Implications of Local Friendliness violation for quantum causality

“For me … this is the real problem with quantum theory: the apparently essential conflict between any sharp formulation [of quantum theory] and fundamental relativity …. It may be that a real synthesis of quantum and relativity theories requires not just technical developments but radical conceptual renewal”. (John S. Bell, 1984 Bell (1987)).

In a recent work Bong et al. (2020), we (the present authors together with co-authors) proved what we called a “strong” no-go theorem on the Wigner’s friend paradox—the “Local Friendliness” (LF) no-go theorem. In this paper, we discuss further in what sense it is stronger than Bell’s theorem and give a broader picture of some of its implications. In particular, we discuss how it presents a challenge to a popular resolution of Bell’s theorem that goes back to Shimony Shimony (1984).

Shimony proposed to separate Bell’s notion of Local Causality into two independent assumptions, which he called “Parameter Independence” (PI) and “Outcome Independence” (OI). His idea was that, while violation of PI was undoubtedly a case of action-at-a-distance—and was thus contrary to the letter of the theory of relativity—violation of OI was a far milder affliction—“passion at a distance” Shimony (1984)—that was contrary only to the spirit of the theory of relativity. This allowed, the argument goes, for a “peaceful coexistence” Shimony (1984); Myrvold (2002); Berkovitz (2007) between quantum theory and relativity.

Shimony’s proposal has been criticized for missing the mark. For Bell, it left open the question of causal explanation of correlations Bell (1990). The program of quantum causal models Leifer and Spekkens (2013); Cavalcanti and Lal (2014); Pienaar and Brukner (2015); Costa and Shrapnel (2016); Allen et al. (2017); Barrett et al. (2019) has provided a candidate answer to this challenge by generalising the classical framework of causal models Pearl (2000) to accommodate quantum correlations. This program shares important similarities with Shimony’s early proposal, and as we will show, is subject to the same challenge in light of the LF no-go theorem.

To better put these results and discussions in context, we present the Local Friendliness theorem within an updated version of the conceptual framework introduced by us in Wiseman and Cavalcanti (2017), where we reformulated the two theorems of John Bell (the 1964 Bell (1964) and 1976 Bell (1976) theorems – see Ref. Wiseman (2014) by one of us for a detailed discussion of the history and controversy surrounding the distinction.) in terms of fundamental metaphysical principles concerning events, space-time, and causality. This work was partly motivated by disagreements between two broad interpretational camps we referred to (following Ref. Wiseman (2014)) as “realists” and “operationalists”. (As an aside: some whose positions are not in what we call the “realist” camp would nevertheless claim to be realists in some sense. We will keep the scare quotes when referring to these two camps here to avoid debates about the meaning of those terms.)  Many of these disagreements arose from (often implicit) assumptions about the meaning of terms like “local”, and by using deeper principles, the foundations of these different perspectives could be made clearer. Moreover, we could present a “conciliatory” reformulation (Theorem 8 of Wiseman and Cavalcanti (2017)), in which the two camps could agree on the meaning of all the assumptions involved but only disagree about which assumptions were to be rejected in light of the theorem.

As we will discuss in detail in this paper, the new challenge alluded to in the opening paragraph here is a challenge for the operationalist camp. In our Bell-conciliation theorem Wiseman and Cavalcanti (2017), the “realist” camp would reject the principle of Relativistic Causality, while the “operationalist” camp would reject that of Decorrelating Explanation. This latter principle is the “quantitative” part of Reichenbach’s Principle of Common Cause  (RPCC), which we split into two parts following the analysis proposed by EGC and Lal Cavalcanti and Lal (2014). Rejection of this quantitative part of RPCC is also, implicitly or explicitly, the approach to resolve Bell’s theorem taken within the frameworks of quantum causal models Pienaar and Brukner (2015); Costa and Shrapnel (2016); Allen et al. (2017); Barrett et al. (2019). However—and this is the new challenge—the assumption of Decorrelating Explanation is not required for the derivation of LF inequalities.

What are the implications of this? Firstly, there is of course the possibility that the LF inequalities derived in Bong et al. (2020) fail to be violated with “genuine” observers (the “friends” of Wigner). We discuss this question at length in another paper Wiseman et al. (2021). For simplicity, here, we just acknowledge this as an open question and consider the implications of taking seriously the possibility that the LF violations demonstrated for very simple “observers” in recent experiments Proietti et al. (2019); Bong et al. (2020) can be maintained at the level of systems we may be strongly inclined to consider genuine observers—e.g., human-level artificial intelligences running in a very large quantum computer Wiseman et al. (2021).

Assuming that nature violates Local Friendliness, then, this implies that a popular class of resolutions of Bell’s theorem—encompassing both “passion at a distance” and quantum causal models—is not available to resolve the LF theorem. Is a “peaceful coexistence” between quantum theory and relativity therefore impossible?

A “realist” would say yes. This camp would see the LF no-go theorem as a strengthening of the “nonlocal” position (rejecting Relativistic Causality), since it demonstrates more clearly than with Bell’s theorem the depth of the “essential conflict” between quantum theory and relativity, making it sharper than ever the need for “radical conceptual renewal”, in Bell’s words.

On the other hand, an “operationalist” may also find hope in a different sort of radical conceptual renewal. We will argue that Leibniz’s Principle of the Identity of Indiscernibles Spekkens (2019)—a methodological principle underlying Einstein’s principles of relativity and of equivalence, as well as the program of quantum causal models—can only be maintained by rejecting Absoluteness of Observed Events. This last is one of the assumptions of the LF no-go theorem, as well as a (typically implicit) assumption in Bell’s theorem. Maintaining Leibniz’s methodological principle, in other words, seems to require a kind of strengthening of relativity: that not only space-time but events themselves be described as relative.

This paper is organised as follows. In Section II, we review Bell’s two theorems (from 1964 and 1976) and the implicit and explicit assumptions behind them. In Section III, we break down Bell’s 1976 assumption of Local Causality into more primitive concepts to see how this can be given up while still holding to the existence of a Relativistic Causal Arrow, as Leibniz’s methodology requires. In Section IV, we show that this route, which is the one taken by quantum causal models, does not work when it comes to the LF theorem, because Local Causality is not part of that theorem. In Section V, we break the assumptions down to the deepest principles, to display the options allowed by the theorems we consider. Critically, the LF theorem leaves fewer options, and we conclude in Section VI by suggesting that cleaving to Leibniz’s principle requires rejecting Absoluteness of Observed Events.

Bell’s theorem, and the LF theorem, are about quantum violations of different conjunctions of metaphysical assumptions through violations of constraints those assumptions imply for empirical correlations between events in certain experimental scenarios. Following Wiseman and Cavalcanti (2017), we start by stating two fundamental assumptions about events and space-time, typically left implicit in discussions of Bell’s theorem. In Wiseman and Cavalcanti (2017), we separated the various assumptions into “Axioms”, “Postulates” and “Principles”. The “Axioms” were so named because they were left as background assumptions in the statement of some or all of the theorems discussed, with their logical implications left implicit. This terminology is accurate for this section, but the LF theorem explicitly uses one of our Axioms in its formulation. Here and in subsequent definitions, whenever a word appears in small caps, it indicates a term whose meaning may be further specified or modified by other assumptions or principles.

We called this Axiom “Macroreality” in Wiseman and Cavalcanti (2017), and it plays a crucial role in the LF theorem, as we will see.

In Wiseman and Cavalcanti (2017), we called this assumption “Minkowski Space-Time”, but a flat space-time is not strictly required, only a time-orientable pseudo-Riemannian manifold.

In a Bell-type experiment, the observable events under consideration are choices of measurement settings (which we may label $X,Y$ for the case of two distant parties), and their corresponding outcomes (which we may label $A,B$). We will use the same symbol (e.g., $A$) to refer interchangeably to a variable that ranges over possible values of this outcome and to the event corresponding to a particular outcome (say $A=a$) having been observed. We will thus talk about “correlations between events $A$ and $B$” as a shorthand for “correlations between the variables associated with events $A$ and $B$”.

The second of the two assumptions above, Space-Time, implies that those types of events can be located in space-time to sufficient precision in order to ascertain, e.g., that a pair of events $(X,A)$ is contained in a space-like region separated from a region containing a pair of events $(Y,B)$. The first assumption, Absoluteness of Observed Events, implies that these variables take well-defined values in every experimental run, and that it is therefore possible to define a conditional probability $p(A,B|X,Y)$ for those events (which in quantum mechanics will be given by the Born rule).

Bell’s theorem demonstrates that certain phenomena predicted by quantum mechanics (i.e., the quantum predictions for certain sets of $p(A,B|X,Y)$) cannot be explained by models simultaneously satisfying certain sets of metaphysical assumptions. In our notation, Bell’s 1964 theorem can be expressed as:

These notions are precisely defined as follows.

This is a rigorous definition for the loose concept sometimes called “Freedom of Choice”, and what we call “interventions” here are usually called “free choices”. We prefer the term “intervention” here as it has a more precise meaning in the literature on causality Pearl (2000); Hausman and Woodward (1999). That is, we emphasise that an intervention does not require the free will of a human agent, nor does it need to be itself entirely uncaused. The only important requirement is that an intervention can be chosen via external variables not a priori causally related with any of the other variables relevant to the experiment at hand.

This is a rigorous version of the concept called “Parameter Independence” by Shimony, also discussed (in the same year, and with much the same motivation) by Jarrett Jarrett (1984), who called it, as here, “Locality”. Bell’s use of this term in their 1964 theorem Bell (1964) also accords with this definition Wiseman (2014).

Here we again use the abstract noun employed by Bell in their 1964 paper Bell (1964), but note that “Determinism” is often also used for the same concept. It is also similar to the concept of “Outcome Determinism” used by Spekkens Spekkens (2005) in relation to contextuality, the main difference being that in discussions of contextuality relativistic space-time concepts typically do not play a role.

As an aside: here we are using definitions based on those in our earlier work Wiseman and Cavalcanti (2017), with changes in terminology or definitions noted and motivated when they arise, from Section III onwards. With regard the current section, we have realised that the definitions for No-Superdeterminism and Predetermination do not match well with natural language expectations when applied to models that require a preferred foliation, such as Bohm’s Bohm (1952a, b). We will address that problem in a future publication. However, we note here that: (i) the theorems using these assumptions are still valid; (ii) Predetermination plays no part in the (more interesting) version of Bell’s theorem, from 1976, nor in the LF theorem; (iii) both No-Superdeterminism and Locality are replaceable by the single more natural assumption of Local Agency, as discussed in Section V.

The celebrated Bell experiments of Aspect and co-workers Aspect et al. (1981, 1982) in the early 1980s led to almost universal acceptance of the veracity of the phenomena referred to in Bell’s theorem (theorem 1). One of the initial responses Rohrlich (1983) to this was to advocate giving up Predetermination (or “hidden variables” Rohrlich (1983)), by pointing out that “standard quantum mechanics” was, after all, an indeterministic theory. Many still hold to this simple argument today. As far as Bell’s 1964 theorem alone is concerned, this could allow one to keep Locality, with the hope of thereby maintaining compatibility with relativity. This was, however, not satisfactory for Bell Bell (1990):

“Do we then have to fall back on “no signalling faster than light” as the expression of the fundamental causal structure of contemporary theoretical physics? That is hard for me to accept. For one thing we have lost the idea that correlations can be explained, or at least this idea awaits reformulation”.

As Bell proved in 1976, the Bell inequalities can be derived from a principle specifying (what Bell took to be) a necessary condition for causal explanation in a relativistic space-time, the principle of Local Causality, which we reformulate as:

With this definition, Bell’s 1976 theorem can be formulated as follows.

The logical implications of Bell’s 1964 and 1976 theorems are illustrated in Figure 1. In light of Bell’s 1976 theorem, most physicists conclude (for different reasons, as we will discuss later) that it is Bell’s notion of Local Causality that needs to be rejected. That said, there are research programs pursuing theories violating No-Superdeterminism, and we note that the assumption of No-Superdeterminism given here can be violated by retrocausal Price (2008); Wharton and Argaman (2020)—as well as self-identifiedly superdeterministic Hossenfelder and Palmer (2020); Sen and Valentini (2020)—approaches.

However, if Local Causality is rejected, how can we make sense of causal explanation of correlations in a relativistic space-time? We consider this question in the next section.

One of the basic principles of classical causal explanation was proposed by Hans Reichenbach in 1956. We reformulate it below, following Cavalcanti and Lal (2014); Wiseman and Cavalcanti (2017).

In more modern terms, Reichenbach’s Principle follows from a general framework of (classical) causal models Pearl (2000) (An introduction to the classical causal model framework applied to quantum foundations can be found in Wood and Spekkens (2015).) In this framework, causal structure is represented by a directed acyclic graph (DAG), with random variables of interest associated with nodes, and arrows between nodes representing an asymmetric cause–effect relationship. The added requirement of acyclicity is intended to exclude causal loops. For example, the causal structure used in the derivation of a Bell inequality in a bipartite scenario has the form shown in Figure 2.

In a classical causal model, any probability distribution over the node variables that is compatible with a given graph must satisfy a constraint called the Causal Markov Condition (CMC). This can be expressed as the requirement that a variable $X$ is independent of its non-effects $\mathrm{Nd}(X)$ (any nodes that are not among its “descendants” in the graph), conditional on its direct causes $\mathrm{Pa}(X)$ (its “parent” nodes). That is,

The Causal Markov Condition implies Reichenbach’s Principle of Common Cause as a special case Hitchcock and Rédei (2020). The converse, however, does not hold. To see this, consider a causal graph with three nodes in a chain, $X\rightarrow Y\rightarrow Z$. The CMC implies that the middle node screens off the end nodes, i.e., $p(Z|Y,X)=p(Z|Y)$. RPCC does not imply this condition.

The classical causal model framework (as the formulation of RPCC above) does not require any a priori assumptions about the causal structure in a Bell scenario. It allows, for example, that the causal structure may include a direct causal connection between a choice of setting (such as $X$ in Figure 2) and a space-like separated outcome (such as $B$). To obtain Local Causality from this framework, it must be supplemented by some principle relating space-time and causal structure. It is easy to see that the following principle, taken in conjunction with RPCC, is sufficient to imply Local Causality:

We note that this concept was not explicitly defined in ref. Wiseman and Cavalcanti (2017) but, as will be discussed in Section V, it is a simple consequence of deeper principles introduced there.

The relationships between the various concepts defined so far are represented in Figure 3. Thus, if in light of Bell’s 1976 theorem, one chooses to reject Local Causality, one is faced with a dilemma: reject Relativistic Causal Arrow or Reichenbach’s Principle of Common Cause.

Wigner’s friend paradox is the quintessential intuition pump for the measurement problem. Some recent results have proposed no-go theorems based on extensions of the WFS including multiple observers and entanglement, such as the work by Frauchiger and Renner Frauchiger and Renner (2018). However, unlike Bell’s theorem, the Frauchiger-Renner theorem is not theory-independent—that is, it makes assumptions specifically related to quantum theory. (As an aside: more recent work generalises that theorem Nurgalieva and del Rio (2019), highlighting the modal-logical nature of some of its assumptions. It could be perhaps said that the Frauchiger–Renner theorem is to the Kochen–Specker theorem Kochen and Specker (1967) as the LF theorem is to Bell’s theorem, in the sense that the first two are expressed in terms of assumptions about (modal) logic, whereas the latter two are expressed in terms of metaphysical assumptions.) The Local Friendliness no-go theorem, by contrast, is theory-independent. It is based on the Extended Wigner’s Friend scenario introduced by Časlav Brukner Brukner (2017, 2018).

In this scenario, depicted in Figure 4, there are two labs, controlled by Alice and Bob, each containing a perfectly isolated vault, with a respective friend inside. The friends share a pair of particles prepared in an entangled state, on which they each perform a measurement on a fixed basis, obtaining outcomes $C$ and $D$, respectively. Alice and Bob have choices of measurements labelled by $X$ and $Y$, with respective outcomes $A$ and $B$.

In that protocol, as illustrated in Figure 5, if Alice chooses $X=1$ (Fig. 5a), she opens the vault and asks Charlie what he observed and sets her own outcome to be equal to that of Charlie, i.e., $A=C$. If $X\neq 1$, she performs a measurement on the contents of the vault—including Charlie—in an incompatible basis. This can be done via reversing the unitary evolution that entangled Charlie (and his device, etc.) with his particle (Fig. 5b) and proceeding to perform a measurement on the particle alone, corresponding to a different observable from that observed by Charlie (Fig. 5c). A similar protocol is followed by Bob and Debbie.

For brevity, we will not review all of the details of the LF theorem here but refer the reader to Ref. (Bong et al., 2020). For the present purposes, the following summary is sufficient. We call Local Friendliness the conjunction of Locality, No-Superdeterminism and Absoluteness of Observed Events. We then show that in an Extended WFS, with a space-time arrangement of events as shown in Fig. 5d, Local Friendliness implies constraints on the class of phenomena $p(A,B|X,Y)$ that can be observed by Alice and Bob. These constraints can be put in the form of “LF inequalities”. We then show that LF inequalities can in principle be violated by quantum mechanics, if the requisite quantum operations can in principle be performed on observers. This leads to the following theorem:

As depicted in Figure 6, the set of LF correlations (which for a particular scenario has the form of a polytope—the “LF polytope”) strictly contains the Local Hidden Variable polytope (the set of correlations satisfying the Bell inequalities for a given scenario). In Bong et al. (2020), we illustrated this hierarchy in a proof-of-principle experiment involving polarisation-entangled photons, with the path a photon takes playing the role of the “friend” and a polarising beam splitter the role of the observation. This hierarchy is a reflection of an important fact: the Local Friendliness assumptions are strictly weaker than the assumptions needed to derive a Bell inequality. This means that violation of LF inequalities has strictly stronger implications than violations of Bell inequalities.

A brief look at Figure 3 shows that it is not possible to resolve the LF theorem by dropping Decorrelating Explanation (and the other assumptions in red), as it was in the “peaceful coexistence” resolutions of Bell’s theorem—those assumptions are not used in the theorem. As prefigured, this undermines the narrative of the quantum causal models program, and related ideas (such as Shimony’s) that quantum mechanics and relativistic causality can be reconciled. To better understand what options for reconciliation remain open, we now turn to an analysis using deeper principles.

In this section, we clarify the implications of the LF theorem for quantum causality, and how it differs from Bell’s theorem, by refining many of the concepts introduced so far as consequences of more fundamental principles. Again, we largely follow Wiseman and Cavalcanti (2017), with some modifications noted along the way.

We first define the notion of causal past of an event as a set of events containing all of its causes.

The next two principles impose spatio-temporal constraints on the causal past.

This principle is closely related to what we called Temporal Order in Wiseman and Cavalcanti (2017), except that here we use causal past instead of past, and further specify that it is in the same temporal direction as the past light-cone, to avoid a potential ambiguity. The following principle is a relaxation of the homonymous principle of Refence Wiseman and Cavalcanti (2017):

In Wiseman and Cavalcanti (2017), the principle of the same name specified that the past is the past light cone, but here the role of picking a direction of time for the causes is done by Temporal Causal Arrow instead. (The formulation we adopt here seems likely to be useful for disentangling retrocausal and superdeterministic approaches, which we will explore in future work.) In any case, the last two principles imply that the causal past is indeed in the past light cone. Thus, Temporal Causal Arrow and Relativistic Causality together imply Relativistic Causal Arrow.

Next we define the principle of Independent Interventions, a more cautious reformulation of “Free Choice” in Ref. Wiseman and Cavalcanti (2017) (see Hausman and Woodward (1999) for a detailed discussion of the rationale for this criterion and the role it plays in a manipulability account of causation).

Similarly to how Relativistic Causal Arrow arises from the conjunction of two more fundamental principles, we also define a principle which arises from the conjunction of Independent Interventions and Principle of Common Cause. We call this Interventionist Causation  (in (Wiseman and Cavalcanti, 2017) it was called “Agent-Causation”):

We now note that the conjunction of Relativistic Causal Arrow and Interventionist Causation imply both Locality and No-Superdeterminism, as depicted in Figure 7a. Indeed, that conjunction also implies the more natural concept of Local Agency Wiseman and Cavalcanti (2017).

This can be used in place of Locality and No-Superdeterminism  in both Bell’s 1964 theorem and the LF theorem, as discussed in Refs. Wiseman and Cavalcanti (2017); Bong et al. (2020), respectively. In Bell’s 1976 theorem, Local Agency can be used to replace No-Superdeterminism. All of this is depicted in Fig. 7b.

The violation of Bell inequalities thus requires the rejection of at least one of the deeper principles in the top row of Fig. 7. In the case of the program for “peaceful coexistence” discussed in Sec. III.2, resolution of Bell’s theorem is achieved by rejecting Decorrelating Explanation, and consequently all of the concepts shown in grey in Fig. 7, while keeping all the remaining ones. However, the main point of this paper is that none of the concepts in grey in Fig. 7 are required for deriving LF inequalities, and thus their rejection is not sufficient to resolve the LF no-go theorem. Only the blue assumptions go into the LF no-go theorem—in the simplest terms, only Absoluteness of Observed Events and Local Agency.

Given the conclusion of the preceding section, what is the way forward for the program of quantum causal models? Firstly, it is important to point out that just as Bell’s theorem does not invalidate classical causal models as a useful tool in its regime of applicability, our results do not invalidate quantum causal models as a useful tool in its own regime of applicability—namely, in the description of the vast majority of quantum experiments where one can assume, for all practical purposes, a fixed Heisenberg cut—with “observers” on one side and “quantum systems” on the other.

One response could thus be to clearly state and accept the limited validity of each of these frameworks, rather than to attempt to resolve the conflict with the LF no-go theorem. This seems somewhat defeatist; a similar response in regards to classical causal models would have precluded the development of quantum causal models. A more interesting response could be to search for a further generalisation of quantum causal models to accommodate Wigner’s Friend scenarios.