A proof of Bell’s theorem without inequalities is presented in which distant local setups do not need to be aligned, since the required perfect correlations are achieved for any local rotation of the local setups.

Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum “paradoxes”, such as that of Hardy.

Bell non-locality and Kochen–Specker contextuality are logically independent concepts, fuel different protocols with quantum vs classical advantage, and have distinct classical simulation costs. A natural question is what are the relations between these concepts, advantages, and costs. To address this question, it is useful to have a map that captures all the connections between Bell non-locality and KS contextuality in quantum theory. The aim of this work is to introduce such a map. After defining the theory-independent notions of Bell non-locality (...) and KS contextuality for ideal measurements, we show that, in quantum theory, due to Neumark’s dilation theorem, every quantum Bell non-local behavior can be mapped to a formally identical KS contextual behavior produced in a scenario with identical relations of compatibility but where measurements are ideal and no space-like separation is required. A more difficult problem is identifying connections in the opposite direction. We show that there are “one-to-one” and partial connections between KS contextual behaviors and Bell non-local behaviors for some KS scenarios, but not for all of them. However, there is also a method that transforms any KS contextual behavior for quantum systems of dimension d into a Bell non-local behavior between two quantum subsystems each of them of dimension d. We collect all these connections in map and list some problems which can benefit from this map. (shrink)

We introduce a two-party communication complexity problem in which the probability of success by using a particular strategy allows the parties to detect with certainty whether or not some forbidden communication has taken place. We show that theprobability of success is bounded by nature; any conceivable method which gives a probability of success outside these bounds is impossible. Moreover, any conceivable method to solve the problem which gives a probability success within these bounds is possible in nature. This example suggests (...) that a suitaby chosen set of communication complexity problems could be the basis of an information-theoretic axiomatization of quantum mechanics. (shrink)

If quantum mechanics is correct and there is a finite upper bound for the speed of causal influences (e.g., the speed of light), then quantum mechanics is complete (i.e., it does not admit a more detailed description in terms of hidden variables). Here I show that the conclusion holds if we replace the assumption of bounded velocity by the assumption that there is a finite upper bound to the memory a finite physical system can store (e.g., the Holevo bound). On (...) the way to this conclusion I first show that, although the quantum violation of an inequality valid for any non-contextual model can be explained with a classical contextual model, the inequality can be promoted to a Bell inequality in which, if the model is contextual, then it must be also non-local. This suggests that there is something non-classical in any contextual explanation of the individual systems, and leads us to the question of which are the minimum resources (and specifically memory) any contextual explanation should consume. (shrink)