Computability limits nonlocal correlations

Abstract

If the no-signalling principle was the only limit to the strength of non-local correlations, we would expect that any form of no-signalling correlation can indeed be realized. That is, there exists a state and measurements that remote parties can implement to obtain any such correlation. Here, we show that in any theory in which some functions cannot be computed, there must be further limits to non-local correlations than the no-signalling principle alone. We proceed to argue that even in a theory such as quantum mechanics in which non-local correlations are already weaker, the question of computability imposes such limits.Bell's seminal work [1] on non-local correlations, not only allowed us to distinguish classical from quantum mechanics, but also spurred a multitude of useful applications in quantum information theory (see e.g. [2,3]). As a result it is important to know which non-local correlations can indeed be physically realized.Let us now explain more carefully what is meant by non-local correlations. For simplicity, we thereby consider a bipartite system, Alice and Bob. Let X and Y denote a set of possible measurements for Alice and Bob respectively, and let A and B denote corresponding set of outcomes. Furthermore, letdenote the probability that Alice and Bob obtain outcomes a ∈ A and b ∈ B when making measurements x ∈ X and y ∈ Y respectively. When it comes to the study of non-local correlations, we are interested in the set of allowed values for Pr [a, b|x, y]. For example, the famous CHSH inequality [4] tells us that for any classical theory 1 4x,y∈{0,1} a∈{0,1}for any possible measurements labeled X = {0, 1} and Y = {0, 1}, and any state shared by Alice and Bob [31]. Quantumly, however, there exist a shared state and measurements such that the sum attains the value of 1/2 + 1/(2 √ 2) ≈ 0.853. This is the highest value possible in quantum mechanics [5]. It demonstrates that in a quantum world, non-local correlations can be strictly stronger than is allowed classically. Much work has gone into determining bounds on quantum correlations [5][6][7][8][9][10][11][12], and investigating the difference between the classical and quantum regime (see e

Agent Inaccessibility as a Fundamental Principle in Quantum Mechanics: Objective Unpredictability and Formal Uncomputability

“…Significantly, Berendsky et al [ 63 ] concluded with the message that their findings are not in “…conflict with the different interpretations of quantum mechanics”, and they further noted that in “…the Copenhagen interpretation, the measurement process is postulated as random, whereas, for example, in Bohmian mechanics, it is deterministic but the initial conditions are randomly distributed and fundamentally unknowable.” For quantum theories operating in a universally deterministic universe (see Figure 3 and Figure 4 ), such as dBB-theory and Bohmian mechanics, the quantum randomness would be generated by uncomputable processes, whether they be effectively uncomputable (see the effective non-signaling constraint in Section 5.2 ), or (ii) objectively uncomputable in the strong sense of dynamical chaos and/or undecidable dynamics, e.g., in the form of Turing incomputability (see the objective non-signaling constraint in Section 5.3 ); only the strong option of objective ignorance in deterministic systems could entail objective or true unpredictability. However, the specific topic of self-referential dynamics in formal uncomputability (see Table 1 ) was not addressed in the work by Berendsky et al [ 63 ], although these workers did make the important point that “...in Bohmian mechanics… the initial conditions are… fundamentally unknowable.” Previously, Islam and Wehner [ 105 ] had also suggested that quantum mechanics must entail the presence of (agent-inaccessible) uncomputable states as otherwise a violation of the non-signaling constraint would inevitably ensue, and these researchers noted that “…in any theory in which the Church-Turing principle holds, certain states and/or measurements are not available to us as otherwise any (approximate) no-signaling computation could be performed.” To employ the present terminology, in order (i) to prevent superluminal Shannon-type signaling in nonlocal quantum ontologies or, alternatively, (ii) to prohibit (future-to-past) retro-signaling in time-symmetric quantum ontologies, these “states and/or measurements” must be subject to an AIP as a fundamental principle in quantum mechanics.…”

Introduction to the book "Quantum Theory: Informational Foundations and Foils"

“…Such frameworks are increasingly being used to attempt to describe nonclassical phenomena in a language that does not presume the correctness of quantum theory. Not only is this pursued for the question of Bell inequality violations [33][34][35][36], but also for a number of applications to computer science and physics, including the study of communication complexity [37,38], non-local computation [39], measurement-based computation [40][41][42][43][44], games and interactive proof systems [45][46][47][48][49][50], randomness amplification [51][52][53][54], causal networks [55][56][57], computability [58], complexity [59], key distribution [60], bit commitment [61][62][63], complementarity [64,65], no cloning [63,66,67], teleportation [63,68,69], state discrimination [70][71][72], entropy [73][74][75], thermody...…”