Specker's principle, the condition that pairwise orthogonal propositions must be jointly orthogonal (or rather, the 'exclusivity principle' that follows from it), has been much investigated recently within the programme of finding physical principles to characterize quantum mechanics. Specker's principle, however, largely appears to lack a physical justification. In this paper, I present a proof of Specker's principle from three assumptions (made suitably precise): the existence of 'maximal entanglement', the existence of 'non-maximal measurements' and no-signalling. I discuss these three assumptions and describe canonical examples of non-Specker sets of propositions satisfying any two of them. These examples display analogies with various approaches to the interpretation of quantum mechanics, including retrocausation. I also discuss connections with the work of Popescu & Rohrlich. The core of the proof (and the main example violating no-signalling) is illustrated by a variant of Specker's tale of the seer of Nineveh, with which I open the paper. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
Specker's principle, the condition that pairwise orthogonal propositions must be jointly orthogonal (or rather, the 'exclusivity principle' that follows from it), has been much investigated recently within the programme of finding physical principles to characterize quantum mechanics. Specker's principle, however, largely appears to lack a physical justification. In this paper, I present a proof of Specker's principle from three assumptions (made suitably precise): the existence of 'maximal entanglement', the existence of 'non-maximal measurements' and no-signalling. I discuss these three assumptions and describe canonical examples of non-Specker sets of propositions satisfying any two of them. These examples display analogies with various approaches to the interpretation of quantum mechanics, including retrocausation. I also discuss connections with the work of Popescu & Rohrlich. The core of the proof (and the main example violating no-signalling) is illustrated by a variant of Specker's tale of the seer of Nineveh, with which I open the paper. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'.
What principle excludes superquantum correlations that don’t violate the nosignaling principle? In an argument using PR boxes in a 2009 Nature paper, Pawlowski and colleagues show that the Tsirelson bound – the limit achievable in a simulation of a Popescu–Rohrlich correlation with quantum resources – follows from an information-theoretic principle they call “information causality.” For three qubits, it is now known that there are superquantum correlations that cannot be excluded by a bipartite principle like information causality, so this principle does not completely characterize quantum correlations. An intriguing principle, developed by Cabello as the “exclusivity principle” or “Specker’s principle” (from Ernst Specker’s story about a seer from Nineva), and separately by Fritz and colleagues as the principle of “local orthogonality,” yields the Tsirelson bound, and also rules out correlations beyond the quantum limit for the Klyachko correlation and other cases where contextuality or nonlocality is relevant.
I draw attention to the fact that three recently proposed physical principles, namely "local orthogonality", "global exclusive disjunction", and "compatible orthogonality" are not new principles, but different versions of a principle that Ernst Specker noticed long ago. I include a video of Specker stating this principle in 2009 in the following terms: "Do you know what, according to me, is the fundamental theorem of quantum mechanics? (...) That is, if you have several questions and you can answer any two of them, then you can also answer all of them". I overview some results that suggest that Specker's principle may be of fundamental importance for explaining quantum contextuality. Specker passed away in December 10, 2011, at the age of 91.
The totalitarian principle establishes that 'anything not forbidden is compulsory'. The problem of quantum correlations is explaining what selects the set of quantum correlations for a Bell and Kochen-Specker (KS) contextuality scenario. Here, we show that two assumptions and a version of the totalitarian principle lead to the quantum correlations. The assumptions are that there is a non-empty set of correlations for any KS contextuality scenario and a statistically independent realization of any two KS experiments. The version of the totalitarian principle says that any correlation not forbidden by these assumptions can be produced. This paper contains a short version of the proof (presented elsewhere) and explores some implications of the result. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
The question concerning the physical realizability of a probability distribution is quite important in quantum foundations. Recent studies emphasized a principle which was first indicated by Specker. This principle, called Specker's principle, bounds physically realizable correlations. We study Specker's observation in the scenario that involves three inputs each with two outputs. Then using only linear constraints imposed on joint probabilities by this principle, we reveal the unphysical nature of Garg-Mermin (GM) correlation. Interestingly, GM correlation was proposed to falsify the following suggestion by Fine: if the inequalities of Clauser and Horne (CH) holds, then there exists a deterministic local hidden-variable model for a spin-1/2 correlation experiment of the Einstein-Podolsky-Rosen type, even when more than two observables are involved on each side. Our result establishes that, unlike in the CH scenario, the local orthogonality principle at single copy level is not equivalent to the no-signaling condition in the GM scenario.
Nonlocality is arguably one of the most fundamental and counterintuitive aspects of quantum theory. Nonlocal correlations could, however, be even more nonlocal than quantum theory allows, while still complying with basic physical principles such as no-signaling. So why is quantum mechanics not as nonlocal as it could be? Are there other physical or information-theoretic principles which prohibit this? So far, the proposed answers to this question have been only partially successful, partly because they are lacking genuinely multipartite formulations. In Fritz et al. [Nat. Commun. 4, 2263 (2013)], we introduced the principle of local orthogonality (LO), an intrinsically multipartite principle which is satisfied by quantum mechanics but is violated by nonphysical correlations. Here we further explore the LO principle, presenting additional results and explaining some of its subtleties. In particular, we show that the set of no-signaling boxes satisfying LO is closed under wirings, present a classification of all LO inequalities in certain scenarios, show that all extremal tripartite boxes with two binary measurements per party violate LO, and explain the connection between LO inequalities and unextendible product bases.
The Kochen Specker theorem revealed contextuality as a fundamental nonclassical feature of nature. Nonlocality arises as a special case of contextuality, where entangled states shared by space like separated parties exhibit nonlocal correlations. The notion of maximality in correlations, analogous to maximal entanglement, is less explored in multipartite systems. In our work, we have defined maximal correlations in terms of contextual models, which are analogous to absolutely maximally entangled (AME) states. Employing the sheaf theoretic framework, we introduce maximal contextual correlations associated with the corresponding maximal contextual model. The formalism introduces the contextual fraction CF as a measure of contextuality, taking values from 0 (noncontextual) to 1 (fully contextual). This enables the formulation of a new class of correlations termed absolutely maximal contextual correlations (AMCC), which are both maximally contextual and maximal marginals. In the bipartite setting, the canonical example is the Popescu Rohrlich (PR) box, while in the tripartite case, it includes Greenberger Horne Zeilinger (GHZ) correlations and three way nonlocal correlations. In this work, we extend these findings to four party correlations. Notably, no AME state exists for four qubits, which introduces a subtle difference between AMCC and AME. The construction follows the constraint satisfaction problem (CSP) and parity check methods. In particular, the explicit realization of a non AMCC correlation that is maximally contextual yet not maximal marginal is obtained within the CSP framework.