We recently proved that Relational Quantum Mechanics (RQM), including its central concept, relative facts, is incompatible with Quantum Mechanics [1], originally published on arXiv [2]. We proved this by deriving a Greenberger-Horne-Zeilinger (GHZ) contradiction within a Wigner-Friend scenario typical of RQM arguments. The contradiction arises from the characteristic GHZ correlations among measurement outcomes on three-qubit systems (on which we shall elaborate below). We emphasise that our proof is based on the postulates of RQM as presented in 2021 by the founder, Carlo Rovelli [3]. His presentation now appears as chapter 43 in “The Oxford Handbook of the History of Quantum Interpretations” published in June 2022 by Oxford University Press.

A critique of our article was published subsequently by Aurélien Drezet [4], originally published on arXiv [5]. In this article, Drezet reconstructs our GHZ proof based on his own formulation of the postulates, which does not correspond to Rovelli’s presentation [3]. In this way, he misrepresents both our paper and RQM itself. Therefore his critique is not valid, but it provides a good example of the potential misuses of RQM. Below we discuss several specific instances in the order of their appearances in his paper:

  1. 1.

    A repeated misconstruction of RQM first appears in Section II of Drezet’s paper, where he asserts that:

    “... in RQM the main issue concerns the interpretation of the full wavefunction \(|{\Psi _{SO}}\rangle\) involving observer (O) and observed system (S). In RQM the fundamental object relatively to (O) is not \(|{\Psi _{SO}}\rangle\) but the reduced density matrix

    $$\hat{\rho }_{S|O}^{(red.)} = \hbox {Tr}_O\left[ \hat{\rho }_{SO}\right] = \hbox {Tr}_O \left[ |\Psi _{SO} \rangle \langle \Psi _{SO}|\right] . {''}$$
    (1)

    This statement misrepresents RQM in three ways:

    (i) In RQM, the reduced density matrix is not the fundamental object relative to (O). Quoting Rovelli [3], RQM has “an ontology based on facts (or events), not quantum states,” and, “In RQM, a ‘quantum state’ is a bookkeeping of known facts, and a tool for predicting the probability of unknown facts, on the basis of available knowledge.”Footnote 1

    (ii) In RQM the reduced density matrix does not occupy a more fundamental status than the state vector. State vector encodes the relative fact which the observer obtains in an interaction with the system.

    (iii) Drezet’s reduced density matrix is defined as the trace over states of the observer (O). In two specific implementations described below, this treats the observer as an environment which introduces decoherence. This in turn contradicts RQM postulates regarding the concept of relative facts. We shall explain exactly how this inconsistency arises in items 2 and 3 below.

    We are aware of no precedent for Drezet’s trace operation in RQM. A different kind of trace is found in [6]. Here, one traces over the initial system states, and this has no effect on the final observer correlations. In Drezet’s case, on the other hand, one traces over final observer states, and this removes correlations otherwise present in those states. We know of no instances of the latter type of trace in the RQM literature.

  2. 2.

    Based on the above (inappropriate) use of the reduced density matrix, Drezet compares our GHZ correlation equations with his own, which, not surprisingly, show less correlations and do not pose a contradiction. The two cases in point are Eqs. (14) and (15) for the Friend (portrayed here by Alice), and later, Eqs. (19) and (27) for Wigner (played here by Bob). To be more explicit, Drezet states [immediately prior to Eq. (14)] “Due to decoherence, i.e., entanglement with the environment (Alice) we have lost coherence and correlations between spins.” This contradicts the description of Wigner-Friend scenarios given by Rovelli, dating back to the original 1996 work [7], and developed more comprehensively in his Handbook article [3]. In these examples, a relative fact (a definite value of a physical variable) is realized by Alice, and has not yet been converted into a stable fact by decoherence. Again quoting Rovelli regarding the Wigner-Friend scenario [3]:

    • For instance, in the Wigner’s friend scenario, the friend interacts with a system and a fact is realised with respect to the friend. But this fact is not realised with respect to Wigner, who was not involved in the interaction, and the probability for facts with respect to Wigner (realised in interactions with Wigner) still includes interference effects.

    Drezet’s approach does not allow for relative facts, because when decoherence occurs in the realization of a fact, it becomes a stable fact [8].

  3. 3.

    In criticizing our treatment of Bob’s RQM measurements on the compound system \(S \otimes A\), Drezet again introduces the reduced density matrix to erroneously describe the situation given by RQM, relative to Bob. Using this, he derives the set of equations (19), which show no GHZ correlations in three of the four cases. This derivation again contradicts RQM, whose postulates tell us that, after Alice’s measurements, Bob faces an entangled state of \(S \otimes A\). Ironically, Drezet’s would be correct quantum mechanically, were one to assume that Alice had made proper quantum mechanical measurements on the three qubits ( unlike the prescribed Wigner-Friend scenario) [9].

  4. 4.

    Furthermore, with regard to Bob’s RQM measurements, Drezet misrepresents our argumentation by saying that: “they consider that Bob only measures one of the 3 qubits belonging to SA” [p. 6, between Eqs (19) and (20)]. In fact, in our scenario, Bob performs measurements on each of the three qubits in \(S \otimes A\) (see top paragraph, right column of p. 4 in [1]).

  5. 5.

    Drezet challenges our assertion that relative facts are non-contextual. He argues that we introduce non-contextuality arbitrarily, saying [p. 7, starting three lines after Eq. (24)]: “Since the 3 operations ... are acting ‘locally’ only on one of the subsystems \(SA_m\) their meaning should be non-contextual and absolute.” This statement is incorrect; firstly it misrepresents the actual experimental situation. In GHZ demonstrations of noncontextuality [10, 11], separate measurements are made on each of the subsystems. Secondly, our paper contains a claim that Alice’s relative facts, which (according to RQM) come into existence with Alice’s RQM measurements, are non-contextual hidden variables with respect to the context of Bob’s future measurements. What we mean by context is the future measurement by the second observer Bob who plays the role of Wigner, and not context of measurement on one of the qubits given by the measurement setting for the other ones (this applies both to Friend and Wigner actions). We apply only quantum rules, and do not assume that the qubits are spatially separated. The quantum predictions for three qubits do not depend on their separation. The Alice-Friend’s relative results are hidden variables for Bob-Wigner as they determine Bob’s results if he chooses to measure with the same settings as Alice.

To conclude, let us re-emphasize that we have adhered to the postulates of RQM in constructing our GHZ proof. We reviewed these postulates explicitly in Sec. II of our paper. In contrast, , Drezet used arguments from quantum mechanics while violating key postulates of RQM – most notably the concept of relative facts. Hence it is not surprising that he obtains a result opposite to ours. Ironically, this outcome only reinforces our claim that Relational Quantum Mechanics is incompatible with Quantum Mechanics!

Since writing the paper under discussion here, we have extended our critique of RQM to account for amendments to Rovelli’s Handbook article [3]. This includes amendments described in the papers [8] and [12]. An example of these amendments is the “cross-perspective links” postulate [12], which was formulated to ensure the consistency of perceived outcomes of measurements by different observers. This new postulate, while striving for consistency at a superficial level, only deepens the underlying inconsistency between the principles of RQM and of quantum mechanics. We refer the reader to a new e-print by two of us [13] that discusses this development.