Closing objectivity loophole in Bell tests on a public quantum computer

Here we reply to to the criticism, indicating changes.

Referee 1:

Weaknesses: "It is questionable how novel the introduced method is,"

The method is no sophisticated experimental protocol, but it points out a very important aspect of the Bell test - the problem of objectivity. In this sense it is novel as none of the previous experiments addressed this issue, not even by simple outcome copying, not to mention quantum cloning (in a preferred basis).

" How far the proposal goes towards closing aforesaid loophole, and whether the task cannot be achieved by simpler means."

The loophole has not been closed by any means earlier, relying on the trust in the time-tags and the saved values of the outcomes. Moreover, simple copying would occur AFTER the measurement, which destroys coherence.

"It is also unclear how important such a loophole is and whether closing it has any practical relevance, say for device-independent cryptography or tests of quantum entanglement."

Closing the objectivity loophole is yet another test of the limitation of entanglement. The test requires all friends to be entangled and so the level of the violation is a quantitative measure of multipartite entanglement, directly extracted for the experimental test in contrast to other quantities (e.g. Schmidt measure).

"The objectivity loophole requires a clear definition and a discussion on its importance. The idea would be that an unspecified sub-luminal interaction (or an adversary in a device-independent scenario) could, in principle, change Alice’s registered value after the readout time tag based on the information about Bob’s input choice. In this sense, the violation of the CHSH inequality would be explainable by an LHV model supplemented with communication since the final time tag would already be within the lightcone."

The Referee is right, we have added a separate paragraph motivated by this suggestion.

"But then, a simpler (classical) solution would be to simply copy the readout of Alice’s measurement to an independent classical memory along with the final time tag and store both copies. It would be helpful if the authors could discuss why their method is a preferred solution to closing the objectivity loophole."

In principle, yes, but a) none of the previous experiments realized even this simple scenario b) classical copying occurs after the measurement, when the state has already collapsed and lost coherence. The adversary then has an easy task just to change classical values. On the other hand, when the copying happens in the quantum regime, as cloning in some basis, the adversary can be tricked out by the unitary reversal of the operation and changing the setting. This is connected with the local friendliness scenario and is one of the further directions of research. Testing objectivity strengthens the trust in the security of the entanglement.

"how good the two additional qubits per site are as observers in such a fundamental (aiming to be loophole-free) Bell test. For instance, wouldn’t the proposal in the paper require Alice and Bob to trust the quantum nature of their device (i.e., that it implements specific control-NOT operations and majority vote measurements)."

No. They don't need to trust. The C-NOT operation is expected to copy the state (in the 0-1 basis, consistent with no-cloning theorem), but can be imperfect. The restriction is put on the inequality (i.e. value assignments), not the assumptions or additional trust. Then the Bell value can be smaller, but once it exceeds 2, the test is passed, and no trust is necessary. Of course there can be total cheating, e.g. projective measurements and classical copying instead of quantum reversible gate, but even then, the conclusion remains the same, and only the possible disagreements with quantum predictions could be worrying.

"Closing the objectivity loophole in the present setup would require a more involved analysis to show that no possibility exists for an LHV model supplemented with sub-luminal communication or an adversary in a DI protocol to simulate the observed statistics."

The family of LHV models excluded here extends to the multioutcome case as we have formally 8 instead of 2 values. The rest of Bell assumptions remain unchanged.

"A second issue (related to comment 1 above) that needs clarification is the issue of apparent signalling. The presence of apparent signalling seems to undermine the attempt to close the objectivity loophole, some clear analysis is needed to explain how the objectivity loophole has been closed to statistical significant levels despite the apparent signalling. Some results such as Quantum 9, 1760 (2025), Nat. Comm. 16, 4390 (2025), Phil. Trans. R. Soc. A.38220230011 (2024) may be useful in this regard. "

Early apparent signaling was due to postselection as explained in Quantum 9, 1760 (2025), but is harder to accept in new loophole-free tests. No-signaling is a consequence of locality, not viceversa, and so its violation means the locality is invalid, but locality can be dropped, while keeping no-signaling. This is known for the Bell test, but the practical question is which level of signaling is still acceptable in comparison to the Bell parameter. In our data, signaling is ~10^-2- 10^-3 and absent in ibm_kingston while B=2.1-2.5. Then B is order of magnitude larger. On the other hand other "loophole-free tests" [17,20] also show moderate signaling (p-value ~5%) (in [20] there is a typo in Table SVI with missing 0 in row 2 and 4, see detailed discussion in [42,62]) We have added a new section discussing no-signaling and its experimental relevance.

I"t was also unclear if the authors are post-selecting on outcomes 000 and 111, and ignoring rounds in which other outcomes were observed. It would be helpful to clarify this."

We do not make any postselection. Other outcomes are assigned zero, not discarded (i.e. the probability is inferred from the fraction with the denominator equal the sum of all events including 001, 010, etc.) We added a comment.

"In Reference [24], Nature Physics 16, 1199 (2020), Bong et al. introduced an Extended Wigner's Friend Scenario using a photonic qubit to play the role of each of the friends'' and measured violations of a generalisation of Bell’s inequality that they term “Local Friendliness” inequalities. These inequalities are derived based on the assumptions: (i) “absoluteness of observed events” (every observed event happens for all observers), (ii) freedom-of-choice (free choices can be made uncorrelated with other events outside their future light cone) and (iii) locality (the probability of an event is unchanged by conditioning on a free choice made at a space like separated location). There again, a photonic qubit is taken to be a physical system that counts as an “observer”. In principle, the experiment in that paper could also be interpreted as a Bell test with additional observers. The novelty in the current paper of considering 3 observers instead of 2 does not seem to be of major significance."

That experiment tested a different scenario and is not equivalent to ours even if restricting to two parties. Firstly, there are indeed pairs of observers AC and BD. However, either A or C measured (analogously B or D), not both, since a movable mirror decides which one. Secondly, there is no free choice, rather a fixed setting (i.e. the experiment is run separately for seach set of settings not mixing them), while we shuffle randomly settings. There is one setting for C and two settings for A (then A reverses any unitary changes by C). Thirdly, it's a photonic experiment with postselection on coincidences. Lastly, it addressed the the hypothetical LHV model where a hidden value is measured by C and then passed to A (absoluteness). We do not make any other assumption on LHV other than in standard Bell, only restricting the counting event to the unanimous ones. In other words we check the absoluteness while [24] assumes it.

"It would be helpful to include a discussion on the relationship between the present manuscript and the literature, especially with regards to the Local Friendliness inequalities. For instance, couldn’t the authors claim a stronger result by testing for a violation of the LF inequalities instead of the CHSH Bell inequality in the present paper (i.e., relaxing the assumption of absoluteness of observed events in the LF inequalities)?"

One can run an experiment combining LF and objectivity, and it is an interesting further possible development. Our result is already stronger than the usual Bell as it requires unanimity, and indeed it does not assume absoluteness. We expanded a discussion of this relation and suggestion in the paper.

"A number of typos in the manuscript need fixing: p1 column 1 “Quantum mechanics is incompatible with local realism”, p2 c2 “it needs some criterion that is has been completed”, p3 c2 “which gives makes the friends unable to receive identical outcomes”, p4 c4 “violation of no-signaling in part of the groups”, etc. and many more which can be improved in a revised version by a careful reading of the manuscript."

We apologize and corrected.

Referee 2:

" sending circuits to publicly available quantum computers — a relatively straightforward task."

The circuits have to be ogranized in the way that correspond to the Bell test with friends 1) transfer of the state to friends by C-NOT gates 2) choice of setting in the way minimizing signaling and maximizing randomness 3) make the data easy to analyze

"the manuscript is not always easy to read, and the presentation could be clarified in some parts."

We have added a few paragraphs in particular explaining objectivity and no-signaling in more detail.

"My main criticism concerns the choice to implement the objectivity test by creating redundancy of the measurement information quantumly rather than classically. While the redundancy occurs after the unitary representing the measurement, and is therefore related to classical information in principle, encoding it in quantum systems introduces significant drawbacks: the stored informa- tion is subject to decoherence, gate errors, crosstalk, and other implementation imperfections. In contrast, a standard Bell test could be performed with a single system per party, and the outcome (which is a classical information) could then be reliably replicated for multiple observers. Such a procedure would be far more robust and less sensitive to the limitations of current quantum hard- ware. Given these points, I cannot see a clear practical or theoretical advantage in the approach taken by the authors"

We have chosen to create quantum redundancy instead of classical one exactly because of the referee's "drawbacks". Simple classical copying would occur after the projective measurement when the state has lost coherence. This is insecure given an adversary who wants to change the outcome exploiting some postmeasurement exchange of information (subluminal), see also the referee's 1 report and our reply. On the other hand changing the quantum state, especially cloned (in some basis, in agreement with no-cloning) is beyond the capability of a standard enemy. The protocol can be even improved adding an optional unitary reversal and choice of a different basis. Anyway, passing this test DESPITE imperfections like errors and crosstalk, is a benchmark for quantum machines and proves that they can create a stable multiparty entanglement.

Moreover, the test is original, none of previous (also loophole-free) experiments covers our concept and there was no hint that the inequality would be still violated, except theoretical belief.

Other comments: We have made the corrections. As regards objectivity and freedom of choice, we maintain that there is a relation between the two, as they focus on opposite time endpoints of the experiment, but we tried to emphasize a difference.

"A genuine violation of nonsignalling under space-like separation would instead imply a breakdown of special relativity."

The Referee is right but we remind that the Bell inequality was invented in the non-relativistic framework. Of course, combining it with postulates of relativity (e.g. Wightman axioms), leads to the Referee conclusion, but putting the Bell test, especially measurements, into the framework of relativistic quantum field theory is a formidable task, see our paper, Phys. Rev. D 108, 056020 (2023). Moreover, the performed loophole-free experiments operate in non-relativistic framework (preferred frames and communication speeds smaller than vacuum light speed). The Referee's statement is then rather an axiom (widely accepted), not a property derived from any theory.