Abstract
In a recent no-go theorem [Bong et al., Nature Physics (2020)], we proved that the predictions of unitary quantum mechanics for an extended Wigner’s friend scenario are incompatible with any theory satisfying three metaphysical assumptions, the conjunction of which we call “Local Friendliness”: Absoluteness of Observed Events, Locality and No-Superdeterminism. In this paper (based on an invited talk for the QBism jubilee at the 2019 Växjö conference) I discuss the implications of this theorem for QBism, as seen from the point of view of experimental metaphysics. I argue that the key distinction between QBism and realist interpretations of quantum mechanics is best understood in terms of their adherence to different theories of truth: the pragmatist versus the correspondence theories. I argue that a productive pathway to resolve the measurement problem within a pragmatist view involves taking seriously the perspective of quantum betting agents, even those in what I call a “Wigner bubble”. The notion of reality afforded by QBism, I propose, will correspond to the invariant elements of any theory that has pragmatic value to all rational agents—that is, the elements that are invariant upon changes of agent perspectives. The classical notion of ‘event’ is not among those invariants, even when those events are observed by some agent. Neither are quantum states. Nevertheless, I argue that far from solipsism, a personalist view of quantum states is an expression of its precise opposite: Copernicanism.
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Notes
I am aware that philosophers who work on metaphysics are called “metaphysicians”. I use “metaphysicist” to refer to physicists who work on (experimental) metaphysics.
In the same sense as “metamathematics”—the field that has mathematical theories as its object of study, and Gödel’s incompleteness theorem as its most prominent result.
In fact, this is most likely impossible, a fact that philosophers refer to as the “underdetermination of theory by evidence”. For example, any set of publicly shared observations can always be described within a deterministic single-world theory, simply by adding a sufficient number of hidden variables and allowing the laws of physics to have a sufficiently complex mathematical form. There would no doubt be reasons to reject ad-hoc theories of this kind via some philosophical criteria, but they cannot be ruled out as impossible.
Note that I am not advocating beauty as a criterion for truth, a view that has been recently criticised as unproductive for physics [4]. Every theory in the landscape is probably “ugly” by some physicists’ aesthetic judgement. I’ll return to the question of truth later in this paper.
Fuchs acknowledges in [7] that he is “deeply indebted to Eric G. Cavalcanti for suggesting the imagery of “truly private worlds” in my discussion of Wigner’s friend”.
And in practice LF inequalities have already been violated by the experiments of [8, 9], with single quantum systems playing the role of “observers”. For interpretations where any physical system can be considered an observer (e.g. relational quantum mechanics [10]), this is sufficient evidence. For QBism, I suggest that a conclusive demonstration would be one that involves agents who are “users of quantum theory” [6, 11].
Incidentally, Wigner neglects another alternative, compatible with his interpretation of the quantum state as encoding information: a psi-epistemic [14] theory. If instead of quantum states we were talking about classical probability distributions, there would be no contradiction between the friend observing a coin landing heads and thus assigning probability 1 to that event, while at the same time Wigner assigning it probability 1/2. This way out, if applied to the quantum case, would amount to a theory where quantum states represent states of information about an underlying reality (unlike in QBism, in which they are an agent’s information about the outcomes of potential observations on the system). A psi-epistemic theory would however contradict Wigner’s premise that a pure quantum state provides a complete description of the system. In any case, we now have several no-go theorems and experimental results putting strong constraints on such theories [15,16,17,18]. Furthermore, the collapse theory advocated by Wigner undermines his premise that the quantum state represents subjective information, since in such a theory the wavefunction that Wigner ought to attribute to the situation changes with the friend’s observation, even before that information is communicated to Wigner. In such a theory the wavefunction seems to acquire a much more objective character.
Recall that Local Causality can be parsed as the conjunction of two assumptions: “Parameter Independence” and “Outcome Independence” [23]. Locality is equivalent to “Parameter Independence”.
In [22] this notion was called “Local Agency”.
Based on Eq. (2), this should produce the same result as if Alice measured the system \(S_A\) directly on the computational basis, which she can do subsequently to check for consistency with Charlie’s report.
In a scenario with a single friend, this is the class of “semi-Brukner” inequalities [8].
The LF polytopes were studied independently by Erik Woodhead [24] (who called them “partially deterministic polytopes”) in the context of device-independent randomness certification in the presence of no-signalling adversaries. The term “partially deterministic” refers to the fact that unlike in a Bell-local model (where all observables in the scenario can be modelled via a local deterministic response function), only a subset of the observables have deterministic values in the model—in our scenario, those that are always measured by the friend in every run.
Reichenbach’s principle says that if two events A and B are correlated, then either one is the cause of the other, or they share a common cause C such that A and B are uncorrelated when conditioned on C.
I will discuss the implications of the LF theorem for quantum causality in more detail in an upcoming paper.
A more recent paper [33] proposes a few different alternative rules, within a Page-Wootters formalism for relational time. I will not analyse all of those proposals here, but the main lesson from the Local Friendliness theorem applies to any proposal that specifies a joint probability distribution to the outcomes of Charlie and Alice.
AOE implies that there exists a joint probability distribution P(a, b, c|x, y) for the observed outcomes of Alice, Bob and Charlie such that the observations by Alice and Bob are consistent with the marginals \(P(a,b|x,y)=\sum _c P(a,b,c|x,y)\). Marginalising over Bob’s outcomes instead, one obtains \(P(a,c|x,y) = \sum _b P(a,b,c|x,y)\), and assuming Local Action, this distribution must be independent of Bob’s choice of measurement y, \(P(a,c|x,y)=P(a,c|x)\).
In [32], this is Alice herself, and the description is based on what may be interpreted as an Everettian universal wavefunction. Since in QBism there are no free-floating wavefunctions, this could be reinterpreted from the point of view of a further superobserver. For simplicity, I here describe it as a measurement apparatus under Alice’s control.
Since F is initially assigned a fixed ready state \(|R\rangle _F\), it is sufficient to consider the action of \(V_x\) on the d-dimensional subspace spanned by \(|c,O_c\rangle _{S_AF}\).
Note that Deutsch’s appeal to the Everett interpretation does not imply a belief in this joint probability.
Note that the same argument does not hold in the case of violations of Bell inequalities. There, the probabilities P(a, b|x, y) may well correspond to gambling commitments of some agent, for example a referee Rob who receives the outcomes a and b in the common future of Alice and Bob. A QBist Rob who can locate those events within their space-time (as they must if they are sufficiently pragmatic, see Sect. 3.2) cannot therefore dismiss Bell’s theorem simply by rejecting AOE, as (essentially) suggested by Fuchs, Mermin and Schack in [29]. Instead, I suggest that a QBist Rob must also reject the principle of Decorrelating Explanation, as in the operationalist version of Theorem 8 in [22]. This alternative corresponds to Rob using quantum causal models to describe their gambling commitments about events which they can effectively take to exist (relative to their own perspective and powers) without any practical risk of contradiction.
My use of the capital \(\varPsi _A\) of course follows the notation for a quantum state, since in QBism the quantum state is the encapsulation of an agent’s information. However, more generally, it can refer to whatever mathematical object encapsulates an agent’s pragmatic degrees of belief, including matters beyond quantum theory, for which a quantum state may not be a convenient or suitable description. \(\varPsi _A\) must be understood as an agent’s information from a certain perspective.
Another pragmatist approach to quantum theory is that proposed by Healey [34]. According to Healey, “The key difference [between QBism and Healey’s pragmatism] is that while, for the QBist, quantum state ascriptions depend on the epistemic state of the agent who ascribes them, on the present pragmatist approach what quantum state is to be ascribed to a system depends only on the physical circumstances defining the perspective of the agent (actual or merely hypothetical) that ascribes it”.
Since Franz is an AI, and thus genderless, I refer to them with the neutral singular pronouns they/them/their/themself.
We assume the dimension of the Hilbert space of C to be sufficiently large so that it can encode the necessary values to whatever desired precision.
Though one may question whether this gamble is fair from Eugene’s perspective if he cannot confirm the sale price before seeing the final outcome, and if Franz’ wallet is not in a well-defined state relative to Eugene at all times. Those are interesting questions that may lead to difficulties in some scenarios, but I will leave them for future work. For the purpose of this example, let us suppose that Eugene has set a lower bound on the sale price, and that this lower bound depends on the information that he takes Franz to have when placing their bet. In the first case, he is happy to sell it for \(\$p_0\), but he would not accept less than \(\$1\) in the second case.
It might be interesting to consider what happens if the wallet could be measured in a basis different from the cash basis. This doesn’t make any classical sense, but might not be completely meaningless in the quantum case, as long as all agents can agree on the rules of the game.
Although this paper was based on my 2019 Växjö talk, some aspects of it have evolved since then. The appeal to a Copernican principle, and the argument for how it supports a personalist view of quantum states is essentially the same, as is the argument against joint probabilities of the type considered by Baumann and Brukner. But the analysis in Sect. 5 is largely new.
But more generally, I do not see why this should be required in a pragmatist view, lest it excludes some prosaic classical situations, e.g. an agent considering their beliefs about what a medical scan of the agent’s own body may find. This can even make sense when the agent undergoes total anaesthesia, and does not have any memories of the procedure, similarly to a Wigner’s friend scenario.
Those works consider changes between perspectives associated with different physical systems, where a system’s “perspective” is taken to be a reference frame in which the system’s position is at the origin. However, this seems insufficient for the analysis of a Wigner’s friend scenario, where we consider physical systems that do not have a unique preferred perspective. Rather, a perspective in the sense I use here is not simply associated with a system, but with a system in a particular state of information.
I put this in scare quotes to emphasise that it is not an objective process.
In the talk this paper was based on, I referred to this as a “Wigner hole”. Since this is a very fragile process, however, I now think “bubble” conveys a better imagery.
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Acknowledgements
I acknowledge useful discussions with Howard Wiseman, Aníbal Utreras-Alarcón, Yeong-Cherng Liang, Mateus Araújo, Fabio Costa, Peter Evans, Jacques Pienaar, Veronika Baumann, Caslav Brukner, Chris Fuchs, Jon Barrett, Matt Pusey, Frida Trotter, Markus Frembs, and thank an anonymous referee for constructive suggestions. This work was supported by Grant Number FQXi-RFP-1807 from the Foundational Questions Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Community Foundation, and ARC Future Fellowship FT180100317.
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Cavalcanti, E.G. The View from a Wigner Bubble. Found Phys 51, 39 (2021). https://doi.org/10.1007/s10701-021-00417-0
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DOI: https://doi.org/10.1007/s10701-021-00417-0