Can observers of quantum mechanical phenomena agree to disagree?

The concept of quantum mechanics is well-known for making its inventors uncomfortable. It is very paradoxical, and over a century after its inception, it continues to elicit much philosophical and conceptual debate.

We define two disagreement concepts based on Aumann's theorem. The first is a straightforward equivalent of the classical agreement theorem, while the second relaxes the shared certainty constraint while requiring the probability estimates to disagree maximally. We discovered that neither classical nor quantum systems had this type of disagreement. In postquantum situations, however, both types of disagreement can arise. In fact, we identify no-signaling distributions that exhibit these characteristics. After that, we combine our two characterizations and look for distributions that fulfil both concepts of disagreement: The PR box3 is of this type, displaying extreme disagreement in the meaning described above. Our findings reveal a deeper relationship between the quantification of disagreement and the quantification of non-locality, because the PR box is an extreme example of a no-signaling box as a non-local resource.

If a physical theory allowed agents to agree to disagree, this could have unfavourable effects in the context of references. This is why we believe that the inability to agree to disagree is an important element of all physical theories, and that it should be raised to a fundamental principle. Its simplicity makes it ideal for vetting new postquantum theories for consistency.

We begin with an intuitive understanding of the classical agreement theorem's setting. Assume Alice and Bob share a classical system, which may be characterised by a local hidden-variable model (each value of the variable reflects a state of the world in Aumann's terminology). The observers, on the other hand, have no idea which value of the hidden variable is correct (i.e., which is the true state of the world). Instead, each observer is limited to making a single local measurement on the system. Each measurement represents a split of the variable's values, and the outcome tells which partition element contains the value that holds. The sum of the probabilities of the values in the appropriate partition element determines the chance of each outcome.

Assume that Alice wants to calculate the probability of an event (i.e., a collection of values for the variable) that does not correspond to a partition element. Then, using Bayesian inference, she can only calculate the event's conditional probability based on the results of her local measurement. Bob is in the same boat. Assume Alice and Bob are both interested in perfectly connected occurrences (i.e., with probability 1, either both happen or neither happens). The classical agreement theorem thus states that their estimations must be comparable if they are of common certainty.

Alice is certain of Bob's estimate (i.e., assigns probability 1), Bob is certain of Alice's estimate, Alice is certain of Bob's certainty of Alice's estimate, and so forth.

A probability space with some given partitions is referred to as a (classical) ontological model when formalising these concepts. In the literature, ontological models often include a series of preparations that underpin the distribution over the probability space, with the partitions commonly expressed in terms of measurements and outcomes. However, to bridge the gap between classical probability spaces and no-signaling boxes more seamlessly, we consider preparations implicit and employ the language of partitions.

We limit our study to two observers, Alice and Bob, for the sake of simplicity and in accordance with Aumann. In Aumann's original theorem, both observers have common knowledge about a single event of interest. With common certainty, we propose a small generalisation regarding two perfectly correlated occurrences of interest, one for each observer. This lets us to move on to the no-signaling box framework, which we'll use later. The classical agreement theorem is what we name it. This language is further justified by the fact that both statements—the original Aumann's theorem and our formulation with perfectly linked events—can be demonstrated to be identical in purely classical settings (as long as null probability states of the universe are ignored).

References:

1. Ferrie, C. Quasi-probability representations of quantum theory with applications to quantum information science. Rep. Prog. Phys. 74, 116001 (2011).
2. Aaronson, S. The complexity of agreement. In Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’05, pages 634–643, New York, NY, USA, 2005. Association for Computing Machinery.
3. https://www.nature.com/articles/s41467-021-27134-6#Sec7