We present here what we consider the fundamental problems with the proposal in [1] and discuss perspectives for fixing them.
Unitary Measurements and Evaluating Probabilities
In order to test whether quantum mechanics violates the objectivity of physical outcomes in the sense of Healey, one would need to find an operational way that would enable, at least in principle, to collect statistics from which the correlation functions could be computed and inserted into the Bell’s expression. In the protocol there is no space–time region where the measurement counts for all four measurement settings—which would enable the computation of the four correlations functions—are accessible in principle, because the data of Carol’s or/and Dan’s measurements are erased before the data of Alice’s and Bob’s measurements get established.
Moreover, the computed correlation functions do not all correspond to correlations between measurement outcomes because the “measurements” of Carol and Dan are considered as unitary transformations. This gives rise to multiple problems in the argument. Firstly, the correlation functions E(a, d) and E(c, b) are computed from two different reference frames, namely the one of Alice and of Bob respectively (see Fig. 1). It is known that the notion of quantum state, and hence of entanglement as well as the computed correlation functions for a given state are reference-frame dependent. This is a consequence of the relativity of simultaneity in different reference frames [6]: observers in different reference frames assign different quantum states on their respective slices corresponding to t = const. It is thus not clear to us what combining expression (1)–(4), which are computed for different reference frames and thus for different quantum states, in a CHSH-like inequality is supposed to signify. We note that there would be no ambiguity, if the correlation functions would refer to measurement data, since those are of course reference frame independent.
Secondly, one could equally assign different values to the reference frame dependent correlation functions E(b, c) and E(a, d) by an argument similar to that proposed in Ref. [7]. More concretely, with respect to Alice’s reference frame, the calculated correlation function \(E(c,b) = 0\) for all times. For times between \(t_1\) and \(t_3\) the register b is in a fixed pre-measurement state \(|r\rangle _b\) and the results \(c=+1\) and \(c=-1\) occur with equal probability, whereas for times between \(t_3\) and \(t_6\) the result of c is erased and the register is reset to it initial state \(|r\rangle _c\) and the results \(b=+1\) and \(b=-1\) occur with equal probability. The mean value of the product of c and b (i.e. the correlation function) is zero independently of choice of the value assigned to the fixed states of the registers. Similarly, Bob would predict \(E(a,d)=0\), since in his reference frame result d is erased before a is measured. Combining either of the two with (1), (4) and either (2) or (3) respectively still gives a violation of the CHSH-inequality, but with a different value of \(3/\sqrt{2}\). However, combining the predictions of both reference frames, as done in [1], but using both \(E(c,b)=0\) and \(E(a,d)=0\) instead of (2) and (3) results in no violation of Bell’s inequality:
$$\begin{aligned} \begin{aligned}&|E(a,b)+E(b,c)+E(c,d)-E(a,d)|\\&\quad =|E(a,b)+0+E(c,d)-0|\le 2. \end{aligned} \end{aligned}$$
(5)
Note that we are not claiming that Eq. (5) as opposed to using Eqs. (1)–(4) is the correct way to evaluate correlation functions for the Bell test, but rather that there is an ambiguity in how to evaluate unobservable, reference-frame dependent quantities. Without specifying how the computed expressions are related to observable quantities, there seems to be no physical motivation to favour the one combination of values which leads to the alleged violation, over the one that predicts no violation of the inequality. Clearly, all these ambiguities appear because the computed expressions for correlation functions are not associated to observable quantities. If the four measurements would be identified with four space–time points in which counts are registered, then the correlations between these counts will be reference frame independent and there would be no problems. However, then no violation would be observed in the protocol.
A Modified protocol that Gives the Expected Probabilities Suffers the Same Criticism that Healey Raises
The protocol proposed in Ref. [1] allows to observe in principle only the correlations between a and b (since at the end of the protocol the measurement results of Carol and Dan have been erased). For the “superobservers” Alice and Bob to evaluate all the correlation functions and test the proposed CHSH-like inequality in Ref. [1], one is forced to adapt the protocol to one wherein Alice can choose to measure either a or c and Bob either b or d by deciding whether or not to erase the measurement outcome of their “friends”. In the latter, each “superobserver” (Alice and Bob) performs the same measurement as their associated observer (Carol and Dan, respectively). Since in that case one allows for a “choice of setting” one has to ensure “locality” and “freedom of choice” as required in a standard Bell-inequality setup. In the original protocol of Healey, “locality” is not required, since all four measurements are fixed and performed in each round. Any assignment of four definite outcomes of course satisfies a Bell-like inequality by construction.
In the adapted protocol, Alice not erasing Carol’s measurement, but rather measuring Carol’s observable, allows for computing E(b, c), whereas Bob not erasing Dan’s measurement, but performing the same measurement instead will make E(a, d) experimentally accessible; whereas, both the “superobservers” not undoing their friends’ measurements allows for measuring E(c, d). This adapted protocol is illustrated in Fig. 2. Note that in the rounds where Alice, Bob or both decide to measure the observables of Carol or Dan or both respectively, and register counts in the respective measurements, they cannot erase these counts in a unitary fashion to continue with the subsequent measurements. Upon registering the counts in the first measurements, Alice and Bob will observe a statistics in the subsequent measurement, which is compatible with the state-update rule and not with a unitary transformation. The continuation of the protocol in its original form [1] but with the application of the state-update rule would not lead to a violation of Bell’s inequalities. It appears therefore that the only way to give an operational meaning to accessing the correlation functions and hence to the verification of Healey’s argument is to assume that in each run Alice and Bob chooses one of the two measurement settings—a step that was criticised by Healey.
It, therefore, seems to us that this setup is subject to the same criticism as [2] when one requires the actual computation of the terms violating the CHSH-inequality to correspond to measured values. Otherwise, we think Eqs. (1)–(4) need further justification.
In conclusion, the only possibility for the correlations to be in principle measured is to revise the scheme in a way that does not allow to be sure that the four values coexisted in each round, therefore suffering the very same criticism that was addressed towards proposal [2]. However, one can make a stronger assumption that these values are fixed even when they are not all co-measured, which then allows to derive the no-go theorem of Ref. [2].