOgUTZMEu2AeHwAfHoAQuQRlUAQYxuAcP4NG5c56cF+d1Uppxpj074A+c9x/nj6Cc</latexit><latexit sha1_base64="Dfq6iTzXZhde6iJPe2zGdIq4Pco=">AAACDnicbVC7SgNBFJ2NrxgfWbUUZDAIVmHXRgvFgI1lAuYBSQizk9lkyDyWmVklLCnt7W31F8RGxM5f8A8E/QhnkxSaeGDgcM59zQkiRrXxvA8ns7C4tLySXc2trW9s5t2t7ZqWscKkiiWTqhEgTRgVpGqoYaQRKYJ4wEg9GFykfv2aKE2luDLDiLQ56gkaUoyMlTpu/gaewRZHpq94Igejjlvwit4YcJ74U1I4f/u63XuufJc77merK3HMiTCYIa2bvheZdoKUoZiRUa4VaxIhPEA90rRUIE50OxkfPoIHVunCUCr7hIFj9XdHgrjWQx7YyvREPeul4r+eoJiECuGZ/SY8aSdURLEhAk/WhzGDRsI0G9ilimDDhpYgrKj9AcR9ZMcYm2DORuPPBjFPakdF3yv6Fb9QOgUTZMEu2AeHwAfHoAQuQRlUAQYxuAcP4NG5c56cF+d1Uppxpj074A+c9x/nj6Cc</latexit><latexit sha1_base64="hkXs2M9+E0TYm36SCFRDix+4n0Q=">AAACDnicbVDLSgMxFM3UV62Pjrp0EyyCqzLjRhcKBTcuK9gHtEPJpHfa0EwyJBmlDP0H9271F9yJW3/BP/AzzLSz0NYDgcM595UTJpxp43lfTmltfWNzq7xd2dnd26+6B4dtLVNFoUUll6obEg2cCWgZZjh0EwUkDjl0wslN7nceQGkmxb2ZJhDEZCRYxCgxVhq41Ud8jfsxMWMVZ3IyG7g1r+7NgVeJX5AaKtAcuN/9oaRpDMJQTrTu+V5igowowyiHWaWfakgInZAR9CwVJAYdZPPDZ/jUKkMcSWWfMHiu/u7ISKz1NA5tZX6iXvZy8V9PMAqRInRpv4kug4yJJDUg6GJ9lHJsJM6zwUOmgBo+tYRQxewPMB0TO8bYBCs2Gn85iFXSPq/7Xt2/82uNqyKkMjpGJ+gM+egCNdAtaqIWoihFz+gFvTpPzpvz7nwsSktO0XOE/sD5/AFyb5wZ</latexit>

w = ok<latexit sha1_base64="wwQsQWwR+GHS2U+9q9T1nk18iN4=">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</latexit><latexit sha1_base64="Di+F2ikHmKMU1hb1EIVg3vQyjeg=">AAACbHicbVFNaxRBEO2Z+BHXj2zUW4gMLooXl5lczEEx4MVjAtkkkFmWmt6abLP9MXTXJCzNHP1PXv0NnvwHgYh3b/bs5JBsLGh4vPeqqvt1UUnhKE1/RfHavfsPHq4/6j1+8vTZRn/z+ZEzteU44kYae1KAQyk0jkiQxJPKIqhC4nEx/9Lqx+donTD6kBYVjhWcaVEKDhSoSV/lBVh/0XzyPldztHpnuKvq3ISedqTPK2sIOeXn1Qw0GeWp6YzvO6dT4GahF2hmlTfz5pbc3BjaNL1Jf5AO02Uld0F2DQaff1x9e/X94Pf+pH+ZTw2vFWriEpw7zdKKxh4sCS6x6eW1wwr4HM7wNEANCt3YL2NpkjeBmSalseFoSpbszQ4PyrmFKoKzvb5b1Vryv5oWHEsLfGU/lbtjL3RVE2rerS9rmZBJ2uSTqbAhSLkIALgV4QUJn0EYQ+F/2miy1SDugqOdYZYOs4NssPeRdbXOtthr9o5l7APbY1/ZPhsxzn6yv1EcrUV/4pfxVrzdWePouucFu1Xx23+IycNx</latexit><latexit sha1_base64="Di+F2ikHmKMU1hb1EIVg3vQyjeg=">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</latexit><latexit sha1_base64="4NeNA7h0Ft87Ic1cOzsYGWwsvvo=">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</latexit>

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Figure 2: Illustration of the Gedankenexperiment. In each round n = 0, 1, 2, . . . of the experiment,agent F tosses a coin and, depending on the outcome r, polarises a spin particle S in a particulardirection. Agent F then measures the vertical polarisation z of S. Later, agents W and W measure theentire labs L and L (where the latter includes S) to obtain outcomes w and w, respectively. For theanalysis of the experiment, we assume that all agents are aware of the entire procedure as specified inBox 1, but that 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 drawconclusions about r.

obtain a strengthening of Bell’s theorem [10] (cf. the discussion section).

Analysis of the Gedankenexperiment

We analyse the experiment from the viewpoints of the different agents, who have access to different piecesof information (cf. Fig. 2). We assume, however, that all agents are aware of the entire experimentalprocedure as described in Box 1. We also suppose that they all employ the same theory. That is, theirreasoning is based on the same set of rules, which we specify in the following as assumptions.

The first such assumption is that the agents “use quantum theory”, by which we basically meanthat their conclusions about measurement outcomes are inferred from the Born rule. In fact, for ourpurposes, it suffices to consider the special case where the state of the system that is to be measured liesin the image of only one single measurement operator. In this case, the Born rule assigns probability 1to the corresponding outcome, i.e., it predicts the outcome with certainty.

Assumption (Q)

Suppose that agent A has established that

Statement A(i): “System S is in state |ψ〉S at time t0.”

(Here S is an arbitrary system around the agent and |ψ〉S a unit vector of the Hilbert space of S.)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 {πt0x }x∈Xof Heisenberg operators relative to time t0, which is completed at time t.”

(Here πt0x , for x ∈ X , are positive operators on S that sum up to the identity.)If 〈ψ|πt0ξ |ψ〉 = 1 for some ξ ∈ X then agent A can conclude that

Statement A(iii): “I am certain that x = ξ at time t.”

4

time interval time evolutionwithin round n F’s lab L F’s lab L

before n:00set R to|init〉R =

√1/3|heads〉R +

√2/3|tails〉R

[irrelevant]

from n:00to n:10

U00→10R→LS

=

{|heads〉R 7→ |h〉L ⊗ |↓〉S|tails〉R 7→ |t〉L ⊗ |→〉S

[irrelevant]

from n:10to n:20

U10→20L→L

= 1L U10→20S→L =

{|↓〉S 7→ |−

12〉L

|↑〉S 7→ |+12〉L

from n:20to n:30

[irrelevant] U20→30L→L = 1L

Table 1: Time evolution. The two labs, L and L, are assumed to be isolated quantum systems.Technically, this means that their time evolution is described by norm-preserving linear maps, i.e.,isometries. The second protocol step, for instance, in which F measures S, induces an isometry U10→20

S→L

from S to L. The vectors |− 12 〉L and |+ 1

2 〉L are defined as the outputs of this isometry, i.e., the statesof lab L at the end of the protocol step depending on whether the incoming spin was |↓〉S or |↑〉S,respectively. For concreteness, one may think of them as states of the form (2) — although theirstructure is irrelevant for the argument. Analogously, |h〉L and |t〉L are defined as the states of lab L atthe end of the first protocol step, depending on whether r = heads or r = tails, respectively.

agent valuemeasured

systemmeasurementcompleted at

relevant vectors ofmeasurement basis

Heisenberg projectorsused for reasoning via (Q)

F r R n:01 |heads〉R |tails〉R πn:10w=ok =

[(U10→20

S→L )†|ok〉L][·]†

πn:10w=fail = 1− πn:10

w=ok

F z S n:11 |↓〉S |↑〉Sπn:10z=− 1

2

= |↓〉〈↓|Sπn:10z=+ 1

2

= |↑〉〈↑|S

W w L n:21 |ok〉L =√

1/2(|h〉L − |t〉L

) πn:00(w,z)=(ok,− 1

2)

=[(U00→10

R→LS)†|ok〉L|↓〉S

][·]†

πn:00(w,z)6=(ok,− 1

2)

= 1− πn:00(w,z)=(ok,− 1

2)

W w L n:31 |ok〉L =√

1/2(|− 1

2〉L−|+ 1

2〉L

)πn:00

(w,w)=(ok,ok)

=[(U00→10

R→LS)†(U10→20

S→L )†|ok〉L|ok〉L][·]†

Table 2: Measurements carried out by the agents. Each of the four agents observes a value, definedas the outcome of a measurement on a particular system at a particular time. The measurement basisvectors |ok〉L and |ok〉L shown in the last two rows are expressed in terms of states, such as |− 1

2 〉L and|+ 1

2 〉L, which are defined in Table 1. The last column shows the measurement operators that the agentsinsert into statement A(ii) when reasoning according to Assumption (Q). These operators are givenin the Heisenberg picture, referring to the system’s state at a particular time, which is specified by asuperscript. The bracket [ · ]† stands for the adjoint of the preceding expression.

5

For our argument, we take this rule to be valid universally. It shall in particular be applicable tolarge and complex systems, which themselves contain agents. This is, for instance, the case for lab L,which in any round n of the experiment is measured with respect to the Heisenberg operators πn:10

w=ok andπn:10w=fail defined in Table 2. For our analysis, we suppose that agent F wants to predict the outcome w

of this measurement. To this aim, she may start her reasoning with a statement that describes thecorresponding measurement.

Statement Fn:00: “The value w is obtained by a measurement of L w.r.t. {πn:10w=ok, π

n:10w=fail},

which is completed at time n:31.”

Here and in the following, we specify for each statement a time, denoted as a superscript, indicatingwhen the agent could have inferred the statement. Agent F’s statement Fn:00 above does not depend onany observations, so the time n:00 we have assigned to it is rather arbitrary. This is however differentfor the next statement, which is based on knowledge of the value r. Suppose that agent F got r = tailsas the output of the random number generator in round n. According to the experimental instructions,she will then prepare the spin S in state |→〉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 hermeasurement at time n:10.

Statement Fn:01: “The spin S is in state |→〉S at time n:10.”

Agent F could conclude from this that the later state of the lab L, U10→20S→L |→〉S =

√1/2(|− 1

2 〉L+|+ 12 〉L),

will be orthogonal to |ok〉L. An equivalent way to express this is that the state |→〉S has no overlap withthe Heisenberg measurement operator corresponding to outcome w = ok, i.e.,

〈→|πn:10w=fail|→〉 = 1− 〈→|πn:10

w=ok|→〉 = 1 . (4)

The two statements Fn:00 and Fn:01, inserted into (Q), thus imply that w = fail. We may assume thatagent F draws this conclusion at time n:02 and, for later use, put it down as statement Fn:02 in Table 3.Similarly, agent F’s reasoning may be based upon a description of her spin measurement, which isdefined by the operators πn:10

z=− 12

and πn:10z=+ 1

2

given in Table 2.

Statement Fn:10: “The value z is obtained by a measurement of S w.r.t. {πn:10z=− 1

2

, πn:10z=+ 1

2

},which is completed at time n:11.”

Suppose now that agent F observed z = + 12 in round n. Since, by definition,

〈↓|πn:10z=− 1

2|↓〉 = 1 (5)

it follows from (Q) that S was not in state |↓〉 and hence that the random value r was not heads. This isstatement Fn:12 of Table 3. We proceed with agent W, who may base his reasoning upon his knowledgeof how the random number generator was initialised.

Statement Wn:21: “System R is in state |init〉R at time n:00.”

Consider the event that w = ok and z = − 12 , as well as its complement. The Heisenberg operators of the

corresponding measurement are given in Table 2. It is straightforward to verify that U00→10R→LS

|init〉R =√1/3|h〉L ⊗ |↓〉S+

√2/3|t〉L ⊗ |→〉S is orthogonal to |ok〉L ⊗ |↓〉S, which implies that

〈init|πn:00(w,z)6=(ok,− 1

2 )|init〉 = 1− 〈init|πn:00(w,z)=(ok,− 1

2 )|init〉 = 1 . (6)

Agent W, who also uses (Q), can hence be certain that (w, z) 6= (ok,− 12 ). This implies that state-

ment Wn:22 of Table 3 holds whenever w = ok. Furthermore, because agent W announces w, agent Wcan be certain about W’s knowledge, which justifies statement Wn:26 of the table. We have thusestablished all statements in the third column of Table 3.

6

=)(C)according to T

<latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit>

according to T<latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit>

according to T<latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit><latexit sha1_base64="dXpnCnll1BUwLPSKDsdZaxfgY7g=">AAACAnicbVC7SgNBFJ31GeNr1UpsBhPBKuym0cIiYGMZIS9IljA7O5sMmccyMyuEJdj4KzYWitj6FXb+jbPJFpp4YOBwzr3MPSdMGNXG876dtfWNza3t0k55d2//4NA9Ou5omSpM2lgyqXoh0oRRQdqGGkZ6iSKIh4x0w8lt7ncfiNJUipaZJiTgaCRoTDEyVhq6pwhjqSIqRtBIWB1wZMaKZ61ZdehWvJo3B1wlfkEqoEBz6H4NIolTToTBDGnd973EBBlShmJGZuVBqkmC8ASNSN9SgTjRQTaPMIMXVolgLJV9wsC5+nsjQ1zrKQ/tZH6iXvZy8T+vn5r4OsioSFJDBF58FKcsT5v3ASOqCDZsagnCitpbIR4jhbCxrZVtCf5y5FXSqdd8r+bf1yuNm6KOEjgD5+AS+OAKNMAdaII2wOARPINX8OY8OS/Ou/OxGF1zip0T8AfO5w+eDpbj</latexit>

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z = + 12


z = + 12


Figure 3: Consistent reasoning as required by Assumption (C). If a theory T (such as quantumtheory) enables consistent reasoning (C) then it must allow any agent A to promote the conclusionsdrawn by another agent A′ to his own conclusions, provided that A′ has the same initial knowledgeabout the experiment and reasons within the same theory T. A classical example of such recursivereasoning is the muddy children puzzle (here T is just standard logic; see [11] for a detailed account).The idea of using a physical theory T to describe agents who themselves use T has also appeared inthermodynamics, notably in discussions around Maxwell’s demon [12].

For later use we also note that a simple calculation yields

〈init|πn:00(w,w)=(ok,ok)|init〉 = 1/12 (7)

where πn:00(w,w)=(ok,ok)

is the Heisenberg operator belonging to the event that w = ok and w = ok, asdefined in Table 2. Hence, according to quantum mechanics, agent W can be certain that the outcome(w, w) = (ok, ok) occurs after finitely many rounds. This corresponds to the following statement (whichcan indeed be derived using (Q), as shown in the appendix).

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

The agents may now obtain further statements by reasoning about how they would reason fromthe viewpoint of other agents. To enable such nested reasoning we need another assumption, which isillustrated by Fig. 3.

Assumption (C)


Statement A(i): “I am certain that agent A′, upon reasoning within the same theory as theone 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.”

Agent F may insert agent F’s statement Fn:02 into Fn:12, obtaining statement Fn:13 of Table 3. Byvirtue of Assumption (C), she may then conclude that statement Fn:14 holds, too. Similarly, W maycombine this latter statement with his statement Wn:22 to obtain Wn:23. He could then, again using (C),conclude that statement Wn:24 holds. Finally, agent W can insert this into his statement Wn:26 to obtainstatement Wn:27 and, again with (C), statement Wn:28. This completes the derivation of all statementsof Table 3.

For the last part of our analysis, we take again agent W’s perspective. According to statement W0:00,the experiment has a final round n in which the halting condition will be satisfied, meaning in particularthat agent W announces w = ok. Agent W infers from this that statement Wn:28 of Table 3 holds inthat round, i.e., he is certain that he will observe w = fail at time n:31. However, in this final round,

7

agent assumedobservation

statement inferredvia (Q)

further impliedstatement

statement inferredvia (C)

Fr = tails

at time n:01Statement Fn:02: “I amcertain that W will observew = fail at time n:31.”

F z = + 12

at time n:11Statement Fn:12: “I amcertain that F knows thatr = tails at time n:01.”

Statement Fn:13: “I amcertain that F is certainthat W will observe w =fail at time n:31.”

Statement Fn:14: “I amcertain that W will observew = fail at time n:31.”

Ww = ok

at time n:21Statement Wn:22: “I amcertain that F knows thatz = + 1

2at time n:11.”

Statement Wn:23: “I amcertain that F is certainthat W will observe w =fail at time n:31.”

Statement Wn:24: “I amcertain that W will observew = fail at time n:31.”

W

announcementby agent Wthat w = okat time n:21

Statement Wn:26: “I amcertain that W knows thatw = ok at time n:21.”

Statement Wn:27: “I amcertain that W is certainthat I will observe w = failat time n:31.”

Statement Wn:28: “I amcertain that I will observew = fail at time n:31.”

Table 3: The agents’ observations and conclusions. The statements that the individual agentscan derive from quantum theory depend on the information accessible to them (cf. Fig. 2). Agent F,for instance, if she observes r = tails, can use this information to infer w, which will later be observedand announced by W.

he will nevertheless observe w = ok! We have thus reached a contradiction — unless agent W wouldaccept that w admits multiple values. For our discussion below, it will be useful to introduce an explicitassumption, termed (S), which disallows this.

Assumption (S)


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 6= 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 statementswhen applied to the Gedankenexperiment of Box 1.

To illustrate the theorem, we consider in the following different interpretations and modifications ofquantum theory. Theorem 1 implies that any of them must violate either (Q), (C), or (S). This yieldsa 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 concernedwith the special case of probability-1 predictions only, it is largely independent of interpretationalquestions, such as the meaning of probabilities in general. However, the non-trivial aspect of (Q) is that

8

(Q) (S) (C)

Copenhagen X X ×HV theory applied to subsystems X X ×HV theory applied to entire universe × X XMany-worlds ? × ?Collapse theories × X XConsistent histories X X ×QBism X X ×Relational quantum mechanics X X ×CSM approach × X XETH approach × X X

Table 4: Interpretations of quantum theory. The proposed Gedankenexperiment can be employedto study the various interpretations of quantum theory. Theorem 1 implies that each of them mustviolate at least one of the Assumptions (Q), (C), and (S) (indicated by ×). For hidden variable (HV)theories, it is relevant whether agents who are using the theory apply its laws (e.g., the guiding equationin the case of Bohmian mechanics) to subsystems around them or to the universe as a whole.

it regards the Born rule as a universal law. That is, it demands that an agent A can apply the ruleto 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 the agent A can describe herselfas a quantum system. Such a requirement would indeed be overly restrictive (see [13]) for it wouldimmediately rule out interpretations in the spirit of Copenhagen, according to which the observedquantum system and the observer must be distinct from each other [14, 15].

Assumption (Q) is manifestly violated by theories that postulate a modification of standard quantummechanics, such as spontaneous [16, 17, 18, 19, 20] and gravity-induced [21, 22, 23] collapse models (cf.[24] for a review). These deviate from the standard theory already on microscopic scales, although theeffects of the deviation typically only become noticeable in larger systems.

In some approaches to quantum mechanics, it is simply postulated that large systems are “clas-sical”, but the physical mechanism that explains the absence of quantum features remains unspeci-fied [25]. In the view described in [3], for instance, the postulate says that measurement devices areinfinite-dimensional systems whereas observables are finite. This ensures that coherent and incoherentsuperpositions 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 todistinguish coherent from incoherent superpositions once a measurement has been completed. Generalmeasurements on systems that count themselves as measurement devices are thus ruled out. Anotherexample is the “CSM ontology” [27], according to which measurements must always be carried out ina “context”, which includes the measurement devices. It is then postulated that this context cannotitself be treated as a quantum system. Within all these interpretations, the Born rule still holds “for allpractical purposes”, but is no longer a universally applicable law in the sense of Assumption (Q) (seethe discussion in [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 example [29,30, 31]. According to the common understanding, Bohmian mechanics is a “theory of the universe”rather than a theory about subsystems [32]. This means that agents who apply the theory must inprinciple 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 ofAssumption (C). However, because (S) is satisfied, too, it follows from Theorem 1 that (Q) must beviolated (see the appendix for more details).

9

Theories that violate Assumption (C)

If a theory satisfies (Q) and (S) then, by Theorem 1, it must violate (C). This conclusion applies toa wide range of common readings of quantum mechanics, including most variants of the Copenhageninterpretation. One concrete example is the “consistent histories” (CH) formalism [33, 34, 35], which isalso similar to the “decoherent histories” approach [36, 37]. Another class of examples are subjectivisticinterpretations, 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 [9] (seethe appendix for a discussion of the CH formalism as well as QBism).

The same conclusion applies to hidden-variable (HV) interpretations of quantum mechanics, pro-vided that we use them to describe systems around us rather than the universe as a whole (contrastingthe paradigm of Bohmian mechanics discussed above). In this case, both (Q) and (S) hold by construc-tion. This adds another item to the long list of no-go results for HV interpretations: they cannot belocal [10], they must be contextual [41, 42], and they violate freedom of choice [43, 44]. Theorem 1entails that they also violate (C). In particular, there cannot exist an assignment of values to the hiddenvariables 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” [45, 46, 47, 6, 48]. According to the latter, any quantum measurement results ina 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 “parallellives theory” [51]. A related concept is “quantum Darwinism” [52], whose purpose is to explain theperception 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 beviolated (cf. the example in Sec. 10 of [53]). It is also questionable whether (Q) can be upheld ifbranches do not persist over time (cf. the no-histories view described in [54]).

Implicit assumptions

Any no-go result, as for example Bell’s theorem [10], is phrased within a particular framework that comeswith a set of built-in assumptions. Hence it is always possible that a theory evades the conclusions ofthe no-go result by not fulfilling these implicit assumptions. Here we briefly discuss how Theorem 1compares in this respect to other results in the literature.

Bell’s original work [10] treats probabilities as a primitive notion. Similarly, many of the modernarguments in quantum foundations employ probabilistic frameworks [55, 56, 57, 58, 59, 60, 61, 62]. Incontrast, probabilities are not used in the argument presented here — although Assumption (Q) is ofcourse motivated by the idea that a statement can be regarded as “certain” if the Born rule assignsprobability 1 to it. In particular, Theorem 1 does not depend on how probabilities different from 1 areinterpreted.

Another distinction is that the framework used here treats all statements about observations assubjective, i.e., they are always defined relative to an agent. This avoids the a priori assumption thatmeasurement outcomes obtained by different agents simultaneously have definite values. (Consider forexample Wigner’s original setup described in the 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.) Theassumption of simultaneous definiteness is otherwise rather common. It not only enters the proof ofBell’s theorem [10] 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 isconceivable that the conclusions of Theorem 1 can be avoided by theories that provide a non-standard

10

understanding of these concepts. We are however not aware of any concrete examples of such theories.

Discussion

In the Gedankenexperiment proposed in this article, multiple agents have access to different pieces ofinformation, and draw conclusions by reasoning about the information held by others. The rules forsuch reasoning may be ambiguous in the general context of quantum theory where, from the viewpointof one agent, the information held by another agent can be in a superposition of different “classical”states. Crucially, however, in the argument presented here, the agents’ conclusions are all restricted tosupposedly unproblematic “classical” cases. For example, agent W only needs to derive a statementabout agent F in the case where, conditioned on his own information w, the information z held by Fhas a well-defined value (Table 3). Nevertheless, as we have shown, the agents arrive at contradictorystatements.

Current interpretations of quantum theory do not agree on the origin of this contradiction (cf.Table 4). To compare the different views, it may therefore be useful to rephrase the experiment as aconcrete game-theoretic decision problem. Suppose that a casino offers the following gambling game.One round of the experiment of Box 1 is played, with the gambler in the role of agent W, and the rolesof F, F, and W taken by employees of the casino. The casino promises to pay AC1000 to the gamblerif F’s random value was r = heads. Conversely, if r = tails, the gambler must pay AC500 to the casino.It could now happen that, at the end of the game, w = ok and w = ok, and that a judge can convinceherself of this outcome. The gambler and the casino are then likely to end up in a dispute, puttingforward arguments taken from Table 3.

Gambler: “The outcome w = ok proves, due to (4), that S was not prepared in state |→〉S. This means that

r = heads and hence the casino must pay me AC1000.”

Casino: “The outcome w = ok implies, due to (6), that our employee observed z = + 12. This in turn proves

that S was not prepared in state |↓〉S. But his means that r = tails, so the gambler must pay us AC500.”

How should the judge decide on this case? Could it even be that both assertions must be acceptedas two “alternative facts” about what the value r was? We leave it as a task for further research toexplore what the different interpretations of quantum mechanics have to say about this game.

Theorem 1 may be compared to earlier no-go results, such as [41, 42, 10, 7, 8, 43, 9], which also useassumptions similar to (Q) and (S) (although the latter is often implicit). These two assumptions areusually shown to be in conflict with additional assumptions about reality, locality, or freedom of choice.For example, the result of [9], which is as well based on an extension of Wigner’s argument, asserts thatno theory can fulfil all of the following properties: (i) be compatible with quantum theory on all scales,(ii) simultaneously assign definite truth values to measurement outcomes of all agents, (iii) allow agentsto freely choose measurement settings, and (iv) be local. Here we have shown that Assumptions (Q)and (S) are already problematic by themselves, in the sense that agents who use these assumptions toreason about each other as in Fig. 3 will arrive at inconsistent conclusions.

Another noticeable difference to earlier no-go results is that the argument presented here does notemploy counterfactual reasoning. That is, it does not refer to choices that could have been made buthave not actually been made. In fact, in the proposed experiment, the agents never make any choices(also no delayed ones, as e.g., in Wheeler’s “delayed choice” experiment [63]).

We conclude by suggesting a modified variant of the experiment, which may be technologicallyfeasible. The idea is to substitute agents F and F by computers. Specifically, one would program themto carry out the tasks prescribed in Box 1, process the information accessible to them, and outputstatements such as “I am certain that W will observe w = fail at 1:31.” To account for the requirementthat F and F’s labs be isolated, one would need to ensure that the computers used for their simulationdo not leak any information to their environment — a property which is necessarily satisfied by quantumcomputers. Such an experiment could then be used to verify the statements in Table 3. For example,aborting the experiment right after 1:13, one could, in the case when z = + 1

2 , read out statement F1:13

11

made by agent F together with statement F1:02 that agent F has made just before. This would be atest for the correctness of statement F1:13. Note that all statements in the fourth column of Table 3could in the same way be tested experimentally. In this sense, quantum computers, motivated usuallyby applications in computing, may help us answering questions in fundamental research.

Appendix

Information-theoretic description

The experimental protocol described in Box 1 may be represented as a circuit diagram, Fig. 4. Thediagram emphasises the information-theoretic aspects of the experiment. While all agents have fullinformation about the overall evolution (the circuit diagram itself), they have access to different data(corresponding to different wires in the diagram).

Derivation of statement W0:00 using Assumption (Q)

In the analysis of the Gedankenexperiment we argued that the event (w, w) = (ok, ok) must occur afterfinitely many rounds n, which is statement W0:00 described shortly after (7). While this is a prettyobvious consequence of the Born rule, we now show that it already follows from Assumption (Q), whichcorresponds to the special case of the Born rule when it gives probability-1 predictions.

We consider Heisenberg operators relative to time t0 = 0:00, i.e., right before the experiment starts.For any round n, let Wn be the isometry from C to L⊗ L that includes the initialisation of system Rin state |init〉R as well as U00→10

R→LSand U10→20

S→L (cf. Table 1), i.e.,

WnC→LL = (1L ⊗ U10→20

S→L )U00→10R→LS |init〉R . (8)

The Heisenberg operator of the event (w, w) = (ok, ok) in round n relative to time t0 can thus be writtenas

π(n)

(w,w)=(ok,ok)= (Wn

C→LL)†(|ok〉〈ok|L ⊗ |ok〉〈ok|L)(WnC→LL) . (9)

We may now specify a Heisenberg operator π0:00halt for the halting condition, i.e., that the event (w, w) =

(ok, ok) occurs in some round n,

π0:00halt =

∞∑n=0

π(n)

(w,w)=(ok,ok)

n−1∏m=0

(1C − π(m)

(w,w)=(ok,ok)

). (10)

Note that these are operators on C, i.e., πn(w,w)=(ok,ok)

= p for some p ∈ C. It follows directly from (7)that p = 1/12 > 0. We thus have

π0:00halt =

∞∑n=0

p(1− p)n = 1C . (11)

Inserting this measurement operator into the corresponding statement of Assumption (Q) yields state-ment W0:00.

Analysis within Bohmian mechanics

According to Bohmian mechanics, the state of a system of particles consists of their quantum-mechanicalwave function together with an additional set of variables that specify the particles’ spatial positions.While the wave function evolves according to the Schrodinger equation, the time evolution of theadditional position variables is governed by another equation of motion, sometimes referred to as the“guiding equation”. The general understanding is that these equations of motion must always be applied

12

tn:00 n:10 n:20 n:30 n:40

〈init|RR

(U00→10R→LS

)†

D

F

R

S

|heads〉〈heads|R〈h|D〈h|F〈↓|S+|tails〉〈tails|R〈t|D〈t|F〈→|S

(U10→20S→L )†

S

F

D

|↓〉〈↓|S〈− 12 |D〈−

12 |F

+|↑〉〈↑|S〈+ 12 |D〈+

12 |F

L

E

L

W

|ok〉〈ok|L〈ok|E〈ok|W+|ok〉〈ok|⊥L〈fail|E〈fail|W

L

W

L

E

|ok〉〈ok|L〈ok|E〈ok|W+|ok〉〈ok|⊥L〈fail|E〈fail|W

Figure 4: Circuit diagram representation of the Gedankenexperiment. The actions of theagents during the protocol correspond to isometries (boxes) that act on particular subsystems (wires).For example, the measurement of S by agent F in the second protocol step, which starts at time n:10,induces an isometry U10→20

S→L from S to F’s lab L, analogous to the one defined by (2). The subsystemslabelled by F, F, W, and W contain the agents themselves. Similarly, D, D, E, and E are “environment”subsystems, which include the agents’ measurement devices. The states of these subsystems depend onthe measurement outcome, which is indicated by their label. For example, |+ 1

2 〉F is the state of F whenthe agent has observed z = + 1

2 .

to the universe as a whole. As noted in [32], “if we postulate that subsystems [rather than the universe]must obey Bohmian mechanics, we ‘commit redundancy and risk inconsistency.’”

The Gedankenexperiment presented in this work shows that this risk is real. Indeed, if the agentsapplied the Bohmian equations of motion directly to the relevant systems around them, rather than tothe universe as a whole, their reasoning would be the same as the one prescribed by (Q). But sinceBohmian mechanics also satisfies (S), this would, by virtue of Theorem 1, imply a violation of (C),i.e., the agents’ conclusions would contradict each other. (This finding should not be confused withthe known fact that, if the spatial position of a particle is measured, the Bohmian position of themeasurement device’s pointer is sometimes incompatible with the Bohmian position of the measuredparticle [64, 65, 66, 67, 68].)

The directive of [32] that Bohmian mechanics should be applied to the entire universe means thatthe agents must model themselves from an outside perspective. This ensures that they all have thesame view, so that reasoning according to (C) is unproblematic. But then, because of Theorem 1,(Q) is necessarily violated. This is indeed confirmed by an explicit calculation in Bohmian mechanics,which reveals that statement Fn:02 of Table 3 does not hold there. Furthermore, the time order ofthe measurements carried out by agents W and W is relevant within Bohmian mechanics. If agent Wmeasured before agent W then, according to Bohmian mechanics, statement Wn:22 would be invalidwhereas Fn:02 would hold. This is a clear departure from standard quantum mechanics, where the timeorder in which agents W and W carry out their measurements is irrelevant, because they act on separatesystems.

This violation of (Q) raises the question under what circumstances Bohmian mechanics still endorsesthe use of the quantum-mechanical Born rule for predicting the outcome of a measurement. A candidatecriterion could be that such a prediction is only valid if a memory of the prediction is available uponcompletion of the measurement. One may then be tempted to argue that agent F’s statement Fn:02, forinstance, is invalid because F is herself subject to a measurement, which may destroy her memory of

13

the prediction for w before that value is measured. This argument does however not work. The reasonis that, in the relevant case when w = ok, the value r and hence also agent F’s prediction for w is, byvirtue of statements Wn:22 and Fn:12, retrievable at the time when w is measured.

Analysis within the CH formalism

In the consistent histories (CH) formalism, statements about measurement outcomes are phrased interms of “histories”. These must, by definition, be elements of a whole family of histories, called a“framework”, that satisfies certain consistency conditions. In the Gedankenexperiment proposed in thiswork, a possible history would be

History h1: “In round n the outcomes r= tails, z=+ 12 , w=ok, and w=ok were observed.”

To verify that h1 is indeed a valid history, one has to construct a framework containing this history.It is straightforward to check that one such framework is the set consisting of h1 together with theadditional histories

History h2: “In round n the outcomes r= tails, z=+ 12 , w=ok, and w= fail were observed.”

History h3: “In round n the outcomes r=heads, z=+ 12 , and w=ok were observed.”

History h4: “In round n the outcomes z=− 12 and w=ok were observed.”

History h5: “Outcome w= fail was observed.”

The CH framework contains the Born rule as a special case and hence fulfils Assumption (Q). Sinceit also satisfies (S), it follows from Theorem 1 that it violates (C). To illustrate how this violationmanifests itself, we may consider a shortened version of history h1, which leaves the values z and wunmentioned:

History h′1: “In round n the outcomes r= tails and w=ok were observed.”

The CH formalism provides a rule to assign probabilities to these histories, which turn out to be

Pr[h1] = 1/12 and Pr[h′1] = 0 . (12)

Note that these probabilities disagree with the fact that h′1 is just a part of history h1, i.e., h1 =⇒ h′1.(This finding may be compared to the “three box paradox” [69], where calculations in three differentconsistent frameworks yield mutually incompatible probability assignments; see Sec. 22 of [35] as wellas [70] for a discussion.)

The CH formalism accounts for this disagreement by imposing the rule that logical reasoning mustbe constrained to histories that belong to a single framework, which is not the case for h1 and h′1. Toillustrate what this means, it is useful to return to the casino example described in the discussion sectionabove. Within a framework that contains history h′1, the gambler’s reasoning is correct, for Pr[h′1] = 0.That is, w = ok implies that r = heads. Conversely, considering the framework above, which containshistory h1, it is readily verified that the other histories, h2, h3, h4, and h5, have probabilities 1

12 , 0, 0,and 5

6 , respectively. That is, all non-zero probability histories of this framework that agree with theobservation w = ok also assert that z = + 1

2 and r = tails. This seems to be in agreement with thecasino’s argument, i.e., w = ok implies that z = + 1

2 and r = tails. However, because the frameworkdoes not include a history that talks about r alone, it disallows the — seemingly obvious — implication(r = tails and z = + 1

2 ) =⇒ r = tails. In other words, within the CH formalism, the casino can provethat r = tails and z = + 1

2 , but not that r = tails.

Analysis within QBism

QBism is one of the most far-reaching subjectivistic interpretations of quantum mechanics. It regardsquantum states as representations of an agent’s personal knowledge, or rather beliefs, about the out-comes of future measurements, and it also views these outcomes as personal to the agent.

14

To reflect these tenets of QBism in the analysis of the Gedankenexperiment, it is useful to imag-ine that the agents write their observations and conclusions into a personal notebook. For example,according to Table 3, when agent F gets r = tails in round n = 1, she may put down the following

Statement F1:02: “r = tails at time 1:01, hence I am certain that I will hear W announcingw = fail at the end of this round.”

Here the phrase “is certain that” expresses a degree of belief and may also be replaced by somethinglike “would bet an arbitrarily large amount on”. Similarly, agent F, when she gets z = + 1

2 , may writeinto her notebook

Statement F1:12: “z = +12 at time 1:11, hence I am certain that, if I now checked F’s

notebook, I would read that r = tails at time 1:01.”

Agent F may as well write about agent F’s conclusions, i.e.,

Statement F1:13: “z = + 12 at time 1:11, hence I am certain that, if I now checked F’s

notebook, I would read that she is certain that she will hear W announcing w = fail at theend of this round.”

If one takes QBism to be a working tool for any agent to make predictions, it should in particular bethe case that agent F’s prediction is accurate — at least for her. That is, agent F’s later observation ofW’s announcement, which she may also put down into her notebook, should be in agreement with herearlier prediction. Agent F may thus be tempted to conclude from the above that

Statement F1:14: “z = + 12 at time 1:11, hence I am certain that, if I will check F’s notebook

at the end of this round, I will read that she heard W announcing w = fail.”

However, permitting such implications is akin to assuming (C). Because QBism satisfies (Q) and (S),it would result in the agents issuing contradictory statements such as “I heard W announcing w = okbut I am certain that my friend heard him announcing w = fail.” The Gedankenexperiment is thusan example of a multi-agent scenario where, to ensure consistency of QBism, implications of the typeF1:13 =⇒ F1:14 must be disallowed. Nevertheless, there should be ways for agents to consistentlyreason about each other. One may therefore ask whether (C) could be substituted by another (weaker)rule that enables such reasoning but does not lead to contradictions. This question is currently beinginvestigated (J.B. DeBrota, C.A. Fuchs, and R. Schack, manuscript in preparation).

Acknowledgments

We would like to thank Yakir Aharonov, Mateus Araujo, Alexia Auffeves, Jonathan Barrett, VeronikaBaumann, Serguei Beloussov, Charles Bennett, Hans Briegel, Caslav Brukner, Harry Buhrman, AdanCabello, Giulio Chiribella, Roger Colbeck, Patricia Contreras Tejada, Giacomo Mauro D’Ariano, JohnDeBrota, Lıdia del Rio, David Deutsch, Artur Ekert, Michael Esfeld, Philippe Faist, Aaron Fenyes,Hugo Fierz, Jurg Frohlich, Christopher Fuchs, Shan Gao, Svenja Gerhard, Edward Gillis, Nicolas Gisin,Sheldon Goldstein, Sabrina Gonzalez Pasterski, Gian Michele Graf, Philippe Grangier, Bob Griffiths,Arne Hansen, Lucien Hardy, Aram Harrow, Klaus Hepp, Pawe l Horodecki, Angela Karanjai, AdrianKent, Gijs Leegwater, Matthew Leifer, Seth Lloyd, John Loverain, Markus Muller, Thomas Muller,Hrvoje Nikolic, Travis Norsen, Nuriya Nurgalieva, Jonathan Oppenheim, Sandu Popescu, MatthewPusey, Gilles Putz, Joseph Renes, Jess Riedel, Valerio Scarani, Rudiger Schack, Robert Spekkens,Cristi Stoica, Antoine Suarez, Tony Sudbery, Stefan Teufel, Roderich Tumulka, Lev Vaidman, VlatkoVedral, Mordecai Waegell, Andreas Winter, Stefan Wolf, Filip Wudarski, Christa Zoufal, and WojciechZurek for comments and discussions.

This project was supported by the Swiss National Science Foundation (SNSF) via the NationalCentre of Competence in Research “QSIT”, by the Kavli Institute for Theoretical Physics (KITP) atthe University of California in Santa Barbara, by the Stellenbosch Institute for Advanced Study (STIAS)in South Africa, by the US National Science Foundation (NSF) under grant No. PHY17-48958, by theEuropean Research Council (ERC) under grant No. 258932, and by the European Commission underthe project “RAQUEL”.

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References

[1] E. Schrodinger. Die gegenwartige Situation in der Quantenmechanik. Naturwissenschaften, 23:823–828, (1935).

[2] E.P. Wigner. Remarks on the mind-body question. In Symmetries and Reflections, pages 171–184.Indiana University Press, (1967).

[3] K. Hepp. Quantum theory of measurement and macroscopic observables. Helv. Phys. Acta,45(2):237–248, (1972).

[4] J.S. Bell. On wave packet reduction in the Coleman-Hepp model. Helv. Phys. Acta, 48(1):93–98,(1975).

[5] C.A. Fuchs. QBism, the perimeter of quantum Bayesianism. Preprint athttp://arxiv.org/abs/1003.5209, (2010).

[6] D. Deutsch. Quantum theory as a universal physical theory. Int. J. Theor. Phys., 24:1–41, (1985).

[7] L. Hardy. Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories.Phys. Rev. Lett., 68:2981–2984, (1992).

[8] L. Hardy. Nonlocality for two particles without inequalities for almost all entangled states. Phys.Rev. Lett., 71:1665–1668, (1993).

[9] C. Brukner. On the quantum measurement problem. In R. Bertlmann and A. Zeilinger, editors,Quantum [Un]Speakables II: Half a Century of Bell’s Theorem, pages 95–117. Springer, (2017).

[10] J.S. Bell. On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys., 38:447–452,(1966).

[11] R. Fagin, J.Y. Halpern, Y. Moses, and M.Y. Vardi. Reasoning About Knowledge. MIT press,(2004).

[12] C. H. Bennett. The thermodynamics of computation—a review. Int. J. Theor. Phys., 21:905–940,(1982).

[13] M.L. Dalla Chiara. Logical self reference, set theoretical paradoxes and the measurement problemin quantum mechanics. J. Philos. Logic, 6:331–347, (1977).

[14] W. Heisenberg. Ist eine deterministische Erganzung der Quantenmechanik moglich? In A. Her-mann, K. von Meyenn, and V.F. Weisskopf, editors, Wolfgang Pauli. Wissenschaftlicher Briefwech-sel mit Bohr, Einstein, Heisenberg, u.a., volume II, pages 409–418. Springer, (1935).

[15] N. Bohr. Discussions with Einstein on epistemological problems in atomic physics. In P.A. Schilpp,editor, Albert Einstein: Philosopher-Scientist, pages 200–241. Cambridge University Press, (1949).

[16] G.C. Ghirardi, A. Rimini, and T. Weber. Unified dynamics for microscopic and macroscopicsystems. Phys. Rev. D, 34:470–491, (1986).

[17] N. Gisin. Stochastic quantum dynamics and relativity. Helv. Phys. Acta, 62:363–371, (1989).

[18] P. Pearle. Combining stochastic dynamical state-vector reduction with spontaneous localization.Phys. Rev. A, 39:2277–2289, (1989).

[19] R. Tumulka. On spontaneous wave function collapse and quantum field theory. In Proc. R. Soc.A, volume 462, pages 1897–1908. The Royal Society, (2006).

[20] S. Weinberg. Collapse of the state vector. Phys. Rev. A, 85:062116, (2012).

16

[21] F. Karolyhazy. Gravitation and quantum mechanics of macroscopic objects. Il Nuovo Cimento A(1971-1996), 42:390–402, (1966).

[22] L. Diosi. Models for universal reduction of macroscopic quantum fluctuations. Phys. Rev. A,40:1165–1174, (1989).

[23] R. Penrose. On gravity’s role in quantum state reduction. Gen. Rel. Gravit., 28:581–600, (1996).

[24] A. Bassi, K. Lochan, S. Satin, T.P. Singh, and H. Ulbricht. Models of wave-function collapse,underlying theories, and experimental tests. Rev. Mod. Phys., 85:471–527, (2013).

[25] L.D. Landau and E.M. Lifshitz. Quantum Mechanics: Non-Relativistic Theory, chapter The basicconcepts of quantum mechanics. Elsevier, (2013).

[26] J. Frohlich and B. Schubnel. Quantum probability theory and the foundations of quantum me-chanics. In P. Blanchard and J. Frohlich, editors, The Message of Quantum Science: AttemptsTowards a Synthesis, pages 131–193. Springer, (2015).

[27] A. Auffeves and P. Grangier. Contexts, systems and modalities: A new ontology for quantummechanics. Found. Phys., 46:121–137, (2015).

[28] J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag, (1932).

[29] L. De Broglie. La mecanique ondulatoire et la structure atomique de la matiere et du rayonnement.J. Phys. Radium, 8:225–241, (1927).

[30] D. Bohm. A suggested interpretation of the quantum theory in terms of “hidden” variables. I.Phys. Rev., 85:166–179, (1952).

[31] D. Durr and S. Teufel. Bohmian Mechanics: The Physics and Mathematics of Quantum Theory.Springer, (2009).

[32] D. Durr, S. Goldstein, and N. Zanghı. Quantum equilibrium and the origin of absolute uncertainty.J. Stat. Phys., 67(5):843–907, (1992).

[33] R.B. Griffiths. Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys.,36:219–272, (1984).

[34] R. Omnes. Consistent interpretations of quantum mechanics. Rev. Mod. Phys., 64:339–382, (1992).

[35] R.B. Griffiths. Consistent Quantum Theory. Cambridge University Press, (2002).

[36] M. Gell-Mann and J.B. Hartle. Quantum mechanics in the light of quantum cosmology. InW. Zurek, editor, Complexity, Entropy and the Physics of Information. Addison Wesley, (1990).

[37] J.B. Hartle. The quasiclassical realms of this quantum universe. Found. Phys., 41:982–1006, (2011).

[38] C. Rovelli. Relational quantum mechanics. Int. J. of Theor. Phys., 35:1637–1678, (1996).

[39] C.A. Fuchs and R. Schack. Quantum-Bayesian coherence. Rev. Mod. Phys., 85:1693–1715, (2013).

[40] C.A. Fuchs, N.D. Mermin, and R. Schack. An introduction to QBism with an application to thelocality of quantum mechanics. Am. J. Phys., 82:749–754, (2014).

[41] J.S. Bell. On the Einstein Podolsky Rosen paradox. Physics, 1:195–200, (1964).

[42] S. Kochen and E. P. Specker. The problem of hidden variables in quantum mechanics. J. Math.Mech., 17:59–87, (1967).

[43] R. Colbeck and R. Renner. No extension of quantum theory can have improved predictive power.Nat. Commun., 2:411, (2011).

17

[44] R. Colbeck and R. Renner. A short note on the concept of free choice. Preprint athttp://arxiv.org/abs/1302.4446, (2013).

[45] H. Everett. “Relative state” formulation of quantum mechanics. Rev. Mod. Phys., 29:454–462,(1957).

[46] J.A Wheeler. Assessment of Everett’s “relative state” formulation of quantum theory. Rev. Mod.Phys., 29:463, (1957).

[47] B.S. DeWitt. Quantum mechanics and reality. Phys. Today, 23:155–165, (1970).

[48] D. Deutsch. The Fabric of Reality: The Science of Parallel Universes and Its Implications. AllenLane Science, (1997).

[49] H.D. Zeh. On the interpretation of measurement in quantum theory. Found. Phys., 1:69–76, (1970).

[50] D. Albert and B. Loewer. Interpreting the many worlds interpretation. Synthese, 77:195–213,(1988).

[51] G. Brassard and P. Raymond-Robichaud. Can free will emerge from determinism in quantumtheory? In A. Suarez and P. Adams, editors, Is Science Compatible with Free Will? Exploring FreeWill and Consciousness in the Light of Quantum Physics and Neuroscience, pages 41–61. Springer,(2013).

[52] W.H. Zurek. Relative states and the environment: einselection, envariance, quantum Darwinism,and the existential interpretation. Preprint at http://arxiv.org/abs/0707.2832, (2007).

[53] L. Vaidman. On schizophrenic experiences of the neutron or why we should believe in the many-worlds interpretation of quantum theory. Int. Stud. Phil. Sci., 12:245–261, (1998).

[54] J. Butterfield and G.N. Fleming. Quantum theory and the mind. PAS, Supplementary Volumes,69:113–173, (1995).

[55] J. Barrett. Information processing in generalized probabilistic theories. Phys. Rev. A, 75:032304,(2007).

[56] L. Hardy. Reformulating and reconstructing quantum theory. Preprint athttp://arxiv.org/abs/1104.2066, (2011).

[57] G. Chiribella, G.M. D’Ariano, and P. Perinotti. Informational derivation of quantum theory. Phys.Rev. A, 84:012311, (2011).

[58] L. Masanes and M.P. Muller. A derivation of quantum theory from physical requirements. New J.Phys., 13:063001, (2011).

[59] R. Colbeck and R. Renner. Is a system’s wave function in one-to-one correspondence with itselements of reality? Phys. Rev. Lett., 108:150402, (2012).

[60] M.F. Pusey, J. Barrett, and T. Rudolph. On the reality of the quantum state. Nat. Phys., 8:475–478, (2012).

[61] M.S. Leifer and R.W. Spekkens. Towards a formulation of quantum theory as a causally neutraltheory of Bayesian inference. Phys. Rev. A, 88:052130, (2013).

[62] H. Barnum and A. Wilce. Post-classical probability theory. In G. Chiribella and R.W. Spekkens,editors, Quantum Theory: Informational Foundations and Foils, pages 367–420. Springer, (2016).

[63] J.A. Wheeler. The “past” and the “delayed-choice” double-slit experiment. In A.R. Marlow, editor,Mathematical Foundations of Quantum Theory, pages 9–48. Academic Press, (1978).

18

[64] B.-G. Englert, M.O. Scully, G. Sussmann, and H. Walther. Surrealistic Bohm trajectories. Z.Naturforsch., 47:1175–1186, (1992).

[65] M. Correggi and G. Morchio. Quantum mechanics and stochastic mechanics for compatible ob-servables at different times. Ann. Phys., 296:371–389, (2002).

[66] J. Kiukas and R.F. Werner. Maximal violation of Bell inequalities by position measurements. J.Math. Phys., 51, (2010).

[67] G. Naaman-Marom, N. Erez, and L. Vaidman. Position measurements in the de Broglie-Bohminterpretation of quantum mechanics. Ann. Phys., 327:2522 – 2542, (2012).

[68] N. Gisin. Why Bohmian mechanics? One and two-time position measurements, Bell inequalities,philosophy and physics. Entropy, 20:105, (2018).

[69] Y. Aharonov and L. Vaidman. Complete description of a quantum system at a given time. J. Phys.A, 24(10):2315, (1991).

[70] A. Kent. Consistent sets yield contrary inferences in quantum theory. Phys. Rev. Lett., 78:2874–2877, (1997).

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quantum theory cannot consistently describe the use of itself

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